2012.7.li.Ø©Ar

目录

Frobenius流形的一些例子 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .傅宇龙 1

Galois表示及其在Fermat大定理证明中的初步应用 . . . . . . . . . . . . . . . . . . . . . . . . . .潘锦钊 37

局部域与整体域的欧拉特征公式 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .卢伟 66

Fully Nonlinear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .方汉隆 83

Mirzakhani’s Volume Recursion and Witten-Kontsevich Theorem. . . . . . . . . . .朱艺航 113

Rational Homotopy Theory of Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .潘雯瑜 141

毕业感言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .朱艺航 170

Frobenius流形的一些例子

傅宇龙∗

指导教师：张友金教授

摘要

Dubrovin在1991年由数学物理中的2-D拓扑场论的研究中引进了Frobenius流形的

概念。这是一类复流形，其上的WDVV方程组决定了其切空间上的 Frobenius代数

结构。在得到其上的正交坐标系后，我们可更加方便地研究与旋转系数有关的方程

组，并在此过程中发现 Frobenius流形与某个算子 Λ的单值有密切联系，由此得到两

者间自然的对应。

关键词：Frobenius流形，WDVV方程组，正则坐标，保单值形变

1 引言

在数学物理对 2-D 拓扑场论的研究中，我们寻找这样一类多复变函数 F = F (t) ,

t = (t1, ..., tn)使得其三阶导数

cαβγ(t) :=∂3F (t)

∂tα∂tβ∂tγ

满足下面的性质

1)规范化:

ηαβ := c1αβ(t) (1.1)

构成一个常值非退化矩阵。令

ηαβ := (ηαβ)−1.

我们将使用矩阵 (ηαβ)和 (ηαβ)来进行升降指标操作。

2)结合性:对任何 t，如下函数

cγαβ(t) := ηγεcεαβ(t)1 (1.2)

在一个以 e1, ..., en为基的 n−维空间上定义了一个结合代数 At的结构

∗基数811在本文中，若未作特别说明，均默认当上下脚标同时出现时为对其求和.

1

eα · eβ = cγαβ(t)eγ (1.3)

易知这是一个交换代数。又

cβ1α(t) =

n∑

ε=1

ηβεηεα = δβα (1.4)

因此，∀ eα,有 (e1, eα) = cγ1α(t)eγ = δγαeγ = eα ,所以 e1是代数 At中的单位元。

3) F (t)关于其变量满足准齐次性:

∀c = 0 ,及对某些固定数 d1, ..., dn, dF 成立下式

F (cd1t1, ..., cdntn) = cdFF (t1, ..., tn) (1.5)

上式左右两边对 ti (i = 1, ..., n)求导，得

∂iF (cd1t1, ..., cdntn)cdi = cdF ∂iF (t1, ..., tn) (∗)

在对(1.5)式关于 c两边求导，得

n∑

i=1

∂iF (cd1t1, ..., cdntn)dicditi = dF c

dFF (t1, ..., tn) (∗∗)

将 (∗)式代入 (∗∗)式，得到n∑

i=1

diti∂iF (t1, ..., tn) = dFF (t1, ..., tn)

为方便引入Euler向量场记号: E = Eα(t)∂α，此处 Eα(t) = dαtα,则准齐次条件化为

LEF (t) := Eα(t)∂αF (t) = dF · F (t) (1.6)

这里线性向量场 E = E(t) =∑

α dαtα∂α生成伸缩变换 (1.5)。

注意到李导数 LE 在向量场 ∂i (i = 1, ..., n)上的作用为

LE∂i = [E, ∂i] = [∑

α

dαtα∂α, ∂i] = −di∂i (1.7)

下面的两个对准齐次性和Euler向量场的一般化的推广将在之后的讨论中很重要：

1.在 F (t)上加上一个关于 t1, ..., tn 的二次多项式 P (t)后，并不改变其三阶导数，故

代数 At保持不变。设 P (t) = Aαβtαtβ +Bαt

α + C ,则

LEP (t) =∑

α,β

Aαβ(dα + dβ)tαtβ +∑

α

dαBαtα

由 LE(F (t) + P (t)) = dF (F (t) + P (t)) ⇒LEF (t) − dFF (t) =

∑

α,β

(dF − dα − dβ)Aαβtαtβ +

∑

α

(dF − dα)Bαtα + CdF

设 A′αβ = (dF − dα − dβ)Aαβ , B

′α = (dF − dα)Bα , C

′= CdF ,则上式化为

LEF (t) = dFF (t) +A′αβt

αtβ +B′αt

α + C′

(1.8)

从而得到一般的准齐次条件的形式。

2

特别地，当 ∀α, β ,均有 dF = 0 , dF = dα , dF = dα + dβ 时，可由 (1.8)式反求出多

项式 P (t)，从而对 F (t)加上 P (t)后可消去 (1.8)式中多余的尾项。

2.我们考虑更一般的线性非齐次Euler向量场

E(t) = (qαβ t

β + rα)∂α (1.9)

设 Q = (qαβ ) ,当 Q的特征根都是单的且非零时，此时 Q可对角化，设存在矩阵 A使

得AQA−1 = diag(d1, ..., dn)

通过对 t = (t1, ..., tn)作仿射变换

(s1, ..., sn)T = A(t1, ..., tn)T +AQ−1(r1, ..., rn)T

得到一组新的坐标 s = (s1, ..., sn)，由 ∂∂tα = ∂sβ

∂tα∂

∂sβ ,得到

(∂

∂t1, ...,

∂

∂tn) = (

∂

∂s1, ...,

∂

∂sn)A

从而在新坐标下Euler向量场 E 化为

E = (∂

∂t1, ...,

∂

∂tn)Q(t1, ..., tn)T + (r1, ..., rn)T

= (∂

∂s1, ...,

∂

∂sn)AQA−1(sT −AQ−1(r1, ..., rn)T ) + (r1, ..., rn)T

= (∂

∂s1, ...,

∂

∂sn)diag(d1, ..., dn)sT

=∑

α

dαsα∂α

(1.10)

具有之前的简单形式。

当 Q 的一些特征根是零时，一般来说不能通过对 t 进行仿射变换来消去 (1.9) 式

中的非齐次项。在此情形下，对可对角化的 Q ，可经由对 t 作类似前面的仿射变换，

将Euler向量场约化为下面较简单的形式:

E(t) =∑

α|dα =0

dαtα∂α +

∑

α|dα=0

rα∂α (1.11)

易见次数 d1, ..., dn, dF 在相差一个非零因子的意义下是唯一的，以后我们仅考虑

d1 = 0的情形，因此我们可对这些次数正规化并总假设

d1 = 1. (1.12)

在物理文献中，这些正规化的次数通常由 q1 = 0, q2, ..., qn和 d参数化，使得

dα = 1 − qα , dF = 3 − d . (1.13)

3

2 WDVV方程组和Frobenius流形

2.1 WDVV方程组及其解的约化

引言中提到的代数 At 的结合性给出了函数 F (t)需要满足的WDVV方程组。由代数

乘法的结合性，有

∀α, β, γ , (eα · eβ) · eγ = eα · (eβ · eγ)

将 (1.3)式代入上式得

cγαβciγθ = cγθβc

iγα , ∀α, β, γ, θ ∈ 1, ..., n

上式两边乘以 ηiδ 并对 i求和，得到

cγαβcγθδ = cγθβcγαδ , ∀α, β, δ, θ ∈ 1, ..., n

即

∂3F (t)

∂tα∂tβ∂tεηεγ ∂3F (t)

∂tγ∂tθ∂tδ=

∂3F (t)

∂tθ∂tβ∂tεηεγ ∂3F (t)

∂tγ∂tα∂tδ, ∀α, β, δ, θ ∈ 1, ..., n (2.1)

上述方程组再加上决定其伸缩约化的准齐次性条件 (1.8)及指明函数 F (t)对变量 t1

的依赖性的规范化条件 (1.1) ,称为Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV)方程

组，它的一个解称为自由能函数。

观察发现此方程组关于坐标 t1, ..., tn的线性变换保持不变，我们可用下面的推导更加

显式地描述WDVV方程组。

引理2.1 由Euler向量场 E 生成的伸缩变换作用在矩阵 (ηαβ)上为如下线性共性变换

LEηαβ = (dF − d1)ηαβ (2.2)

上式中 dF 和 d1如 (1.6, 7)两式中定义。

证明: 对等式 (1.8)两边分别关于 t1 , tα和 tβ 求导，对等式左边，依次得到

∂1LEF (t) =∑

ρ

(∂1Eρ)∂ρF (t) +

∑

ρ

Eρ∂1ρF (t)

由 ∂1Eρ = δρ

1 ,有

∂1LEF (t) = ∂1F (t) +∑

ρ

Eρ∂1ρF (t)

∂α∂1LEF (t) = ∂1αF (t) +∑

ρ

qρα∂1ρF (t) +

∑

ρ

Eρηρα

∂β∂α∂1LEF (t) = ηαβ + qραηρβ + qρ

βηρα

再对等式右边求导，最后得到

qραηρβ + qρ

βηρα = (dF − d1)ηαβ = (2 − d)ηαβ

将度量 < , >看作 0, 2−张量，由李导数在张量上的作用法则，有

4

LEηαβ = LE < ∂α, ∂β >

= ∇Eηαβ+ < ∇∂αE, ∂β > + < ∂α,∇∂βE >

= 0+ < (∂αEρ)∂ρ, ∂β > + < ∂α, (∂βE

ρ)∂ρ >

= qραηρβ + qρ

βηρα.

(2.3)

从而引理得证。 推论2.1 1)若 η11 = 0且 E(t)的所有特征根都是单的，则通过对坐标 tα 的线性变换，矩

阵 (ηαβ)能被约化为反对角线形式

ηαβ = δα+β,n+1 (2.4)

2) 在上述坐标下，对某个关于变量 (t2, ..., tn)的函数 f(t2, ..., tn) , F (t)可约化为下

面形式

F (t) =1

2(t1)2tn +

1

2t1

n−1∑

α=2

tαtn−α+1 + f(t2, ..., tn) (2.5)

证明: 由 E(t)的所有特征根都是单的，不妨设 Q = (qαβ ) = diag(d1, ..., dn) . 由引理2.1,有

qραηρβ + qρ

βηρα = (2 − d)ηαβ

从而对矩阵 Q(ηαβ) = (qραηρβ) ,有

(qραηρβ) + (qρ

αηρβ)T = (2 − d)(ηαβ)

⇒ diag(d1, ..., dn)(ηαβ) + (ηαβ)diag(d1, ..., dn) = (2 − d)(ηαβ)

⇒ (dα + dβ)ηαβ = (2 − d)ηαβ

由 d1, ..., dn互不相同及矩阵 (ηαβ)非退化，故 (ηαβ)每一列恰有一个元素非零。

(否则设 ∃ α, γ , α = γ ,使得 ηαβ , ηγβ = 0 ,则 dα + dβ = 2 − d = dγ + dβ ⇒ dα = dγ ，

与 d1, ..., dn互不相同矛盾！)

又由矩阵 (ηαβ)对称及 η11 = 0，易知可在保持坐标分量 t1 不变的情形下通过调换其

余坐标的次序，使得度量矩阵 (ηαβ)具有反对角线的形式，即

(ηαβ) =

η1nη2,n−1···

ηn−1,2ηn1

再对上面得到的每个坐标作适当的伸缩变换，则可将上面矩阵中的反对角线元素均化

为 1，最后得到 ηαβ = δα+β,n+1。至此推论第 1)部分得证。

由上面的证明过程知，和

dα + dn−α+1 = 2 − d (2.6)

5

不依赖 α ,且

dF = 2d1 + dn (2.7)

若这些次数被规范化使得 d1 = 1 ,并设 dα = 1 − qα ,则

dF = 3 − d (2.8)

其中 q1, ..., qn, d满足

q1 = 0, qn = d, qα + qn−α+1 = d. (2.9)

下面证明命题的第 2)部分，先设在 1)中得到的坐标下 F (t)有如下形式

F (t) =1

2(t1)2tn +

1

2t1

n−1∑

α=2

tαtn−α+1 + f(t1, ..., tn) (2.10)

下证 f(t) = f(t1, ..., tn)不依赖于变量 t1，实际上，由 ηαβ = δα+β,n+1 ,推出

∂1αβf(t) = 0, ∀ α, β ∈ 1, ..., n.设函数 g := ∂1f ,则 g 的所有二阶偏导数均恒为零，故 g 为 t = (t1, ..., tn)的一次函

数，即 ∃ a1, ..., an, b ∈ C ,使得

g(t) = a1t1 + · · · + ant

n + b

对 g(t)关于变量 t1积分得

f(t) =1

2a1(t

1)2 + bt1 + t1n∑

α=2

aαtα + h(t2, ..., tn)

注意到在相差 t 的一个二次多项式的意义下 F 的三阶导数保持不变，故 f(t) ∼h(t2, ..., tn)为只依赖于变量 t2, ..., tn的函数。从而有

F (t) =1

2(t1)2tn +

1

2t1

n−1∑

α=2

tαtn−α+1 + f(t2, ..., tn) =1

2t1

n−1∑

α=1

tαtn−α+1 + f(t2, ..., tn)

2.2 低维WDVV方程组的解的例子

例 2.1下面以 n = 3为例求出WDVV方程组的约化为 (18)形式的解。

此时，设

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + f(t2, t3)

为记号简便，下设 t2 = x, t3 = y, f(t2, t3) = f(x, y)此时度量矩阵 (ηαβ) = (ηαβ) = 0 0 1

0 1 01 0 0

，由此得到

e2 · e2 = fxxye1 + fxxxe2 + e3 (2.11)

e2 · e3 = e3 · e2 = fxyye1 + fxxye2 (2.12)

e3 · e3 = fyyye1 + fxyye2 (2.13)

6

又由代数 At的结合性， n = 3时只需考虑 (e2 · e2) · e3 = e2 · (e2 · e3)及 (e3 · e3) · e2 =

e3 · (e3 · e2)，将 (2.11, 12, 13)代入这两个条件中，均得到相同的关于 f 的微分方程

(fxxy)2 = fyyy + fxxxfxyy (2.14)

易见上面方程是 n = 3时 f 关于代数的结合性所需满足的唯一方程。最后还要考虑

F (t)满足的准齐次性条件：由 d1 + d3 = 2d2 = dF − d1 = 2 − d ⇒ d3 = 1 − d, d2 = 1 − d2

，故Euler向量场

E(t) = t1∂1 + ((1 − d

2)t2 + r2)∂2 + ((1 − d)t3 + r3)∂3

当 d = 1, 2时，通过对 t2, t3平移变换可将 E(t)化为

E(t) = t1∂1 + (1 − d

2)t2∂2 + (1 − d)t3∂3

将上式代入 (1.8)式，并记 (1.8)式中右边关于 t的二次多项式尾项为 P (t)，整理后

得到

(1 − d

2)t2

∂

∂t2f(t2, t3) + (1 − d)t3

∂

∂t3f(t2, t3) = (3 − d)f(t2, t3) + P (t)

由上式知多项式 P (t)是只依赖于 t2, t3 的函数，将 t2, t3 替换为 x, y并记 q = − 1−d1− d

2

，

则上式化为

xfx − qyfy = (q + 4)f + P (x, y) (2.15)

先考虑 P (x, y) = 0时的齐次解，由特征线法，解得

f(x, y) = x4+qϕ(yxq)

另外易验证在相差一个二次多项式Q(x, y) = ax2 + bxy+ cy2 + vx+ ey+ s的情形下，

函数 f(x, y) = x4+qϕ(yxq) +Q(x, y)满足方程

xfx − qyfy = (q + 4)f − (q + 2)ax2 − (2q + 3)bxy − (3q + 4)cy2

− (q + 3)vx− (2q + 4)ey − (4 + q)s

= (q + 4)f + P (x, y)

显然 q 不可能为 −2，下面按不同的 q 使得多项式 P (x, y)中不同的项的系数分别为

零的情形进行讨论：

①当 q = −32 ,−4

3 ,−3,−4时，即 d = −2,−1, 4, 3时，

∀二次多项式 P (x, y)，总能找到合适的 a, b, c, v, e, s使得 f(x, y) = x4+qϕ(yxq) +Q(x, y)

是 (28)的解，又相差一个二次多项式意义下视为等同，故此情形下WDVV方程组的解

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + (t2)4+qϕ((t2)qt3). (2.16)

② 当 q = −32 时，由 2q + 3 = 0 ，形如 x

52ϕ(yx− 3

2 ) + Q(x, y) 的函数只能是 (28) 式中

P (x, y)不含 xy项的解，为此，需考虑方程

xfx +3

2yfy =

5

2f + bxy

7

的解。设f(x, y) = x

52ϕ(yx− 3

2 )u(x, y)

则 u满足方程

xux +3

2yuy =

byx− 32

ϕ(yx− 32 )

再由特征线法解得

u(x, y) =byx− 3

2

ϕ(yx− 32 )

log x+ θ(yx− 32 )

⇒ f(x, y) = bxy log x+ x52 (ϕ · θ)(yx− 3

2 )

仍记 ϕ · θ为 ϕ，并可找到多项式Q(x, y)使得 f(x, y)+Q(x, y)为方程 (28)在 q = −32

时的解，又在相差一个 t的二次多项式的意义下视为等同，最后得到此情形下WDVV方

程组的解为

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + bt2t3 log t2 + (t2)

52ϕ(t3(t2)− 3

2 ). (2.17)

③当 q = −43 时，类似情形②中的讨论，最后得到此情形下WDVV方程组的解为

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + c(t3)2 log t2 + (t2)

83ϕ(t3(t2)− 4

3 ). (2.18)

④当 q = −3时，类似情形②中的讨论，最后得到此情形下WDVV方程组的解为

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + vt2 log t2 + t2ϕ(t3(t2)−3). (2.19)

⑤当 q = −4时，类似情形②中的讨论，最后得到此情形下WDVV方程组的解为

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + s log t2 + ϕ(t3(t2)−4). (2.20)

⑥当 d = 1时，Euler向量场可约化为

E(t) = t1∂1 +1

2t2∂2 + r3∂3 (2.21)

代入到 (1.8)式中，仍设 f(t2, t3) = f(x, y)，并简记 r3 = r，整理后得到

1

2xfx + rfy − 2f = P (t) − 1

2r3(t1)2

由于上式左边是变量 (t2, t3) = (x, y)的函数，故 P (t) = 12r

3(t1)2 + P1(t2, t3)，其中

P1(t2, t3) = P1(x, y)是 x, y的二次多项式。下面考虑如下方程的解

1

2xfx + rfy − 2f = P1(x, y) (2.22)

解得 ∀多项式 P1(x, y)， ∃二次多项式 Q(x, y)使得上面方程的解为

f(x, y) = x4ϕ(y − 2r log x) +Q(x, y)

8

又在相差一个 t的二次多项式的意义下视为等同，最后得到此情形下WDVV方程组

的解为

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + x4ϕ(y − 2r log x) (2.23)

⑦当 d = 2时，Euler向量场可约化为

E(t) = t1∂1 + r∂2 − t3∂3 (2.24)

代入到 (1.8)式中，仍设 f(t2, t3) = f(x, y)，整理后得到

rfx − yfy − f = P1(x, y)

解得 ∀多项式 P1(x, y)， ∃二次多项式 Q(x, y)使得上面方程的解为

f(x, y) = y−1ϕ(x+ r log y) +Q(x, y)

又在相差一个 t的二次多项式的意义下视为等同，最后得到此情形下WDVV方程组

的解为

F (t) =1

2(t1)2t3 +

1

2t1(t2)2 + y−1ϕ(x+ r log y) (2.25)

下面考虑 F (t)所满足的代数结合性条件，即 (fxxy)2 = fyyy + fxxxfxyy。

仍按照前面得到的各种情形进行讨论。当 d = 1, 2时，

①当 q = −32 ,−4

3 ,−3,−4时，即 d = −2,−1, 4, 3时， f(x, y) = x4+qϕ(yxq)，令 z = yxq

,对 f 求三阶偏导数得

fxxx = [(4 + q)(3 + q)(2 + q)ϕ+ qz(7q2 + 27q + 26)ϕ′ + q2z2(6q + 9)ϕ′′ + q3z3ϕ′′′]x1+q

= h1(z)x1+q

fxxy = [(4q2 + 14q + 12)ϕ′ + z(5q2 + 7q)ϕ′′ + q2z2ϕ′′′]x2+2q = h2(z)x2+2q

fxyy = [(4 + 3q)ϕ′′ + qzϕ′′′]x3+3q = h3(z)x3+3q

fyyy = ϕ′′′(z)x4+4q

将上面各式代入 (2.14)式中，得到变量 z的函数 ϕ满足如下非线性常微分方程

(h2(z))2 = ϕ′′′(z) + h1(z)h3(z) (2.26)

②当 q = −32 时，为便于区分记

f1(x, y) = bxy log x+ x52ϕ(yx− 3

2 ) = bxy log x+ f(x, y)

这里 f1(x, y) 为 F (t) 的尾项， f(x, y) 如①中形式，并保持意义不变地延用①中记号

hi(z), i = 1, 2, 3。将 f1各三阶偏导数代入 (2.14)式中，整理后得到 ϕ满足方程

(h2(z) + b)2 = ϕ′′′(z) + h1(z)h3(z) − bzh3(z) (2.27)

9

③当 q = −43 时，类似②中的讨论，最后得到此情形下 ϕ满足方程

(h2(z))2 = ϕ′′′(z) + h1(z)h3(z) + 2c(h1(z) + 2zh2(z) + z2h3(z)) (2.28)

④当 q = −3时，类似②中的讨论，最后得到此情形下 ϕ满足方程

(h2(z))2 = ϕ′′′(z) + (h1(z) − v)h3(z) (2.29)

⑤当 q = −4时，类似②中的讨论，最后得到此情形下 ϕ满足方程

(h2(z))2 = ϕ′′′(z) + (h1(z) + 2s)h3(z) (2.30)

⑥当 d = 1时， f(x, y) = x4ϕ(y − 2r log x)，其各三阶偏导数为

fxxx = 2x(12ϕ− 26rϕ′ + 18r2ϕ′′ − 4r3ϕ′′′)

fxxy = x2(12ϕ′ − 14rϕ′′ + 4r2ϕ′′′)

fxyy = x3(4ϕ′′ − 2rϕ′′′)

fyyy = x4ϕ′′′

将上面各式代入 (2.14)式中，整理后得到 ϕ满足方程

0 = ϕ′′′ + 8r3ϕ′′ϕ′′′ + 8r2ϕ′ϕ′′′ + 128rϕ′ϕ′′ − 48rϕϕ′′′

+ 96ϕϕ′′ − 144(ϕ′)2 − 52r2(ϕ′′)2(2.31)

⑦当 d = 2时， f(x, y) = y−1ϕ(x+ r log y)，类似情形⑥求出其各三阶偏导数代入 (2.14)

式中，整理后得到 ϕ满足方程

ϕ′′′(r3 + 2ϕ′ − rϕ′′) − (ϕ′′)2 − 6r2ϕ′′ + 11rϕ′ − 6ϕ = 0 (2.32)

综上， n = 3时针对 d取值的各种情形，WDVV方程组的解可通过解上面各种情形

对应的 ODE而得到。

但当维数 n更高时，WDVV方程组将包含很多个方程，即使是验证它们之间的相容

性也是很困难的，更难于给出其解的明确表达。

2.3 Frobenius代数及其简单性质

为定义 Frobenius流形，我们先来定义与之相关的 Frobenius代数。

定义 2.1 一个 C上的代数 A称为(交换的) Frobenius代数，若

1) A是一个 C上的交换结合代数且有单位元 e；

2) A × A上有一个 C−双线性对称非退化内积 < , >: A × A → C, (a, b) 7→< a, b >，

且关于代数的乘法具有如下不变性：

< ab, c >=< a, bc >, ∀ a, b, c ∈ A.

10

一个简单的 Frobenius代数的例子如下：

例 2.2 A为 n个一维代数的直和，设 Ai = Cei, A = ⊕ni=1Ai。则 e1, ..., en为 A的一组

基，定义基上的乘法为

eiej = δijei, i, j = 1, ..., n (2.33)

则 ∀ i = j，由内积不变性，有

< ei, ej >=< eiei, ej >=< ei, eiej >=< ei, 0 >= 0 (2.34)

从而一组数 < ei, ei >, i = 1, ..., n为确定此种形式的 Frobenius代数的一组参数。另

外易验证 A中无幂零元，故 A的极大可解理想 Rad(A) = 0，从而 A是半单代数。

实际上，有一类 Frobenius代数本质上都具有上述例子的形式，即有下面的引理

引理 2.2 C上无幂零元的 Frobenius代数均具有例 2.2的形式。

证明:设 A为 n维无幂零元的 Frobenius代数， e为其乘法单位元。将 e扩充为 A的一组

基 f1 = e, ..., fn。 ∀ fi , i = 1, ..., n，定义 A上的线性变换

Li : A → A, a 7→ fi · a

下证 ∀ i, Li是半单的。易知

∀ i, j, Li Lj = Lj Li

即 Lini=1之间两两交换，故它们有非零的公共特征向量 x1。设

Lix1 = λ1ix1, λ

1ix1 ∈ C, ∀ i = 1, ..., n

令W1 = x ∈ A : < x, x1 >= 0 ，则W1 为 A的一个 n − 1维线性子空间。下证

x1 /∈ W1。否则，由 x1为公共特征向量，故

∀ y ∈ A, ∃ αy ∈ C, s.t. yx1 = αyx1

⇒ ∀ y ∈ A, < x21, y >=< x1, x1y >=< x1, αyx1 >= 0

由内积非退化⇒ x21 = 0，这与A不含幂零元矛盾！故 x1 /∈ W1。从而A = Cx1⊕W1

(作为线性空间的直和)。又因为 ∀ x ∈ W1，有

< Lix, x1 >=< fix, x1 >=< x, fix1 >=< x, λ1ix1 >= 0

故 Li(W1) ⊂ W1。故 ∀ i = 1, ..., n ， Li|W1 为 W1 上的线性变换，且两两交换，

故在 W1 中它们有非零的公共特征向量 x2 ，类似前面，可得到 W1 的 n − 2 维线性子

空间 W2 = x ∈ W1 : < x, x2 >= 0 ，同理可证 x2 /∈ W2 ，从而 W1 有直和分解

W1 = Cx2 ⊕W2，同样也有 Li(W2) ⊂ W2, ∀ i = 1, ..., n。

如此进行下去，最终得到 Lini=1 的 n个公共特征向量 x1, ..., xn ，易见它们构成 A

的一组基。又

∀ x ∈ A, x = α1f1 + · · · + αnfn ⇒ xxi = (α1λi1 + · · · + αnλ

in)xi = αxxi

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故 ∀ i, ∃ µi ∈ C, s.t. x2i = µixi，取 ei = µ−1

i xi，则 e1, ..., en仍是 A的一组基，且

e2i = (µ−1i )2µixi = µ−1

i xi = ei, ∀ i = 1, ..., n

当 i = j 时，由前面讨论知 ∃ r, s ∈ C, s.t. sei = eiej = rej ，而 ei, ej 线性无关，故

r = s = 0。从而这组基满足条件中的 (2.33)式。

另外，当 i = j 时，不妨设 i < j ，则 ej ∈ Wi ⇒ < ej , ei >= 0，故这组基也满足条

件中的 (2.34)式。并且 ∀ i = 1, ..., n, < ei, ei >= µ−2i < xi, xi >= 0，它们是 Frobenius

代数 A的参数。引理得证。 上述引理将在后面构造 Frobenius流形上的正则坐标时用到。接下来进一步定义分次

Frobenius代数。

定义 2.2 Frobenius 代数称为分次的，若存在一个线性算子 Q : A → A 和数 d，使得

∀ a, b ∈ A，成立

Q(a · b) = Q(a) · b+ a ·Q(b) (2.35)

< Q(a), b > + < a,Q(b) >= d < a, b > (2.36)

算子 Q和数值 d分别称作此 Frobenius代数的分次算子和 charge。

注 2.1 由上面定义，可以进一步考虑分次交换结合环 R 上的分次 Frobenius 代数

(A,<,>)。此时我们有两个分次算子 QR : R → R及 QA : A → A，它们满足如下性质：

QR(αβ) = QR(α)β + αQR(β), ∀ α, β ∈ R (2.37)

QA(ab) = QA(a)b+ aQA(b), ∀ a, b ∈ A (2.38)

QA(αa) = QR(α)a+ αQA(a), ∀ α ∈ R, a ∈ A (2.39)

QR < a, b > +d < a, b >=< QA(a), b > + < a,QA(b) > (2.40)

对于一个非零数 k，可定义含单位元 e的代数的重标度：我们按如下方式改变代数的

乘法法则和单位元

a · b 7→ ka · b, e 7→ ke (2.41)

2.4 Frobenius流形及简单例子

做好了上面的铺垫，我们接下来定义 Frobenius流形。其基本想法是将代数 At 看作

与切空间 TtM 等同

At ∋ eα ↔ ∂α =∂

∂α∈ TtM, ∀ α = 1, ..., n

从而建立起了 Frobenius代数纤维丛与流形M 上的切丛 TM 之间的对应。

定义 2.3 设M 是一个 n−维光滑流形，称M 为 Frobenius流形，若 ∀ t ∈ M ，在点 t处

的切空间均有指定的 Frobenius代数的结构，且光滑地依赖于参数 t ,并满足下列条件：

12

FM1.不变内积< , >t是M 上的一个平坦度量。记∇为流形 (M,< , >)上的 Levi-Civita

联络，单位向量场 e是此联络的协变常量

∇e = 0 (2.42)

这里所说的度量的平坦性，是指 Riemann 曲率张量消失，即存在局部坐标系

(t1, ..., tn)使得度量矩阵 (< ∂α, ∂β >)在此坐标系下为常值。

FM2.令 c是如下定义的M 上的对称 3−张量

c(u, v, w) := < u · v, w > (2.43)

我们要求 4−张量 (∇zc)(u, v, w)关于4个向量场 u, v, w, z ∈ TM 也是对称的。

FM3. M 上固定了一个线性向量场 E ∈ TM ,即

∇(∇E) = 0 (2.44)

并且相应的微分同胚单参数子群在度量 < , >及 Frobenius代数 TtM 上的作用分别

为共形变换和重标度。观察到M 上向量场空间 TM 是M 上光滑解析函数空间 C∞(M)

上的 Frobenius代数，我们可以用分次 Frobenius代数的观点来刻画这一点。为此，我们

取这两个空间上的线性算子

QC∞(M) := E (2.45)

QTM := id+ adE (2.46)

它们按照之前的定义 2.2和注 2.1在 TM 上引入了一个分次环 C∞(M)上给定 charge

d的分次 Frobenius代数结构。

E 称为 Frobenius流形上的Euler向量场。切空间 TtM 上的协变常量算子

Q = ∇E(t) =∂Eα

∂tβ∂α ⊗ dtβ ∈ TM ⊗ TM∗ = End(TM) (2.47)

称为 Frobenius 流形上的分次算子。算子 Q 的特征根 d1, ..., dn 是 M 上的常值函数，

id−Q的特征根 qα = 1 − dα, α = 1, ..., n称为M 的伸缩维数。特别地，由 (2.42)知单位

向量场 e是 Q的对应特征根为 1的特征向量。

条件 FM3还有无穷小量的表达形式

∇γ(∇βEα) = 0 (2.48)

LEcγαβ = cγαβ (2.49)

LEe = −e (2.50)

LEηαβ = (2 − d)ηαβ (2.51)

这里 LE 是沿着Euler向量场的李导数。另外，由前面的分次代数刻画， (2.49, 51)可化为

如下形式

13

LE(u · v) = LEu · v + u · LEv + u · v (2.52)

LE < u, v > = < LEu, v > + < u,LEv > +(2 − d) < u, v > (2.53)

其中 u, v是M 上任意光滑向量场。

明确了相关概念后，我们考虑WDVV方程组的解与 Frobenius流形之间的关系，有

如下引理：

引理 2.3 任何定义在 t ∈ M 的一个邻域上的WDVV方程组 (d1 = 0)的解按下面公式给

出了此区域中的 Frobenius流形结构：

∂α · ∂β := cγαβ(t)∂γ (2.54)

< ∂α, ∂β >:= ηαβ (2.55)

其中

∂α :=∂

∂tα(2.56)

单位向量场

e := ∂1 (2.57)

以及Euler向量场 (1.8)。

反之，局部上任何 Frobenius流形对某个WDVV方程组的解具有 (2.54 ∼ 57)的结

构，同时Euler向量场有 (1.9)式结构。

证明 由于WDVV方程组的一个解按 (2.54, 55)两式在切空间上定义了乘法和度

量，显然度量为常值，故坐标 (t1, ..., tn)是平坦的。从而在此坐标下协变导数就是相应的

偏导。由于单位向量场 e = ∂1，故 ∀ α，我们有

∇αe = (∂α1)∂1 = 0 ⇒ ∇e = 0

从而 FM1得证。又 (2.43)式中张量 c有分量

c(∂α, ∂β , ∂γ) = cαβγ(t) = ∂α∂β∂γF (t) (2.58)

⇒ (∇∂δc)(∂α, ∂β , ∂γ) = ∂δc(∂α, ∂β , ∂γ) = ∂α∂β∂γ∂δF (t) (2.59)

由 F 解析，其偏导数与具体求导顺序无关，故 4−张量 (∇zc)(u, v, w)是完全对称的，FM2

得证。

现在我们证明 FM3，只需要证明 (2.52, 53)两式成立。此时Euler向量场 E是线性的，

算子 Q = ∇E 是对角化常值矩阵

∇E = diag(1 − q1, ..., 1 − qn) (2.60)

14

对准齐次性方程 F (λ1−q1t1, ..., λ1−qntn) = λ3−dF (t1, ..., tn) + quadratic 两边关于

tα, tβ , tγ 求三阶偏导数，得到

∂α∂β∂γF (λ1−q1t1, ..., λ1−qntn) = λqα+qβ+qγ−dcαβγ(t) (2.61)

⇒ ∂ε∂α∂β∂γF (λ1−q1t1, ..., λ1−qntn)λ1−qε = λqα+qβ+qγ−d∂εcαβγ(t) (2.62)

对 (2.61)式两边关于 λ求导得∑

ε

∂ε∂α∂β∂γF (λ1−q1t1, ..., λ1−qntn)(1−qε)λ−qεtε = (qα +qβ +qγ −d)λqα+qβ+qγ−d−1cαβγ(t)

将 (2.62)式代入上式并整理得到

LEcαβγ(t) =∑

ε

[(1 − qε)tε + rε]∂εcαβγ(t) = (qα + qβ + qγ − d)cαβγ(t) (2.63)

由此再加上推论 1.1的证明中得到的等式 (qα + qβ − d)ηαβ = 0，最终我们得到分次

函数环上的分次 Frobenius代数的定义式 (2.52, 53)，条件 FM3得证。

反过来，在一个 Frobenius流形上，我们可以选取局部平坦坐标 (t1, ..., tn)，使得不

变度量 < , >在此坐标下为常值。将对称性条件 (2.43)对 u = ∂α, v = ∂β , w = ∂γ , z = ∂δ

应用，得到 ∂δcαβγ(t)关于 α, β, γ, δ是对称的。这与张量 cαβγ(t)本身的对称性一起，推出

局部存在一个函数 F (t)使得

cαβγ(t) =∂3F (t)

∂tα∂tβ∂tγ(2.64)

又由单位向量场 e 的协变常量性，我们可对坐标做一个线性变换使得 e = ∂1 ，

故(2.57)式成立。下证函数 F (t)满足 (1.8)式。由 (2.44)式，推出Euler向量场 E(t)是一

个具有 (1.9)式形式的线性向量场。由重标度的定义，在坐标 tα下，我们有

[∂1, E] = ∂1 ⇒ ∂1Eα = δα

1 (2.65)

故算子 Q = ∇E 作用在 ∂1上有

∇E(∂1) = (∂βEα)∂α ⊗ dtβ(∂1) = (∂1E

α)∂α = ∂1

即 ∂1 是算子 Q = ∇E 的从属于特征根 1 的特征向量，从而 d1 = 1。由 Frobenius 流

形定义中的条件 FM3 的 (2.52, 53) 两式将 u = ∂α, v = ∂β 带入后，利用等式 LE∂α =

−(∂αEε)∂ε = −qε

α∂ε展开，分别得到

LEηαβ = qεαηεβ + qε

βηεα = (2 − d)ηαβ (2.66)

LEcγαβ = 2 Eε∂εc

γαβ − qγ

ε cεαβ + qε

αcγεβ + qε

βcγαε = cγαβ (2.67)

此时∇E 是对角化的，即 qβα = δαβ(1 − qα)，则上面两式分别化为

2关于李导数在张量上的作用：设 T 是m,n−张量，即 T : (T ∗m)m × (TM)n → C∞(M)，对 α1, ..., αm ∈T ∗M, X1, ..., Xn ∈ TM, f1, ..., fm, g1, ..., gn ∈ C∞(M) ，满足 T (f1α1, ..., fmαm, g1X1, ..., gnXn) =

f1 · · · fmg1 · · · gnT (α1, ..., αm, X1, ..., Xn) ， E 为 M 上光滑向量场，则 (LET )(α1, ..., αm, X1, ..., Xn) :=

(∇ET )(α1, ..., αm,X1, ..., Xn) − ∇T (·,α2,...,αm,X1,...,Xn)α1(E) − · · · − ∇T (α1,...,αm−1,·,X1,...,Xn)αm(E) +

T (α1, ..., αm,∇X1E,X2, ..., Xn) + · · · + T (α1, ..., αm, X1, ..., Xn−1,∇XnE)。此处 cγαβ 实际上是 1, 2−张量 T ，其在基上的定义为 T (dtγ , ∂α, ∂β) = dtγ(∂α · ∂β)。

15

(qα + qβ − d)ηαβ = 0 (2.68)

Eε∂εcγαβ = (qα + qβ − qγ)cγαβ (2.69)

使用 (2.68)式降低 (2.69)式中 γ 的指标，得到

Eε∂εcαβγ = Eε∂εcδαβηδγ = (qα + qβ − qδ)c

δαβηδγ

= (qα + qβ)cαβγ − cδαβ(d− qγ)ηδγ = (qα + qβ)cαβγ − (d− qγ)cαβγ

= (qα + qβ + qγ − d)cαβγ

(2.70)

又 ∂α∂β∂γ(Eε∂εF ) = Eε∂εcαβγ + (3 − qα − qβ − qγ)cαβγ ，结合 (2.70)式推出

∂α∂β∂γ [Eε∂εF − (3 − d)F ] = 0, ∀ α, β, γ = 1, ..., n (2.71)

上式对 tα, tβ , tγ 积分三次，最后得到函数 F 满足准齐次性条件 (1.8)式，引理得证。

定义 2.4 两个 Frobenius流形的(局部)微分同胚 ϕ : M → M 称为(局部)等价，若 ϕ∗

是相应的两个流形上不变度量的线性共形变换，即存在非零常数 c使得

ϕ∗ < , >M

= c2 < , >M

并且 ϕ∗是切空间代数之间的同构

ϕ∗ : TtM → Tϕ(t)M, ∀ t ∈ M

以后将相互等价的 Frobenius流形构成的等价类视为一个对象进行研究。几个经典的

Frobenius流形的例子

例 2.3 (平凡 Frobenius流形)令 A是一个分次 Frobenius代数，即一组权 q1, ..., qn被赋予

到基向量 e1, ..., en，使得下面两组常数 cγαβnα,β,γ=1和 ηαβn

α,β=1满足

cγαβ = 0 for qα + qβ = qγ (2.72)

以及对某个数 d有

ηαβ = 0 for qα + qβ = d (2.73)

在代数 A中有

eαeβ = cγαβeγ , ηαβ =< eα, eβ > (2.74)

上述公式在M = A上定义了一个 Frobenius流形的结构。相应的自由能函数 F (t)是

一个三次多项式

F (t) =1

6cαβγt

αtβtγ (2.75)

设 t = tαeα，则 t3 = tαtβtγeαeβeγ ，从而还有

F (t) =1

6< t3, e > (2.76)

其中 e = e1为单位元。坐标 tα的次数为 dα = 1 − qα，函数 F 的次数为 dF = 3 − d。

例 2.4 (Frobenius流形的直积)设M ′,M ′′ 分别是 n−维和 m−维的两个 Frobenius流形，

则它们的直积M ′ ×M ′′有自然的 Frobenius流形的结构，若它们的伸缩维度满足约束

16

d′1

d′′1

=d′

F

d′′F

(2.77)

若各自的平坦坐标 (t1′, ..., tn′), (t1′′

, ..., tm′′)被规范化为 (2.5)式的形式，即

F ′(t1′, ..., tn′) =

1

2(t1

′)2tn′ +

1

2t1

′n−1∑

α=2

tα′tn−α+1′+ f ′(t2

′, ..., tn′) (2.78)

F ′′(t1′′, ..., tm′′) =

1

2(t1

′′)2tm′′ +

1

2t1

′′m−1∑

β=2

tβ′′tm−β+1′′

+ f ′′(t2′′, ..., tm′′) (2.79)

并且 d′1 = d′′

1 = 1，由 (2.77)式表明 d′n = d′′

m，则乘积流形的前位函数 F 有如下形式

F (t1, t1, t2′, ..., tn−1′

, t2′′, ..., tm−1′′

, tN , tN ) =

=1

2t1

2tN + t1t1tN +

1

2t1

n−1∑

α=2

tα′tn−α+1′+

1

2t1

m−1∑

β=2

tβ′′tm−β+1′′

+

+f ′(t2′, ..., tn−1′

,1

2(tN + tN )) + f ′′(t2

′′, ..., tm−1′′

,1

2(tN − tN )) (2.80)

其中 N = n+m，

t1 =t1

′+ t1

′′

2, t1 =

t1′ − t1

′′

2, tN = tn′ + tm′′, tN = tn′ − tm′′

3 Frobenius流形上的形变联络的和两个度量

3.1 形变联络的平坦性与Frobenius流形的联系

处理 Frobenius流形的一个首要工具是 Levi-Civita联络 ∇的形变∇u(z)v := ∇uv + zu · v (3.1)

这里 u, v是 Frobenius流形M 上的两个向量场， z是这个形变联络依赖的参数。

我们将此推广，得到直积M × C上的一个亚纯联络，其定义由如下公式给出：∇u

d

dz= 0 , ∇ d

dz

d

dz= 0 , ∇ d

dzv = ∂zv + E · v − 1

zµv (3.2)

其中µ := 1 − d

2− ∇E = diag(µ1, ..., µn), µα := qα − d

2(3.3)

这里 u, v是M × C上的切向量场，且沿着 C的分量为零。观察发现这是一个对称联络。

引理 3.1 联络 ∇(z)是平坦的⇔代数 cγαβ(t)是结合的且局部存在函数 F (t)使得 cαβγ(t) =

∂α∂β∂γF (t)。

证明:在平坦坐标下，由 [∂α, ∂β ] = 0 ⇒ R(∂α, ∂β) = [∇α, ∇β ] − ∇[∂α,∂β ] = [∇α, ∇β ]，故

形变联络的曲率的消失性在分量下的表达为

17

∀ γ, 0 = R(∂α, ∂β)∂γ = [∇α(z), ∇β(z)]∂γ = [z(∂βcεαγ − ∂αc

εβγ) + z2(cδβγc

εαδ − cδαγc

εβδ)]∂ε

⇒ ∂βcεαγ = ∂αc

εβγ , c

δβγc

εαδ = cδαγc

εβδ

故以 cγαβ(t)作为基向量的乘法系数的代数是结合的，又由张量 cαβγ = ηγϵcεαβ 的对

称性，故 ∂βcαγδ = ∂αcβγδ 也是关于 α, β, γ, δ 对称的，从而存在函数 F (t)使得 cαβγ(t) =

∂α∂β∂γF (t)。引理得证。 注 3.1 对余切向量 ξ = ξαdt

α + 0dz，形变联络在其分量上作用的表达式为

∇αξβ = ∂αξβ − zcγαβξγ (3.4)

∇ ddzξβ = ∂zξβ − Eγcαγβξα +

1

zM ε

βξε (3.5)

其中M εβ 为线性算子 µ = 1 − d

2 − ∇E 的相应位置的矩阵元。考虑方程组 ∇ξ = 0 (其中协

变导数只是关于变量 t的)，若假设其相容性，即 ∂ϵ∂αξβ = ∂α∂ϵξβ , ∀ α, ε，由

∂αξβ = zcγαβ(t)ξγ⇒ ∂ϵ∂αξβ = z∂εc

γαβ(t)ξγ + z2cγαβ(t)cσγϵ(t)ξσ

交换 α, ε的顺序，由相容性得到 ∂εcγαβ = ∂αc

γεβ 及 cγαβc

σγϵ = cγεβc

σγα，也即曲率的消失

性。这给出了结合性方程组的一个“Lax对”。 ∀ z 此方程组有 n维解空间。这组解与形

变联络 ∇(z)的平坦坐标密切相关。也即，存在一组独立函数 t1(t, z), ..., tn(t, z)，使得在

这组新坐标下，形变联络对应的协变导数就是相应的偏导数

∇α(z) =∂

∂tα(t, z)(3.6)

下面我们来看这组新的平坦坐标 t与方程组 ∇ξ = 0的解的联系。首先，设 t是关于

形变联络 ∇的一组平坦坐标，并设一组函数 ξαβ = ∂β t

α nα,β=1 ，下证它们构成方程组

∇ξ = 0的基础解系。

由于 t是平坦坐标，故在此坐标下协变导数就是相应的偏导数。简记 ∂∂tα

= ∂α ，于

是一方面有∇∂α∂γ = ∇∂α∂γ + z∂α∂γ = zcσαγ∂σ = zcσαγξ

εσ∂ε

另一方面，也有

∇∂α∂γ = ∇ξβα∂β

∂γ = ξβα∇∂β

(ξεγ ∂ε) = ξβ

α(∂βξεγ)∂ε

= ∂αtβ ∂βt

λ(∂λξεγ)∂ε = δλ

α(∂λξεγ)∂ε = (∂αξ

εγ)∂ε

对比上面两式，我们得到 ∂αξεγ = zcσαγξ

εσ ，即 ∀ ε, ξε 是方程组 ∇ξ = 0的一个解，又

(ξεβ)构成坐标变换的 Jacobi矩阵，其行列式非零，故 ξ1, ..., ξn 是此方程组的一个基础解

系。

反之，若 ζ1, ..., ζn是方程组 ∇ξ = 0的一个基础解系，则其中每一个解都是前面得到

的基础解系 ξ1, ..., ξn的线性组合，即存在非退化常值矩阵 P = (Pαβ ) ,使得

18

(ζαβ ) = (ξα

β )P

从而有 ζαβ = ξγ

βPαγ = (∂β t

γ)Pαγ = ∂β(Pα

γ tγ)，记 sα := Pα

γ tγ ,则

(s1, ..., sn) = (t1, ..., tn)P

由矩阵 P 非退化，知 s = (s1, ..., sn)也是形变联络 ∇的一组平坦坐标，且 ζαβ = ∂β s

α。

进一步，方程组 ∇ξ = 0添加进一个含参数 z的方程后，整体的相容性等价于WDVV

方程组。即有如下命题

命题 3.2 WDVV等价于方程组 ∇ξ = 0与下面方程的相容性

z∂zξα = zEγ(t)cβγα(t)ξβ +Qγαξγ (3.7)

其中 Qγα = ∂αE

γ。

证明:由方程组 ∇ξ = 0的相容性，已有 cγαβ 的结合性和 ∂δcγαβ 的对称性，在由添加进方程

z∂zξα = zEγ(t)cβγα(t)ξβ +Qγαξγ 后整体有相容性，故 ∀ i = 1, ..., n，有

∂i∂zξα = ∂z∂iξα (3.8)

具体计算这两项，得到

∂i∂zξα = Qγi c

εγαξε + Eγ(∂ic

εγα)ξε +Qγ

αcεiγξε + zEγcβγαc

εiβξε

∂z∂iξα = cεiαξε + cγiαQεγξε + zcβiαE

γcεγβξε

由之前已得的结合性 cβγαcεiβ = cβiαcεγβ 及对称性 ∂ic

εγα = ∂γc

εiα，我们得到下面等式

Qγi c

εγα + Eγ∂γc

εiα +Qγ

αcεiγ = cεiα + cγiαQ

εγ

上式即为 LE(∂i · ∂α) = LE∂i · ∂α + ∂i · LE∂α + ∂i · ∂α的分量形式,由此得到

LEcεiα = cεiα

因此相应的自由能函数 F (t)为WDVV方程组的解。又上面的推导过程反过来也成

立，故两者等价，命题得证。

3.2 Frobenius流形上的新度量

另一个处理 Frobenius流形的重要工具是其上定义的一个新度量。为得到此度量，先

考虑在余切丛 T ∗M 上定义的一个度量，即作为 1−形式的内积。对两个 1−形式 ω1 和 ω2

，我们定义

(ω1, ω2)∗ = iE(ω1 · ω2) (3.9)

(度量上带脚标 ∗是为强调这是一个在余切丛 T ∗M 上的内积)。这里 iE 是 1−形式关于向量场 E 的收缩算子，即

19

iE : Ω1(M) → C∞(M), ω 7→ ω(E)

由于在 TM 与 T ∗M 之间有基于内积 < , >的同构 TM ↔ T ∗M, ηαβ∂β ↔ dtα ，我

们可定义两个 1−形式 ω1, ω2 的乘积为其在 TM 中对应元素 X,Y 的乘积 X · Y 再映回到T ∗M 中对应的元素，即

(ω1 · ω2)(−) =< X · Y ,− >

于是我们得到新度量的具体定义

(ω1, ω2)∗ = iE(ω1 · ω2) =< X · Y ,E > (3.10)

于是在平坦坐标 tα下，度量 ( , )∗的分量为

gαβ(t) := (dtα, dtβ)∗ =< ηαi∂i · ηβj∂j , Eε∂ε >= ηαiηβjEεcγijηγε = Eε(t)cαβ

ε (t) (3.11)

其中 cαβε (t) := ηασcβσε(t)。当算子 ∇E 可对角化时，设 E = Eε∂ε = (1 − qε)t

ε∂ε ,则

gαβ(t) = Eεcαβε = Eεηiαηjβcijε = ηiαηjβEε∂εFij (3.12)

对 (1.8)式两边关于 ti, tj 求二阶偏导数，得到

LEFij = (1 − d+ qi + qj)Fij +Aij

代入 (3.12)式中，得到

gαβ(t) = (1 − d+ qi + qj)ηiαηjβFij(t) + ηiαηjβAij

= (1 + d− qα − qβ)ηiαηjβFij(t) + ηiαηjβAij

(3.13)

记 Fαβ(t) := ηiαηjβFij(t), Aαβ = ηiαηjβAij ，则上式化为

gαβ(t) = (1 + d− qα − qβ)Fαβ(t) +Aαβ (3.14)

引理 3.3 对充分小的 t1 = 0，度量 (3.9)在 t1−轴附近是非退化的。证明:我们有 cαβ

1 (t) = ηασcβσ1(t) = ηασδβσ = ηαβ ，故对很小的 t2, ..., tn，有

gαβ(t) ≃ t1cαβ1 +Aαβ = t1ηαβ +Aαβ (3.15)

故对充分小的 t1，矩阵 (gαβ)是非退化的。引理得证。 由余切丛上的度量我们可以定义一个切丛上的新的度量。我们已有 T ∗M 上的度量

gij(t) = (dti, dtj)∗

其中 (gij)是可逆对称矩阵，其逆矩阵 (gij) := (gij)−1指定了流形M 上的一个度量，也即

切丛 TM 上的一个非退化内积 ( , )，定义如下

(∂i, ∂j) := gij(t), ∂i :=∂

∂ti(3.16)

设此度量的 Levi-Civita联络为∇，由于此度量不一定是平坦的，设∇k∂i = Γs

ki∂s (3.17)

20

其中 Γkij = 1

2gks(∂igjs + ∂jgis − ∂sgij)是 Christoffel符号，关于指标 i, j 对称。且 ∀ k, ∇k

对度量 g还要满足条件

∇kgij := ∂kg(∂i, ∂j) − g(∇k∂i, ∂j) − g(∂i,∇k∂j) = ∂igij − Γskigsj − Γs

kjgsi = 0 (3.18)

由协变导数在余切丛上作用为 ∇kdti = −Γi

ksdts，上式也等价于 ∇k 对余切丛上的度

量满足

∇kgij := ∂kg

ij + Γiksg

sj + Γjksg

si = 0 (3.19)

为方便，我们使用反变形式的分量，即定义

Γijk := (dti,∇kdt

j)∗ = (dti,−Γjksdt

s) = −gisΓjks (3.20)

则方程组 (3.19)和 Γkij = Γk

ji在反变形式下的表达分别为

∂kgij = Γij

k + Γjik , g

isΓjks = gjsΓik

s (3.21)

为记述方便，引入算子

∇i := gis∇s , ∇iξk = gis∂sξk + Γisk ξs (3.22)

并称算子∇i和相应的系数 Γijk 为反变联络。由算子∇i的非交换性，下式给出此度量的曲

率张量 Rkslt的定义

[∇s,∇l]ξt = (∇s∇l − ∇l∇s)ξt = −Rksltξk (3.23)

其中Rkslt = ∂sΓ

klt −∂lΓ

kst +Γk

srΓrlt −Γk

lrΓrst。若度量的所有曲率都消失，则称该度量是平坦

的。对一个平坦度量局部上存在平坦坐标系使得在此坐标系下度量是常值且 Levi-Civita

联络的分量消失。反之，若对一个度量存在一个满足上述要求的平坦坐标系，则此度量是

平坦的，且此平坦坐标在相差一个常系数仿射变换的意义下是唯一的。此时平坦坐标可由

解下面的方程组得到

∇i∂jp = gis∂s∂jp+ Γisj ∂sp = 0 , i, j = 1, ..., n (3.24)

其 n个线性无关的解 p1, ..., pn 即为该度量的平坦坐标。若我们选择已正交化的平坦

坐标系 (dpa, dpb)∗ = δab，则对此度量和 Levi-Civita联络的分量，成立下面的公式：

gij =∂ti

∂pa

∂tj

∂pa(3.25)

Γijk dt

k =∂ti

∂pa

∂2tj

∂pa∂pbdpb (3.26)

(3.25)式是由于

gij = (dti, dtj)∗ = (∂ti

∂padpa,

∂tj

∂pbdpb)∗ =

∂ti

∂pa

∂tj

∂pbδab =

∂ti

∂pa

∂tj

∂pa

21

(3.26)式是由 Γijk = (dti,∇kdt

j)∗ ⇒

Γijk

∂tk

∂pb=(dti,

∂tk

∂pb∇ ∂

∂tkdtj)∗

=(dti,∇ ∂tk

∂pb∂

∂tk

dtj)∗

=(dti,∇ ∂

∂pb(∂tj

∂psdps)

)∗

=(dti,

∂2tj

∂pb∂ps

∂ps

∂trdtr)∗

=∂2tj

∂pb∂ps

∂ps

∂tr∂tr

∂pa

∂ti

∂pa=

∂2tj

∂pb∂psδsa

∂ti

∂pa

=∂2tj

∂pb∂pa

∂ti

∂pa

从而有

Γijk dt

k = Γijk

∂tk

∂pbdpb =

∂ti

∂pa

∂2tj

∂pa∂pbdpb

现在我们考虑流形上有两个不成比例的度量 ( , )∗1 和 ( , )∗

2 ，在某坐标系下它们有相

应的分量表示 gij1 和 gij

2 。用 Γij1k 和 Γij

2k 标记相应的 Levi-Civita联络 ∇i1和∇i

2。记差

∆ijk = gis2 Γjk

1s − gis1 Γjk

2s (3.27)

是流形上的一个张量。下面我们考虑 Frobenius流形M 上的度量 ( , )∗ 和 < , >∗ （后者

是由原来的平坦度量 < , >诱导的 T ∗M 上的内积）。在这里我们进一步假设Euler向量场

E 在平坦坐标下是线性的（可能非齐次）。关于这两个度量的关系我们有如下命题：

命题 3.4 Frobenius流形上的度量 ( , )∗ 和 < , >∗ 构成一个平坦束，即满足下面两个性

质：

1)度量 gij = gij1 + λgij

2 是平坦的，∀ λ；2) 1)中度量的 Levi-Civita联络具有形式 Γij

k = Γij1k + λΓij

2k。

此时称度量 ( , )∗ 为 Frobenius 流形的交叉形式。已知 Frobenius 流形的交叉形式，

Euler向量场 E 和单位向量场 e，若还有 ∀ 1 6 α, β 6 n, dα + dβ − dF + 2 = 0，则我们

能唯一地重建出这个 Frobenius流形结构。实际上，对比 < dtα, dtβ >∗= ηαβ 及

Le(dtα, dtβ)∗ = ∂1g

αβ − ∇(·,dtβ)∗dtα(e) − ∇(dtα,·)∗dtβ(e)

= ∂1(Eεcαβ

ε ) − gβε∇∂εδα1 − gαε∇∂εδ

β1

= (∂1Eε)cαβ

ε + Eε∂1cαβε = cαβ

1 + Eε∂εcαβ1

= ηαβ + Eε∂εηαβ = ηαβ

我们可令

< , >∗:= Le( , )∗ (3.28)

然后我们可选取坐标 tα为度量 < , >∗的平坦坐标，并且使 E 是齐次的。定义

gαβ := (dtα, dtβ)∗ , deg gαβ :=LEg

αβ

gαβ(3.29)

22

我们可以找到函数 F 使得其 Hessian矩阵的反变分量 Fαβ（如 (3.14)式中定义）满足如

下方程组

gαβ = deg gαβFαβ (3.30)

由此我们可建立 Frobenius流形结构。

4 Frobenius流形上的正则坐标

4.1 正则坐标的构造

为构造 Frobenius流形上的正则坐标系，我们需要流形还要满足半单性假设。我们称

点 t ∈ M 为半单的，若 Frobenius代数 TtM 是半单的（即它无幂零元）。易见半单性是点

的开条件。称 Frobenius流形M 满足半单性假设，若M 上一般的点都是半单的。以后我

们称满足半单性假设的M 为 massive Frobenius流形。关于半单点有如下性质：

主要引理 在半单点的一个邻域上存在局部坐标 u1, ..., un使得

∂i · ∂j = δij∂i , ∂i =∂

∂ui(4.1)

证明:由引理 2.2，在半单点 t的邻域上存在向量场 ∂1, ..., ∂n ，使得 ∂i · ∂j = δij∂i （即代

数 TtM 的幂等性）。我们需要证明这些向量场成对交换。令

[∂i, ∂j ] =: fkij∂k (4.2)

我们重新表述形变联络 ∇(z)在基 ∂1, ..., ∂n 下的平坦性条件。联络 ∇的曲率算子定义为

R(X,Y )Z := [∇X ,∇Y ]Z − ∇[X,Y ]Z (4.3)

我们定义在基 ∂1, ..., ∂n下 Frobenius流形上的欧式联络为

∇∂i∂j =: Γk

ij∂k (4.4)

由 R(·, ·)· 为 3−张量，联络 ∇(z) 的曲率消失等价于曲率各分量 R(∂i, ∂j)∂k =

0 , ∀ i, j, k。固定一组 i, j, k，将其展开得到（只需考虑 i = j 的情形）

0 = R(∂i, ∂j)∂k = (∂iΓljk − ∂jΓ

lik)∂l + (Γl

jkΓril − Γl

ikΓrjl − f l

ijΓrlk)∂r

+ z(Γijk∂i − Γj

ik∂j + (δjk − δik)Γlij∂l − fk

ij∂k)

（其中上下指标同时出现代表对其求和只对指标 l, r 有意义）特别由 z 的系数必须是零，

推出

fkij∂k = Γi

jk∂i − Γjik∂j + (δjk − δik)Γ

lij∂l (4.5)

当 k = i, j 时， (4.5) ⇒ fkij∂k = Γi

jk∂i − Γjik∂j ⇒ fk

ij = 0。

23

当 k = i时，(4.5) ⇒ f iij∂i = Γi

ji∂i − Γjii∂j − Γl

ij∂l ，右边 ∂i 的系数为 Γiji − Γi

ij =

0 ⇒ f iij = 0。

当 k = j 时，(4.5) ⇒ f jij∂j = Γi

jj∂i − Γjij∂j + Γl

ij∂l，右边 ∂j 的系数为 −Γjij + Γj

ij =

0 ⇒ f jij = 0。

综上， ∀ i, j, k，总有 fkij = 0，故 [∂i, ∂j ] = 0，引理得证。

在一个 Frobenius流形M 上，称上面引理中所述的局部坐标 u1, ..., un 为正则坐标。

将正则坐标 u看成是原来坐标 t的函数，则由 ∂∂tα = ∂ui

∂tα∂

∂ui ,∂

∂tβ= ∂uj

∂tβ∂

∂uj ，推出一方面

有∂

∂tα· ∂

∂tβ=∂ui

∂tα∂uj

∂tβ(∂

∂ui· ∂

∂uj) =

∑

i

∂αui∂βu

i ∂

∂ui

另一方面也有∂

∂tα· ∂

∂tβ= cγαβ(t)

∂

∂tγ= cγαβ(t)∂γu

i ∂

∂ui

⇒ cγαβ(t)∂γu = ∂αu∂βu (4.6)

故正则坐标可由上面的 PDEs的解得到。或者等价地， 1−形式 du必须是代数同态

du : TtM → C (4.7)

这是因为

∀ i, α, β, dui( ∂

∂uα· ∂

∂uβ

)= dui

(δαβ

∂

∂uα

)= δiαδαβ = δiαδiβ = dui

( ∂

∂uα

)dui( ∂

∂uα

)

正则坐标下，关于 Frobenius流形上的度量，Euler向量场和单位向量场有如下两个

引理

引理 4.1 在半单点邻域的正则坐标下，Euler向量场 E 具有如下形式：

E =∑

i

ui ∂

∂ui(4.8)

证明:设由 E 生成的重标度参数为 k，则在幂等元 ∂i上的作用为 ∂i 7→ k−1∂i，故对 ui的

变化为 ui 7→ kui，即沿着向量场 E 的流为伸缩，引理得证。 引理 4.2 不变内积 < , >在正则坐标下是对角化的

< ∂i, ∂j >= ηii(u)δij (4.9)

其中 η11(u), ..., ηnn(u)为非零函数。正则坐标下的单位向量场 e具有形式

e =∑

i

∂i (4.10)

此引理证明显然。利用上面两个引理，我们得到关于正则坐标成立如下命题

命题 4.3 在半单点的一个邻域上，特征方程

24

det(gαβ(t) − uηαβ) = 0 (4.11)

的所有根 u1(t), ..., un(t)都是单的，且它们是此邻域上的正则坐标。相反地，若点 t处的

特征方程的根都是单的，则 t为 Frobenius流形上的半单点，且 u1(t), ..., un(t)是点 t邻域

上的正则坐标。

证明:由切空间 TM 和余切空间 T ∗M 之间的对应关系，有 dui ↔ η−1ii ∂i，

⇒ dui · duj ↔ η−1ii η

−1jj ∂i · ∂j = η−2

ii δij∂i ↔ η−1ii δijdu

i

⇒ dui · duj = η−1ii δijdu

i

故交叉形式

gij(u) = (dui, duj)∗ = Eε(u)ηiσcjσε(u) = uεη−1ii c

jiε(u) = uiη−1

ii cjii(u) = uiη−1

ii δij

⇒ det(gij − uηij) = det diag(g11 − uη11, ..., gnn − uηnn) =(∏

i

ηii

)−1∏

i

(u− ui)

由此方程 (4.11)化为 ∏

i

(u− ui) = 0 (4.12)

从而命题的第一部分得证。为证第二部分，我们考虑 TtM 上的线性算子 U = (Uαβ (t))，

其中

Uαβ (t) := gαε(t)ηεβ (4.13)

注意到算子 U 在基向量 ∂i上的作用为

U∂i = U ji ∂j = gjkηki∂j = Eεcjkε ηki∂j = Eεηjlηkrcrlεηki∂j = Eεηjlcrlεδri∂j

= Eεηjlcilε∂j = Eεcjεi∂j = Eε(∂ε · ∂i) = E · ∂i

故 U 的作用就是乘以 Euler向量场 E。又设 U = (gij)(ηij) = AB，故其特征方程

det(λI − U) = (−1)n det(AB − λI) = (−1)n det(A− λB−1) detB

由 detB = 0 ⇒ det(λI − U) = 0 ⇔ det(A− λB−1) = 0，而

A− λB−1 = (gij) − λ(ηij) = (gij − ληij)

由此知算子 U 的特征方程即 (4.11)。由条件，该方程的根都是单的，所以在命题假

设下，在 t 点处乘以 E 的算子是半单的。设其互不相同的特征根为 λ1, ..., λn ，对应

特征向量为 e1, ..., en ，即 Eei = λiei, i = 1, ..., n ，若 TtM 中有幂零元 x = 0 ，设

x = x1e1 + · · · + xnen，且存在正整数 k使得 xk = 0。 i = j 时，由 λi = λj 及

λieiej = (Eei)ej = E(eiej) = E(ejei) = (Eej)ei = λjejei

推出 eiej = 0。又设 E = Ekek，则 Eei = Ekekei = Ei(ei)2 ⇒ (ei)

2 = λi

Ei ei =: µiei。且

由 detAB = 0 ⇒ λi = 0 ⇒ µi = 0。计算 xk 的展开式，利用上面结果，归纳得到

xk = xk1µ

k−11 e1 + · · · + xk

nµk−1n en

上式等于零当且仅当 x1 = · · · = xn = 0，即 x = 0，这与之前假设的 x = 0矛盾！故

TtM 是半单的，命题得证。

25

4.2 正则坐标下的旋转系数

我们使用正则坐标的目的是为了将 massive Frobenius流形的局部分类问题化归为一

个 ODE的可积系统。为得到这样一个系统，我们需要研究在正则坐标下不变度量的性

质。再次重申此度量在正则坐标下具有对角形式，即 u1, ..., un 是（局部）欧式空间关于

（复）欧式度量 < , >的正交曲线坐标。在正交曲线坐标的几何中，我们研究的一个主要

对象是旋转系数

γij(u) :=∂j

√ηii(u)√ηjj(u)

, i = j (4.14)

（这里我们事先选取好√ηii(u)的一个分支）。

引理 4.4 不变度量的系数 ηii(u)具有如下形式：

ηii(u) = ∂it1(u), i = 1, ..., n (4.15)

证明:由度量不变性，对任意两个向量场 a, b，不变内积 < , >具有如下形式

< a, b >=< e · a, b >=< e, a · b >≡ ω(a · b) (4.16)

其中 1−形式 ω的定义为 ω(·) :=< e, · >，故

ω = η1idti = dt1 =

∑

i

∂t1∂ui

dui

⇒ ηii(u) =<∂

∂ui,∂

∂ui>= ω

( ∂

∂ui· ∂

∂ui

)= ω

( ∂

∂ui

)= ∂it1(u)

引理得证。 我们总结在正则坐标下不变度量的性质为如下命题：

命题 4.5 不变度量的旋转系数 (4.14)是对称的，即 γij(u) = γji(u)，这个度量关于对角

平移∑

k

∂kηii(u) = 0 , i = 1, ..., n (4.17)

是不变的。函数 ηii(u)与 γij(u)是正则坐标的对应次数分别为 −d和 −1的齐次函数。

证明:由 (4.15)式，有

γij(u) =∂j

√ηii(u)√ηjj(u)

=1

2

∂jηii(u)√ηii(u)ηjj(u)

=1

2

∂j∂it1(u)√∂it1(u)∂jt1(u)

由此得到 γij(u)的对称性。

又由单位向量场 e =∑∂k 的协变常量性（即 ∀ i, ∇∂i

e = 0），我们有

0 = ∇∂i(∑

k

∂k) =∑

k

∇∂i∂k =

∑

k

(Γjik∂j) = (

∑

k

Γjik)∂j ⇒

∑

k

Γjik = 0, ∀ j

特别 j = i时，有∑

k Γiik = 0，又

26

Γiik =

1

2ηis(∂iηks + ∂kηis − ∂sηik) =

1

2ηii(∂iηki + ∂kηii − ∂iηik) =

1

2ηii∂kηii

⇒∑

k

ηii∂kηii = ηii∑

k

∂kηii = 0

由 ηii = 0，故 ∑

k

∂kηii = 0 , i = 1, ..., n

进一步，推出

0 =∑

k

∂k∂it1(u) = ∂i

∑

k

∂kt1(u) = ∂i

∑

k

ηkk(u),∀ i

故和∑

k ηkk(u)是常值。

由 (2.53)式，将其中的 u, v均换为 ∂i，则左边

LE < ∂i, ∂i >= uk∂kηii

右边

2 < LE∂i, ∂i > +(2 − d) < ∂i, ∂i >= 2 < −∂i, ∂i > +(2 − d)ηii = −dηii

结合这两方面得到 uk∂kηii = −dηii ⇒ ηii(cu) = c−dηii(u), ∀ c。对左式两边关于 uj 求

导，得∂jηii(cu) = c−d−1∂jηii(u)

⇒ γij(cu) =1

2

∂jηii(cu)√ηii(cu)ηjj(cu)

=1

2

c−d−1∂jηii(u)

c−d√ηii(u)ηjj(u)

= c−1γij(u)

至此命题全部得证。 推论 4.6 旋转系数 (4.14)满足下面的方程组

∂kγij = γikγkj , i, j, k互不相同 (4.18)n∑

k=1

∂kγij = 0 (4.19)

n∑

k=1

uk∂kγij = −γij (4.20)

证明: 由 (4.15) 式的对角化度量的平坦性，以及 i = j 时， Γkij = 0 若 k = i.j 和 Γi

ij =12η

ii∂jηii，对任意两两互异的 i, j, k，由

0 = R(∂i, ∂j)∂k = [∇∂i,∇∂j

]∂k − ∇[∂i,∂j ]∂k = ∇∂i(∇∂j

∂k) − ∇∂j(∇∂i

∂k)

展开后得到 ∂j 项的系数

∂iΓjjk + Γj

jkΓjij − Γi

ikΓjij − Γj

jkΓkik = 0 (4.21)

将各 Γiij = 1

2ηii∂jηii式及

27

∂iΓjjk = ∂i(

1

2ηjj∂kηjj) = −1

2(ηjj∂kηjj)(η

jj∂iηjj) +1

2ηjj∂i∂kηjj

代入 (4.21)式，整理后得到

−1

4

∂kηjj∂iηjj

η2jj

− 1

4

∂kηii∂iηjj

ηiiηjj+

1

2

∂i∂kηjj

ηjj=

1

4

∂iηkk∂kηjj

ηkkηjj

上式两边乘以√ηjj/ηii得到

−1

4

ηii(∂kηjj)(∂iηjj)√ηiiηjj

3 − 1

4

ηjj(∂kηii)(∂iηjj)√ηiiηjj

3 +1

2

∂i∂kηjj√ηiiηjj

=1

4

∂iηkk√ηiiηkk

∂kηjj√ηjjηkk

(4.22)

化简 (4.22)式的两边，分别得到

LHS = −1

4

(∂iηjj)∂k(ηiiηjj)√ηiiηjj

3 +1

2

∂k∂iηjj√ηiiηjj

=1

2∂k

( ∂iηjj√ηiiηjj

)= ∂kγij

RHS =(1

2

∂iηkk√ηiiηkk

)(1

2

∂kηjj√ηjjηkk

)= γikγkj

由此得到旋转系数 γij(u)满足方程组的 (4.18)式。直接计算 (4.19)左边，得

n∑

k=1

∂kγij =∑

k

(− 1

4

∂iηjj(ηjj∂kηii + ηii∂kηjj)√ηiiηjj

3 +1

2

∂i∂kηjj√ηiiηjj

)

= −1

4

ηjj∂iηjj√ηiiηjj

3

∑

k

∂kηii − 1

4

ηii∂iηjj√ηiiηjj

3

∑

k

∂kηjj +1

2

1√ηiiηjj

∂i

∑

k

∂kηjj

由命题 4.5中的∑

k ∂kηii = 0知 (4.19)式成立。

最后同样由命题 4.5 中的 γij(cu) = c−1γij(u)，两边对 c 求导，并取 c = 1，得到

(4.20)式。推论得证。 我们已经证明了任意massive Frobenius流形决定了Darboux-Egoroff方程组 (4.18, 19)

的一个伸缩不变 (143)解。反之，我们也可以证明在某种泛型假设下，方程组 (4.18 ∼ 20)

的解都局部决定了一个 massive Frobenius流形。

定义 Γ(u) :=(γij(u)

)为方程组 (4.18 ∼ 20)的解矩阵。

引理 4.7 线性方程组

∂kψi = γikψk, i = k (4.23)n∑

k=1

∂kψi = 0, i = 1, ..., n (4.24)

对辅助向量值函数 ψ =(ψ1(u), ..., ψn(u)

)T有 n−维解空间。

证明:为证引理，只需验证该方程组满足相容性，即 ∀ k = l, ∀ i，有

28

∂l∂kψi = ∂k∂lψi (4.25)

由前面推论，当 i = k, l时，

∂l∂kψi = ∂l(γikψk) = (∂lγik)ψk + γik(∂lψk) = γilγlkψk + γikγlkψl

上式交换 k, l保持不变，故此时 (4.25)式成立。

当 i = k or l时，不妨设 i = k，往证

∂l∂kψk = ∂k∂lψk (4.26)

由 (4.23)式，有∂kψk = −

∑

j =k

∂jψk = −∑

j =k

γjkψj

⇒ ∂l∂kψk = −∑

j =k

[(∂lγjk)ψj + γjk(∂lψj)]

= −∑

j =k,l

γljγlkψj − ∂lγlkψl −∑

j =k,l

γljγjkψl − γlk(∂lψl)

= (∑

j =k,l

∂jγlk + ∂kγlk)ψl −∑

j =k,l

(∂jγlk)ψl + γ2lkψk

= (∂kγlk)ψl + γ2lkψk = ∂k(γlkψl) = ∂k∂lψk

综上，方程组 (4.23, 24)满足相容性，故有 n−维解空间。引理得证。 在特定的泛型假设下。可选取方程组 (4.23, 24)的解空间的一组基，对 u是齐次的。

我们引入 n× n−矩阵

V (u) := [Γ(u), U ] (4.27)

其中 U := diag(u1, ..., un)， [ , ]为矩阵的对易子。

引理 4.8 矩阵 V (u)满足下面的方程组

∂kV (u) =[V (u), [Ek,Γ]

], k = 1, ..., n (4.28)

其中 Ek 是矩阵单位，即 (Ek)ij = δikδjk。相反地，所有微分方程 (4.18 ∼ 20)可由 (4.28)

推出。

证明:首先，由 (4.19)式，当 i = j 时，

∂iγij + ∂jγij = −∑

k =i,j

∂kγij = −∑

k =i,j

γikγkj

又由 (4.20)式，当 i = j 时，

ui∂iγij + uj∂jγij = −γij −∑

k =i,j

uk∂kγij = −γij −∑

k =i,j

ukγikγkj

结合上面两个等式，得到

29

(ui − uj)∂iγij =∑

k =i,j

(uj − uk)γikγkj − γij

ui = uj 时，有

∂iγij =1

ui − uj

( ∑

k =i,j

(uj − uk)γikγkj − γij

)(4.29)

由 V (u) = [Γ(u), U ] =(γij(u

j − ui))及 [Ek,Γ(u)] = (δikγkj − δkjγik)，推出

[V (u), [Ek,Γ(u)] =(∑

l

γil(ul − ui)(δlkγkj − δkjγlk) −

∑

l

γlj(uj − ul)(δikγkl − δklγik)

)

=(γikγkj(u

j − ui) −∑

l

γlk

(γil(u

l − ui)δkj + γlj(uj − ul)δik

))

当 i = j 时，① k = i, j 时， (4.28)式左右两边分量分别为

LHS = ∂k(γij(u)(uj − ui)) = (uj − ui)∂kγij = (uj − ui)γikγkj

RHS = γikγkj(uj − ui) = LHS

② k = i时，将 (4.29)式代入，得 (4.28)式左右两边分量分别为

LHS = −γij + (uj − ui)1

ui − uj

(∑

l =i,j

(uj − ul)γilγlj − γij

)= −

∑

l =i,j

(uj − ul)γilγlj

RHS = γiiγij(uj − ui) −

∑

l

γliγlj(uj − ul) = −

∑

l =i,j

γliγlj(uj − ul) = LHS

③ k = j 时，类似②情形可证 (4.28)式左右两边相等。

i = j 时易验证 (4.28)式左右两边均为零。综上， ∀ k， (151)式成立，引理得证。

推论 4.9 关于矩阵 V (u)有如下性质：

1) V (u)作用在线性方程组 (4.23, 24)的解空间上是封闭的；

2) V (u)的特征根不依赖于 u；

3)程组 (4.23, 24)的一个解 ψ(u)关于 u是齐次的

ψ(cu) = cµψ(u) (4.30)

当且仅当 ψ(u)是矩阵 V (u)的从属于特征根 µ的特征向量

V (u)ψ(u) = µψ(u) (4.31)

证明: 1)由∂kψk = −

∑

l =k

∂lψk = −∑

l =k

γlkψl

可将方程组 (4.23, 24)重新写为矩阵形式

∂kψ = −[Ek,Γ(u)]ψ (4.32)

30

⇒ ∂k(V (u)ψ(u)) = (∂kV (u))ψ(u) + V (u)∂kψ(u)

=[V (u), [Ek,Γ(u)]

]ψ(u) − V (u)[Ek,Γ(u)]ψ(u)

= −[Ek,Γ(u)](V (u)ψ(u))

故 V (u)ψ(u)也是方程组 (4.23, 24)的解。

2)由 V (u)是在方程组 (4.23, 24)的 n−维解空间上的作用，故对任意 V (u)的特征值函数

λ(u)，存在方程组 (4.23, 24)的解 ψ(u)使得

V (u)ψ(u) = λ(u)ψ(u)

∀ k，上式两边对 uk 求导得

(∂kλ(u))ψ(u) − λ(u)[Ek,Γ(u)]ψ(u) = −[Ek,Γ(u)]V (u)ψ(u) = −[Ek,Γ(u)]λ(u)ψ(u)

故 ∂kλ(u) = 0, ∀ k ⇒ λ(u) = const。

3)由 ∂iψ = [Γ(u), Ei]ψ，有∑

i

ui∂iψ =∑

i

[Γ(u), Eiui]ψ = [Γ(u),

∑

i

Eiui]ψ = [Γ(u), U ]ψ = V ψ

故 ψ(cu) = cµψ(u) ⇔ ∑i u

i∂iψ = µψ ⇔ V ψ = µψ，推论得证。 接下来，我们将在方程组 (4.23, 24)中嵌入一个谱参数，从而可以使用单值形变技术

对方程组 (4.18 ∼ 20)进行积分。我们记 V (u)的 n个特征根为 µ1, ..., µn，由于 V (u)是反

对称的，这组特征根可按下面顺序重新排列

µα + µn−α+1 = 0 (4.33)

关于矩阵 V (U)和 Frobenius流形的联系，有如下命题

命题 4.10 对一个与WDVV的伸缩不变解相对应的 massive Frobenius流形，矩阵 V (u)

是可对角化的，其特征向量 ψα =(ψ1α(u), ..., ψnα(u)

)T有如下表达式：

ψiα(u) =∂itα(u)√ηii(u)

, i, α = 1, ..., n (4.34)

相应的特征值为

µα = qα − d

2(4.35)

反之，令 V (u)是任意一个方程组 (4.28)的可对角化的解，且 ψα =(ψiα(u)

)是方程

组 (4.23, 24)的解，并满足

V ψα = µαψα (4.36)

则由下面公式

ηαβ =∑

i

ψiαψiβ, ∂itα = ψi1ψiα, cαβγ =∑

i

ψiαψiβψiγ

ψi1(4.37)

31

局部地决定了一个具有伸缩维度 qα = µα − µ1, d = −2µ1 的 massive Frobenius流形。通

过直接计算可验证该命题。

由命题中的构造，知 Darboux-Egoroff方程组的一个解 γij(u)决定了 n个本质上不同

（等价意义下）的WDVV的解，这来自于在公式 (4.37)中选择 ψi1时的自由性。由此，在

相差一个容许度量（即保持流形旋转系数不变）的意义下，可建立 massive Frobenius流

形到方程组 (4.18 ∼ 20)（其中要求 V (u)可对角化）的解之间的一一对应。在引入谱参数

z后，相应的辅助函数 ψ化为 ψ =(ψ1(u, z), ..., ψn(u, z)

)，并有如下引理

引理 4.11 方程组 (4.18, 19)等价于依赖于谱参数 z的辅助函数 ψ =(ψ1(u, z), ..., ψn(u, z)

)

满足的线性方程组

∂kψi = γikψk, i = k (4.38)n∑

k=1

∂kψi = zψi, i = 1, ..., n (4.39)

的相容性。此引理可通过直接计算验证。此时由公式

∂it(t(u), z)√ηii(u)

= ψi(u, z) (4.40)

给出了方程组 (4.38, 39)的解到 ∂αξβ = zcγαβ(t)ξγ 的解之间的一一对应。

进一步，通过引入关于 z的有理系数微分算子

Λ = ∂z − U − 1

zV (u) (4.41)

（其中矩阵 U 和 V (u)如之前定义），我们可将辅助函数 ψ(u, z)满足的方程组的相容性条

件与旋转系数 γij(u)满足的方程组 (4.18, 19)等价起来，得到如下命题

命题 4.12 方程组 (4.18 ∼ 20)（或等价地，方程组 (4.28)）等价于线性方程组 (4.38, 39)

和 z的微分方程

Λψ = 0 (4.42)

整体的相容性条件。

由引理 4.11，只需再证明此时关于对 z求导的相容性，即

∀ k, i, ∂z∂kψi = ∂k∂zψi

对其直接展开计算便可验证该命题。借由对算子 Λ的研究，可以帮助我们从另一个角度

刻画 massive Frobenius流形。

32

5 算子 Λ的单值

5.1 算子 Λ的 Stokes矩阵

考虑方程组 (4.38, 39, 42)的解 ψ(u, z) =(ψ1(u, z), ..., ψn(u, z)

)T，对固定的一个 u，

它是定义在 C \ z = 0,∞ 上的多值解析函数。这个多值由算子 Λ 的单值描述。现在

我们来研究这个算子单值的更多细节。此时我们固定某个向量 u = (u1, ..., un)，仅要求

uini=1 互不相同。在这里我们仅专注于微分方程 (4.42)的 z−依赖性，而暂不考虑它对 u

的依赖性。

方程 (4.42)在 z−扩充复平面 C∪∞上有两个奇点：正则奇点 z = 0和非正则奇点

z = ∞。其在 z = 0处（此时算子 Λ ≃ ∂z − Vz ）的解矩阵具有形式

ψ(z) ≃ zV ψ0 (5.1)

（ ψ0为任意向量）。若矩阵 V 可对角化且 ∀ α = β，差 µα −µβ /∈ Z，则可建立 (4.42)的

一个基础解系，使得

ψ0jα(z) = zµαψjα(1 + o(z)), z → 0 (5.2)

其中 α是解的序号，向量值函数 (ψjα)nj=1, ∀ α = 1, ..., n是矩阵 V 的从属于特征值

µα 的特征向量。考虑由向量 (5.2)作为列向量构成的 (4.42)的解矩阵 Ψ0 = Ψ0(z)，此矩

阵在 z = 0周围的单值具有如下形式

Ψ0(ze2πi) = 3Ψ0(z) · diag(e2πiµ1 , ..., e2πiµn) (5.3)

在 z = ∞处（此时算子 Λ ≃ ∂z −U ）的单值由一个 n×n的 Stokes矩阵刻画。Stokes

矩阵的一个定义是基于通过规范变换将 Λ约化为正则形式的理论。设矩阵值函数 g(z)是

z = ∞附近的解析函数且满足g(z)g(−z)T = I , g(z) = 1 +O(z) as z → ∞ (5.4)

则 Λ的规范变换为

Λ 7→ g−1(z)Λg(z) (5.5)

由 Birkhoff的理论，该变换的轨道的局部坐标由算子 Λ的 Stokes矩阵决定。为给出

Stokes矩阵的一个构造性定义，我们要研究方程 (4.42)的解在 z = ∞附近的渐进分析。仍然固定一个向量参数 (u1, ..., un)，满足 ui = uj , ∀ i = j，此时称为非震荡情形。

我们先引入 Stokes射线的定义，它们是对 i = j 定义的具有如下形式的射线 Rij

Rij := z | Re(z(ui − uj)) = 0, Re(eiεz(ui − uj)) > 0 对充分小的 ε > 0 (5.6)

由此定义知 Rij 是与−−−−→ui − uj 关于直线 C(1 − i)对称的射线，且 Rij 与 Rji 反向。令

直线 ℓ是一条复平面上过原点且不包含 Stokes射线的任意一条定向直线，它将复平面

3实际上有 ψ0jα(zeiθ) = ψ0

jα(z)eiµαθ .

33

分成两个半平面 Cleft 和 Cright ，在这两个半平面上分别存在 (4.42)的解矩阵 Ψleft(z)和

Ψright(z)，使得它们各自在 Cleft和 Cright上解析，且对 z ∈ Cleft/right，当 z → ∞时，解矩阵有渐进表达

Ψleft/right(z) =(1 +O(z−1)

)ezU (5.7)

这两个函数可被解析地延拓到半平面的某个扇形邻域上，在 Cleft 和 Cright 的扇形邻

域的交集上，解矩阵 Ψleft(z)和 Ψright(z)由一个线性变换联系起来。

更具体的来说，设直线 ℓ = z = ρeiϕ0 | ρ ∈ R, ϕ0 固定 ，具有自然定向，则存在非退化的常值矩阵 S+, S−使得延拓后的 Ψleft/right在直线 ℓ交界处有如下关系：

Ψleft(ρeiϕ0) = Ψright(ρeiϕ0)S+ (5.8)

Ψleft(−ρeiϕ0) = Ψright(−ρeiϕ0)S− (5.9)

边值问题 (5.8, 9)和渐进性 (5.7)一起构成了 Riemann-Hilbert问题的一种特殊情形。

关于矩阵 S+, S−有如下性质

命题 5.1 1)矩阵 S±满足

S+ = S, s− = ST (5.10)

S ≡ (sij), sii = 1, sij = 0若 i = j 且 Rij ⊂ Cright (5.11)

2)矩阵

M := STS−1 (5.12)

的特征值为

(e2πiµ1 , ..., e2πiµn) (5.13)

其中 µ1, ..., µn 是 Frobenius簇的谱值（即矩阵 V (u)的特征值）。又设矩阵 C 的列是

M 的相应的特征向量，则解 Ψ0(z)具有如下形式

Ψ0(z) = Ψright(z)C (5.14)

定义 5.1 (5.10)式中的矩阵 S 称为算子 Λ的 Stokes矩阵。

由 (5.11)式知， Stokes矩阵 S 中除对角线外至多有 n(n− 1)/2个元素非零（实际上

关于矩阵对角线对称位置的两个元素至少其中一个为零），从而可对 u1, ..., un进行适当的

排序使得相应的 Stokes矩阵化为上三角阵。且当直线 ℓ在复平面上旋转穿过一条 Stokes

射线 Rij 时， Stokes矩阵也会随之改变。由上面命题还知道若矩阵M 的特征值都是单

的，则解 Ψ0(z)完全由 Stokes矩阵 S 决定。

5.2 Stokes矩阵与 Frobenius流形的关系

关于 Riemann-Hilbert问题 (5.7 ∼ 9)的解在无穷附近的渐进展开有更加精细的估计，

即当 z → ∞时，解有渐进展开式

Ψleft/right(z) =

(1 +

Γ

z+O

( 1

z2

))ezU (5.15)

34

由此展开式得到 V = [Γ, U ]。

接下来我们假设对一个给定的 Stokes矩阵 S 以及一个给定的 u， Riemann-Hilbert

问题 (5.7 ∼ 9)有唯一解。由于该问题的可解性是一个开条件，对于 S 我们得到一个局部

良定义的反对称矩阵值函数 V (u)，而这恰好就是方程组 (4.28)的解，下面的命题对此有

更加具体的表述。

命题 5.2 若方程组 (4.42)的系数矩阵 V (u)满足方程组 (4.28)，则 Stokes矩阵 S 不依赖

于 u。反之，若 V (u)的 u−依赖性保持矩阵 S 不变，则 V (u)满足方程组 (4.28)。

根据此命题，算子 Λ的 Stokes矩阵 S可看成方程组 (4.28)的解的局部参数化。至此，

我们已经接近了使用算子 Λ的 Stokes矩阵 S 明确的给出WDVV方程组的解的参数化这

一目标。在此之前，还需要明确此对应是建立在什么意义下的。实际上对于两个等价的

massive Frobenius流形，它们相应的方程组 (4.28)的解 V (u) = (Vij(u))可通过下面的坐

标置换联系起来:

(u1, ..., un)T 7→ P (u1, ..., un)T (5.16)

V (u) 7→ ϵP−1V (u)Pϵ (5.17)

其中 P 是这个置换的矩阵表示， ϵ是一个对角元为 ±1的任意对角阵。观察到此置换

在微分算子 Λ上的作用为

Λ 7→ ϵP−1ΛPϵ (5.18)

算子 Λ的 Stokes矩阵 S 的变化为

S → P−1ϵSϵP (5.19)

另一方面注意到 Frobenius流形的 Legendre型坐标变换4仅会如 (5.18)方式改变算子

Λ，故相应的 Stokes矩阵 S 的变换如 (5.19)形式。

总结之前的讨论，我们建立了如下的一个局部一一对应

massive Frobenius流形模掉 Legendre型坐标变换↔微分算子 Λ的 Stokes矩阵模掉变

换 (5.19) 在此意义下，可定义该 Frobenius流形的 Stokes矩阵为算子 Λ的 Stokes矩阵模掉变

换 (5.19)后得到的等价类。

6 结论

通过前面的讨论，我们最后得到在具有给定单值的所有具有如下形式

Λ =d

dz− U − 1

zV

4 Legendre型坐标变换 Sκ 的定义为对一个给定的 κ = 1, ..., n，坐标变换为 tα = ∂α∂κF (t), ∂2F∂tα∂tβ =

∂2F∂tα∂tβ , ηαβ = ηαβ .

35

的微分算子的空间上有一个自然的 Frobenius流形结构，这个结构在相差一个 Legendre

型坐标变换的意义下是唯一确定的。

相反地，任何一个满足半单性假设的 Frobenius流形能由算子 Λ的 Stokes矩阵模掉

一个变换得到的等价类空间而建立，并由此沟通了流形与算子单值的联系。

参考文献

[1] Andre Weil, Numbers of solutions of equations in finite fields, Bulletin of the American

Mathematical Society 55 (1949), 497–508.

[2] R. Hartshorne, Algebraic geometry, Graduate texts in mathematics, Springer, 1977.

[3] E. and Kiehl Freitag R., Etale cohomology and the Weil conjecture, Ergebnisse der

Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1988.

[4] William and MacPherson Fulton Robert, A compactification of configuration spaces,

Annals of Mathematics 139 (1994), 183–225.

[5] Jan Cheah, The Hodge polynomial of the Fulton-MacPherson compactification of con-

figuration spaces, American Journal of Mathematics 118 (1996), 963–977.

[6] J. Phillip Griffiths and Harris, Principles of algebraic geometry, Pure and applied

mathematics, Wiley, 1994.

[7] Shoji Yokura, Motivic characteristic classes, Topology of stratified spaces, 2011, pp.

375-418.

[8] Takehiko Yasuda, Motivic integration over Deligne-Mumford stacks, Advances in

Mathematics 207 (2006), no. 2, 707 - 761, DOI 10.1016/j.aim.2006.01.004.

[9] E.R. and Husseini Fadell S.Y., Geometry and topology of configuration spaces, Springer

monographs in mathematics, Springer, 2001.

[10] Jan and Loeser Denef Francois, Germs of arcs on singular algebraic varieties and

motivic integration, Inventiones Mathematicae 135 (1999), 201-232.

[11] Michael and Lunts Larsen V.A., Rationality criteria for motivic zeta functions, Com-

positio Mathematica 140 (2004), no. 06, 1537-1560.

[12] A. Craw, An introduction to motivic integration, Strings and geometry: proceedings

of the Clay Mathematics Institute 2002 Summer School on Strings and Geometry,

Isaac Newton Institute, Cambridge, United Kingdom, March 24-April 20, 2002, 2004.

[13] M. Kontsevich, Lecture at Orsay (1995).

36

Galois表示及其在Fermat大定理证明中的初步应用

潘锦钊∗

指导教师：印林生教授

摘要

本文综述了Galois表示的基本概念，以及怎样通过模形式和椭圆曲线构造Galois表

示，以及Andrew Wiles给出的Fermat大定理的证明过程中是怎么通过Galois表示将模

形式和椭圆曲线联系起来的。

关键词：Galois表示；Fermat大定理；模形式；椭圆曲线

1 引言

1.1 Fermat大定理的历史

Fermat在大约1637年的时候在他读的《算术》的页边写下了著名的Fermat大定理，

用现在的数学语言描述就是：

命题1 (Fermat大定理) 对于n ∈ Z, n > 3，方程

an + bn = cn

没有满足a, b, c ∈ Z, abc = 0的解。

Fermat的注记写道：“我找到了一个绝妙的证明，但是这里页边空白太小，写不下。”

但是现在看来，Fermat大定理的证明远远没有这么简单，人们都认为Fermat当时并没有

得到一个一般性的证明。

容易看到，只需要证明n = 4或n为奇素数的情况就够了。当n = 4时证明只用到初

等数论([3, §1.5], [7, §2.2])，Fermat本人在1640年左右用他发明的“无穷递降法”证明了。

当n = 3时证明要难一些，虽然有初等的证明[7, §6.5]，但是比较繁琐，最好的方法是利

用Z[ω]的唯一分解性[8, §5.4]，ω = −1+√

3i2 。n = 3的情况最早由Euler在1758–1770年间证

明。因此要使Fermat大定理成立，只需要如下命题成立就够了：

∗基数83

37

命题2 设p > 5是素数，则方程

an + bn + cn = 0 (1)

没有满足a, b, c ∈ Z, abc = 0的解。

在19世纪，为了证明Fermat大定理，数学家发展了代数数论的很多工具，并证明

了此定理的很多特殊的情况，但是一般情况始终没有解决。直到20世纪后期，Gerhart

Frey在1985年考虑了某种椭圆曲线，并指出方程(1)解的存在性将导致某种椭圆曲线的存

在性，但是另一方面，这样的椭圆曲线有非常不寻常的性质，人们猜想它根本不存在。之

后Jean-Pierre Serre, Ken Ribet, Andrew Wiles, Richard Taylor将这一过程中的每一步都

填充完整了。最后Fermat大定理在1995年得到证明。

1.2 本文章的目的

本文章将综述Fermat大定理的证明框架，以及Andrew Wiles的工作，对Fermat大定

理的证明过程中是怎么通过Galois表示将模形式和椭圆曲线联系起来的。为了让更多的人

能了解Fermat大定理的证明过程，本文章将综述模形式、Galois表示和椭圆曲线的定义、

基本性质和重要结论，以及它们之间的联系。对于Fermat大定理的证明细节，由于需要

用到更深入的内容，因此就不能在这里详细介绍，只是陈述证明中的重要定理和结论，详

细步骤可以参见[1], [2]以及其它相关文献。

在1850年左右，Kummer做了大量的工作，他的工作是将Fermat, Euler的方法推广，

试图用此来完全证明Fermat大定理。他利用了分圆域的性质，可以证明当p是正则素

数1时Fermat大定理成立。但是在1915年，K. L. Jensen证明了存在无穷多个素数不是正则

素数，因此这个方法并不能完全证明Fermat大定理。到最后Fermat大定理的解决并没有

使用Kummer的思路。本文章将不再综述Kummer的工作。如果读者想了解Kummer的工

作，可以参阅[3], [9, §3.2.4, §9.1], [8, §6.5, §9.4]等。

2 模形式

我们只介绍经典的，在上半平面H := z ∈ C|Im z > 0上定义的模形式。即使在上半平面定义的模形式，也有几种不同的等价定义，我们只陈述其中一种。此章内容主要来

自[5]和[6]。

1即奇素数p使得分圆域Q(ζp)的类数不能被p整除。

38

2.1 定义

全模群SL2(Z) :=

(a b

c d

)∣∣∣∣∣a, b, c, d ∈ Z, ad− bc = 1

通过分式线性变换作用

在H上：

γ(z) =

(a b

c d

)∗ z :=

az + b

cz + d

其同样通过分式线性变换作用在P1(Q) := Q ∪ i∞上，而且此作用是可迁的。容易知道，ker(SL2(Z) → Aut(H)) = ±I，于是此群作用通过PSL2(Z)分解。SL2(Z)的任意一个子

群也作用在H上。我们定义SL2(Z)的一些子群：

Γ0(N) :=

(a b

c d

)∣∣∣∣∣

(a b

c d

)≡(∗ ∗0 ∗

)(mod N)

Γ1(N) :=

(a b

c d

)∣∣∣∣∣

(a b

c d

)≡(

1 ∗0 1

)(mod N)

ΓN := Γ(N) :=

(a b

c d

)∣∣∣∣∣

(a b

c d

)≡(

1 0

0 1

)(mod N)

其中Γ(N)称为级为N的主同余子群(principal congruence subgroup)。SL2(Z)的子群G称

为同余子群(congruence subgroup)，若存在正整数N使得Γ(N) ⊂ G。如果N是最小的正

整数，使得Γ(N) ⊂ G，则G称为级为N的同余子群。由于Γ(N)是SL2(Z)的指数有限的子

群，因此同余子群也是SL2(Z)的指数有限的子群。反过来不一定对，见[10, Example 2.5]。

(在[6]中，Γ0(N)称为级为N的同余子群。)

设G是SL2(Z)的指数有限的子群，则其在H上的作用诱导商空间Y (G) := H/G，其为

一个Hausdorff空间，有一个Riemann面结构，但不紧致。可以在H中选取一个连通集，使

得其为H/G的等价类代表元集，称为H在G下的基本区域，我们也用符号H/G表示。

全模群SL2(Z)的一个基本区域是(图1)

D =

x+ iy

∣∣∣− 1

26 x <

1

2, x2 + y2 > 1,若x > 0;x2 + y2 > 1,若x 6 0

而且对于群G，其基本区域可以选取为D经过SL2(Z)/G的一个等价类代表元集(此为有限

集)变换之后的并集。可以选取等价类代表元集，使得用此方法得到的G的基本区域是连

通集。

我们看变换γ(z) = az+bcz+d的不动点。若

az+bcz+d = z，则cz2 + (d − a)z − b = 0，判

别式∆ = (d − a)2 + 4bc = (a + d)2 − 4。若∆ > 0，即|a + d| < 2，则γ在R \ Q上有两个不动点。这时γ称为双曲变换。若∆ = 0，即|a + d| = 2，则γ只有一个不动

39

点，为P1(Q)中的点，即有理数或i∞。这时γ称为抛物变换。若∆ < 0，即|a + d| > 2，

则γ有两个不动点，其中有一个在上半平面H中，称为椭圆不动点。这时γ称为椭圆变

换。椭圆不动点z的周期是指其对应椭圆变换在PSL2(Z)中的阶。椭圆不动点只有两种：

z ∈ SL2(Z) ∗ i或z ∈ SL2(Z) ∗ (−12 +

√3

2 i)。两种情况下的周期分别为2和3。

在H/G中添加有限个点后可以使得其成为紧致Hausdorff空间X(G) := H/G。在其

之上有紧致Riemann面的结构(因此其上面还有代数曲线的结构)，称为G的模曲线。这

些添加进去的点是H/G基本区域边界上同时在H边界上的点，是i∞或者实轴上的有理点。这些点称为G的尖点(cusps)。实际上，G的所有尖点集合就是G作用在P1(Q)上的轨

道集合。例如当G = SL2(Z)时，其在P1(Q)上的作用可迁，仅有的尖点是i∞。我们约定X(N) := X(Γ(N)), X0(N) := X(Γ0(N)), X1(N) := X(Γ1(N))。

现在我们可以定义模形式：

定义3 设G是SL2(Z)的指数有限的子群，k ∈ Z。一个亚纯函数f : H→ C称为群G的

权为k的亚纯模形式，如果对所有γ =

(a b

c d

)∈ G, z ∈ H，都有

f(γ(z)) = f

(az + b

cz + d

)= (cz + d)kf(z).

(有时我们还要求f在G的所有尖点处亚纯，定义与下面的在尖点处全纯的定义类似。)

f称为群G的权为k的(全纯)模形式，如果f还是H上的全纯函数，而且在G的所有尖

点处全纯。f称为群G的权为k的尖点形式(cusp form)，如果f满足之上所有条件，而且

在G的所有尖点处取值为0。

我们来解释一下定义中的“在尖点处全纯”的意义。对于G是SL2(Z)的指数有限的子

群，存在最小的正整数N使得z 7→ z +N在G中。这时对于f是群G的权为k的亚纯模形式，

有f(z) = f(z + N)。因此如果f在H上全纯，设q = exp(2πi z

N

)，则f关于q在去心的单位

圆盘内解析。我们说f在i∞处全纯(即解析)，如果f关于q在原点处解析(由复分析，要使

此条件成立，只需要f关于q在原点处连续)。这时f在原点处有幂级数展开

f(z) =∞∑

n=0

anqn,

称为f在i∞的Fourier展开或者q-展开(q-expansion)。an = an(f)称为f的第n个Fourier系

数。类似地，用一个抛物变换将其它尖点变为i∞，可以定义f在其它尖点处的全纯性和Fourier展开。如果f是尖点形式，则有a0 = 0成立。反过来，如果f在每个尖点处

的Fourier展开的第0个系数为0，则f是尖点形式。

40

群G的权为k的模形式全体构成一个线性空间，记为Mk(G)。尖点形式全体构成一个

线性子空间，记为Sk(G)。容易验证权为k和l的模形式相乘得到权为k + l的模形式，因

此M(G) :=⊕∞

k=0Mk(G)构成一个分次环。

我们有一些方法来造G上的权为2k的模形式和尖点形式：Poincare级数和Eisenstein级

数。请参见[6, §8–9]。

2.2 模形式空间的维数

根据Riemann-Roch定理，可以计算出模形式空间和尖点形式空间的维数。我们只陈

述权为偶数时的情况：

定理4 ([5, Theorem 3.5.1]) SL2(Z)的指数有限的子群G的权为2k的模形式空间的维

数是

dimM2k(G) =

0, 若k 6 −1,

1, 若k = 0,

(2k − 1)(g − 1) + σ∞k + σ2

⌊12k⌋

+ σ3

⌊23k⌋, 若k > 1.

权为2k的尖点形式空间的维数是

dimS2k(G) =

0, 若k 6 0,

g, 若k = 1,

(2k − 1)(g − 1) + σ∞(k − 1) + σ2

⌊12k⌋

+ σ3

⌊23k⌋, 若k > 2.

其中g是H/G的亏格，σ∞是H/G的尖点个数，σn是G在H/G中的周期为n的椭圆不动点个

数，n = 2, 3。

作为例子，我们计算SL2(Z)和Γ0(2)的情形。

图1是SL2(Z)的基本区域和椭圆不动点。其中白点表示SL2(Z)的周期为2的椭圆不动

点：i。黑点表示周期为3的椭圆不动点，有两个：±12 +

√3

2 i，但是粘合之后表示的是

同一个点。基本区域的标准三角剖分有3个顶点，3条边，2个面，因此亏格g = 0，以

及σ∞ = σ2 = σ3 = 1。因此得到当k > 1时

dimM2k(SL2(Z)) = 1− k +

⌊1

2k

⌋+

⌊2

3k

⌋=

⌊k6

⌋+ 1, 若k ≡ 1 (mod 6),

⌊k6

⌋, 若k ≡ 1 (mod 6).

同理容易计算当k > 1时，dimS2k(SL2(Z)) = dimM2k(SL2(Z))− 1。

41

图 1 SL2(Z)的基本区域和椭圆不动点 图 2 Γ0(2)的基本区域和椭圆不动点

有Γ0(2) = Γ1(2)，以及Γ0(2)\Γ陪集代表元可以选取为I,

(0 1

−1 0

),

(0 1

−1 1

)。

因此得到图2是Γ0(2)的基本区域和椭圆不动点。其中白点表示周期为2的椭圆不动点：12 + 1

2 i。灰点表示Γ0(2)的与i∞不等价的尖点：0。而Γ0(2)的周期为3的椭圆不动点不存

在。基本区域的标准三角剖分有5个顶点，9条边，6个面，因此亏格g = 0，以及σ∞ =

2, σ2 = 1, σ3 = 0。因此得到当k > 1时dimM2k(Γ0(2)) =⌊

k2

⌋+1，以及dimS2(Γ0(2)) = 0，

当k > 2时dimS2k(Γ0(2)) =⌊

k2

⌋− 1。

2.3 Hecke算子, 新形式

上半平面H上有一个SL2(R)不变测度： dxdyy2 ，对于z = x + iy ∈ H。在此测度下，

SL2(Z)的基本区域D的体积是vol(D) =∫D

dxdyy2 = π

3。对于SL2(Z)的有限指数的子群G，

其基本区域体积为vol(H/G) = [SL2(Z) : G]vol(D) = [SL2(Z) : G]π3。用此不变测度我们可

以定义Petersson内积[6, §12]：

定义5 设G是SL2(Z)的有限指数的子群，在G的权为2k的尖点形式空间S2k(G)上

的Petersson内积⟨·, ·⟩G : S2k(G)× S2k(G)→ C定义为

⟨f, g⟩G :=

∫

H/Gf(z)g(z)y2k dxdy

y2.

可以验证此积分是绝对收敛的(事实上只需要一个是模形式，另一个是尖点

形式)，而且与基本区域的选取无关，而且此内积是正定的。因此Petersson内积

42

是S2k(G)上面的Hermite内积。对于G′ ⊂ G都是SL2(Z)的有限指数的子群，f, g ∈ S2k(G)，

有⟨f, g⟩G′ = [G : G′]⟨f, g⟩G。(有些书上Petersson内积的定义还要除掉H/G的体积，这样

的话就有⟨f, g⟩G′ = ⟨f, g⟩G成立。[5, §5.4]) 在不致引起歧义的情况下，我们将Petersson内

积的下标G省略，写为⟨f, g⟩。我们先定义全模群SL2(Z)上的Hecke算子[6, §15]。对T : z 7→ az+b

cz+d ∈ GL+2 (Q)，定义

JT (z) :=dT (z)

dz=

ad− bc(cz + d)2

,

(f |kT )(z) := JT (z)kf(T (z)),

其中f : H → C。用此记号，我们可以说SL2(Z)的指数有限的子群G上的权为2k的模形

式f满足f |kT = f,∀T ∈ G。设G是SL2(Z)的指数有限的子群，H是GL+

2 (Q)的子集，满足HG = GH = H，而且

其有有限的陪集分解：H =⊔

j GMj ,Mj ∈ H。由H可以造出除子群Div(Y (G))以及模形

式空间M2k(G)上的算子，我们都记为T：

T : Div(Y (G))→ Div(Y (G)),

(z) 7→∑

j

(Mjz),

T :M2k(G)→M2k(G),

f 7→∑

j

f |kMj ,

可以验证，这两个定义与陪集代表元Mj的选取无关。

如果我们取G = SL2(Z)，H = Hn = T ∈ M2(Z)|det(T ) = n，则H满足前面的条

件[6, §15, 引理]，陪集代表元可取为

(a b

0 d

)，其中ab = n, b > 0, d = 0, · · · , d − 1。其

定义出来的算子Tn即为SL2(Z)中的一个Hecke算子。记T为由Tn, n > 1生成的Z-代数，称

为Hecke代数。

命题6 ([6, §15, 定理1]) Hecke代数T是一个交换代数，而且

TmTn =∑

d|(m,n)

dTmnd−2 ,

特别，当(m,n) = 1时，TmTn = Tmn；TpnTp = Tpn+1 +pTpn−1。因此T由p是素数时的Tp生

成。

因此我们可以得到Tn的生成函数可以写为局部因子的乘积：∞∑

n=1

Tnn−s =

∏

p

(1 + p1−2s − p−sTp

)−1.

43

设f ∈ M2k(SL2(Z))。利用Hn的陪集代表元的选取，我们可以通过计算知

道Tnf的Fourier系数与f的Fourier系数的关系[6, §16, 定理2]：

am(Tnf) = n1−k∑

d|(m,n)

d2k−1amnd−2(f),

特别，a1(Tnf) = n1−kan(f)。

在Petersson内积下，Tn是S2k(SL2(Z))上面的自伴随算子[6, §16, 定理4]，即∀f, g ∈S2k(SL2(Z)), ⟨Tnf, g⟩ = ⟨f, Tng⟩。因此S2k(SL2(Z))存在一组正交基g1, · · · , gr，使得每个gj =

∑∞n=1 an(gj)q

n都是T中每个Tn的特征向量，且a1(gj) = 1。gj称为归一化的特征形

式(normalized eigenform)。事实上[6, §17, 定理1]，gj是Tn的特征值为n1−kan(gj)的特征向

量，而且aj(n)满足关系式

am(gj)an(gj) =∑

d|(n,m)

d2k−1amnd−2(gj).

对SL2(Z)的任意指数有限子群G上的尖点形式f，我们可以定义其对应的L-函数

L(f, s) :=

∞∑

n=1

an(f)n−s.

设f是SL2(Z)的权为2k的归一化的特征形式，由其Fourier系数的递推公式，我们得到

其L-函数的局部因子乘积公式[6, §17, 定理2]：

L(f, s) =

∞∑

n=1

an(f)n−s =∏

p

(1 + p2k−1−2s − p−sap(f)

)−1.

为了定义SL2(Z)的同余子群上的Hecke算子，我们先定义一些概念。设G是一个

群，Γ1,Γ2是其子群，对任意g ∈ G，集合Γ1gΓ2 := γ1gγ2|γ1 ∈ Γ1, γ2 ∈ Γ2称为双陪集(double coset)。我们注意到Γ1在Γ1gΓ2上有左乘作用，Γ2在Γ1gΓ2上有右乘作用，于是

我们有陪集分解

Γ1gΓ2 =⊔

j

Γ1βj =⊔

k

δkΓ2 (2)

而且元素βj , δk可分别取在gΓ2和Γ1g中。

定义7 ([6, §29]) Γ1,Γ2称为可公度的(commensurable)，记为Γ1 ≈ Γ2，若[Γi : Γ1 ∩Γ2] < ∞, i = 1, 2。设Γ是G的子群，Γ的可公度化子(commensurator)定义为Γ := g ∈G|g−1Γg ≈ Γ。

可以知道，≈是一个等价关系，Γ的可公度化子Γ是G的子群，而且若Γ1 ≈ Γ2，

则Γ1 = Γ2。

44

命题8 若g ∈ G使得g−1Γ1g ≈ Γ2，则双陪集Γ1gΓ2的陪集分解(2)只有有限多个元

素。特别地，若g ∈ Γ，则双陪集ΓgΓ的陪集分解只有有限多个元素。

证明 可以验证，如下两个映射

φ : Γ1\(Γ1gΓ2)→ (g−1Γ1g ∩ Γ2)\Γ2,

Γ1(γ1gγ2) 7→ (g−1Γ1g ∩ Γ2)γ2,

ψ : (Γ1gΓ2)/Γ2 → g−1Γ1g/(g−1Γ1g ∩ Γ2),

(γ1gγ2)Γ2 7→ (g−1γ1g)(g−1Γ1g ∩ Γ2)

是良好定义的，而且是双射。因此由g−1Γ1g ≈ Γ2立即得到陪集分解(2)只有有限多个元

素。

下面我们设G = GL+2 (Q)，Γ1,Γ2 ⊂ SL2(Z)是两个同余子群。则Γ1和Γ2总是可公

度的[5, Exercise 5.1.2]，因为它们的交集还是同余子群。设Γ是一个同余子群，则∀g ∈GL+

2 (Q)，g−1Γg ∩ SL2(Z)也是同余子群[5, Lemma 5.1.1]。因此容易知道，∀g ∈ GL+2 (Q),

g−1Γ1g ≈ Γ2，特别Γ = GL+2 (Q)。

定义9 ([5, Definition 5.1.3]) 设Γ1,Γ2是两个同余子群，α ∈ GL+2 (Q)，则由双陪

集Γ1αΓ2定义了除子群Div(Y (Γ1))到Div(Y (Γ2))的同态：

T : Div(Y (Γ1))→ Div(Y (Γ2)),

(z) 7→∑

j

(Mjz)

以及模形式空间M2k(Γ1)到M2k(Γ2)的算子：

[Γ1αΓ2]2k :M2k(Γ1)→M2k(Γ2),

f 7→∑

j

f |kMj

其中Mj是Γ1αΓ2的陪集代表元：Γ1αΓ2 =⊔

j Γ1Mj。此定义是良好的，而且不依赖于陪

集代表元的选取，将尖点形式映为尖点形式。

下面我们取Γ1 = Γ2 = Γ1(N)，定义M2k(N) := M2k(Γ1(N))上面的Hecke算子。第

一种Hecke算子是diamond算子⟨d⟩：对d ∈ (Z/NZ)×，⟨d⟩f := [Γ1(N)MΓ1(N)]2kf，其

中M是任意Γ0(N)中的元素，使得其右下角元素模N同余于d。由于Γ1(N)是Γ0(N)的正

规子群(其为将右下角元素模N映射的核)，因此Γ1(N)MΓ1(N)的陪集分解代表元只有一

个元素，就是M。因此我们有⟨d⟩f = f |kM，而且可以知道此定义与M的选取无关。如果(N, d) = 1，则定义⟨d⟩ = 0。容易知道⟨d1⟩⟨d2⟩ = ⟨d2⟩⟨d1⟩ = ⟨d1d2⟩。

45

设χ : (Z/NZ)× → C×是一个模N的特征。我们定义M2k(N)的χ-特征空间：

M2k(N,χ) :=

f ∈M2k(N)

∣∣∣∣∣f |kM = χ(d)f,∀M =

(a b

c d

)∈ Γ0(N)

设1N是模N的平凡特征，显然M2k(N,1N ) = M2k(Γ0(N))。由特征空间的定义，f ∈M2k(N,χ)当且仅当⟨d⟩f = χ(d)f,∀d ∈ (Z/NZ)×。可以知道，M2k(N)是其所有特征空

间的直和，而且若f ∈M2k(N)是所有⟨d⟩的特征向量，则其必定在某个特征空间中。第二种Hecke算子Tn按如下方式定义：我们先定义Tp，其中p是素数：

Tpf := [Γ1(N)

(1 0

0 p

)Γ1(N)]2kf.

我们可以找到一组陪集代表元，从而写出Tpf的显式表达式：

命题10 ([5, Proposition 5.2.1]) 当p | N时，Γ1(N)

(1 0

0 p

)Γ1(N)的陪集代表元可以

取为

(1 b

0 p

), b = 0, · · · , p − 1。而当p - N时陪集代表元除了前面列出的元素之外，还要

加上M

(p 0

0 1

)，其中M是任意Γ0(N)中的元素，使得其右下角元素模N同余于p。

我们看到，当N = 1时⟨n⟩平凡，而Tp和我们前面定义的全模群上的Hecke算子相同。

当n不是素数时Tn是通过递推公式定义的：当(m,n) = 1时，TmTn = Tmn；TpnTp =

Tpn+1 +p⟨p⟩Tpn−1。可以验证，当p = q时，Tp与Tq可交换，而且Tp与⟨d⟩可交换。因此Tn是

良好定义的，而且是交换的。由此递推公式，我们可以得到一般情况下，

TmTn =∑

d|(m,n)

d⟨d⟩Tmnd−2 ,

以及Tn的生成函数可以写为局部因子的乘积：

∞∑

n=1

Tnn−s =

∏

p

(1 + p1−2s⟨p⟩ − p−sTp

)−1.

我们看Hecke算子Tn在f的Fourier系数上的作用。通过Tp的显式表达式，以及Tn的递

推公式，可以得到：

命题11 ([5, Proposition 5.3.1]) 设f =∑∞

n=0 an(f)qn ∈M2k(N)，有

am(Tnf) = n1−k∑

d|(m,n)

d2k−1amnd−2(⟨d⟩f),

46

特别，若f ∈M2k(N,χ)，则有

am(Tnf) = n1−k∑

d|(m,n)

χ(d)d2k−1amnd−2(f).

在S2k(N) := S2k(Γ1(N))中，当(d,N) = 1及p - N时，⟨d⟩和Tp是正规算子[5, Theorem

5.5.3]，而且所有的Hecke算子交换，因此存在S2k(N)的标准正交基，使得它们是所有满

足(d,N) = 1及p - N的⟨d⟩和Tp的特征向量[5, Theorem 5.5.4]。

定义12 ([5, Definition 5.6.1]) S2k(N)的旧形式空间(subspace of oldforms)

S2k(N)old := span

∪

M |NM<N

∪

r| NM

f |k(r 0

0 1

)∣∣∣∣∣f ∈ S2k(M)

= span

∪

p|NS2k(N/p) ∪

f |k(p 0

0 1

)∣∣∣∣∣f ∈ S2k(N/p)

是级较低的尖点形式到级N的尖点形式的提升。S2k(N)的新形式空间(subspace of new-

forms)是旧形式空间的正交补：

S2k(N)new :=(S2k(N)old

)⊥.

可以知道，新形式空间和旧形式空间是所有Hecke算子⟨d⟩和Tp的不变子空间[5, Theo-

rem 5.6.2]。因此前面提到的特征向量都落在新形式空间和旧形式空间的并集中。一个重

要结论是说落在新形式空间中的这些特征向量实际上是所有Hecke算子的特征向量[5, The-

orem 5.8.2]。这些特征向量称为新形式(newform)。这时N称为新形式的导子(conductor)，

其一定落在某个χ-特征空间S2k(N,χ)中，将χ称为新形式的特征(character)。

由于新形式是所有Hecke算子的特征向量，根据Hecke算子在Fourier系数上的作

用，我们可以知道对一个特征为χ的归一化的新形式f =∑∞

n=1 anqn ∈ S2k(N)new，

其Fourier系数有关系式

am(f)an(f) =∑

d|(n,m)

χ(d)d2k−1amnd−2(f),

以及L−函数有局部因子乘积：

L(f, s) =∞∑

n=1

an(f)n−s =∏

p

(1 + χ(p)p2k−1−2s − p−sap(f)

)−1.

47

2.4 模Jacobian, 模Abel簇

此节内容主要来自[2]的§1.2和§1.7。

设Γ是SL2(Z)的指数有限的子群。权为2的尖点形式空间S2(Γ)中的元素f可以定义一

个微分形式fdz，此形式是Γ不变的，因此我们可以由f定义X(Γ)上的微分形式ωf，而且

可以知道这样定义出来的微分形式是全纯的。

我们考虑对偶空间V = S2(Γ)∨ := Hom(S2(Γ),C)，此空间维数与S2(Γ)的维数一样，

是X(Γ)的亏格g。我们可以定义同调群H1(X(Γ)) =: Λ到V的自然映射，把每个闭链c映成

线性泛函ϕc : S2(Γ) → C, f 7→∫c ωf。Λ在V中的像是一个格，即秩为dimR(V ) = 2g的离

散Z-模。

取定H中的一个点z0作为基点，我们有Abel-Jacobi映射ΦAJ : X(Γ) → V/Λ，将z ∈X(Γ)映为线性泛函f 7→

∫ zz0ωf。此定义是良好的，即在相差Λ中的元素的意义下，不

依赖于积分路径的选取。因此V/Λ为一个实2g维环面，其上面有复结构。事实上，其

为X(Γ)的Jacobi簇，是一个Abel簇，我们称其为Γ的模Jacobian，记为J(Γ)。

我们设f =∑∞

n=1 anqn是一个Γ上的(归一化)特征形式，K是Q添加其所有Fourier系数

得到的扩域。则我们有代数满同态λf : T⊗Z Q→ K，将每个Hecke算子映成其作用在f后

得到的尖点形式的第1项Fourier系数。由T在S2(Γ)上的作用，可以诱导T在V = S2(Γ)∨上

的作用。更进一步，可以知道T在J(Γ) = V/Λ上也有作用。

Shimura由f构造了Q上的模Abel簇Af，其维数为[K : Q]。我们简要说明一下构造过

程：设If := ker(λf ) ∩ T ⊂ T是T的一个理想。则If作用在J(Γ)上的像If (J(Γ))是一个Q上的连通的子Abel簇，在T作用下稳定。定义Af := J(Γ)/If (J(Γ))，其为一个Q上的Abel簇，

且只与f在GQ作用下的等价类有关。关于模Abel簇的更多信息，可以参见[2]的相关章节，

以及其他相关书籍。由模Abel簇，可以由模形式定义Galois表示，参见后面的章节。

3 Galois表示

此章内容主要来自[1]和[5]。

3.1 定义与性质

设Q是Q在C中的代数闭包。绝对Galois群GQ := Gal(Q/Q)上面有Krull拓扑：单

位元处的邻域基是GQ的所有指数有限的子群。在此拓扑下，GQ是一个射影有

限(profinite)群(即有限群的逆极限)，特别地，是一个紧拓扑群。

定义13 设A是一个拓扑环。A上的n维Galois表示ρ指的是一个连续群同态：

ρ : GQ → GLn(A).

48

定义14 Galois表示ρ : GQ → GLn(A)称为不可约，若ρ不能写为两个非平凡子表示

的直和。若ρ作为Frac(A)上的Galois表示不可约，其中Frac(A)是A的分式域的代数闭包，

则ρ称为绝对不可约。

定义15 ([1, Chapter I, 2.1]) 一个系数环(coefficient ring)指的是一个完备

的Noether局部环，其剩余域是一个特征为固定素数p的有限域。

数学家常研究三种Galois表示[2, §2.1]：第一种是取A = C，称为Artin表示。

由于GQ是紧的全不连通(totally disconnected)群，因此这时ρ的像只有有限个点，

ker ρ是GQ的指数有限的正规子群，其固定域是Q的有限Galois扩张。第二种是取A = k为

特征p的有限域，称为模p表示。同样地，ker ρ是GQ的指数有限的正规子群，其固定域

是Q的有限Galois扩张。第三种是取A为系数环。任意一个系数环A都有唯一的连续环

同态Zp → A(由唯一环同态Z → A，以及A上的p-adic拓扑导出)，因此这样的表示称

为p-adic表示。

下面如果不加说明，我们的表示总是指2维p-adic Galois表示。

定义16 ([1, Chapter I, 2.2]) 设(A,mA, kA)是一个系数环。Galois表示ρ : GQ →GL2(A)的剩余表示(residual representation)

ρ : GQ → GL2(kA)

指的是ρ与自然映射GL2(A)→ GL2(kA)的复合。

我们说ρ是有限域k上的二维Galois表示ρ0 : GQ → GL2(k)到A的提升，若k = kA，

且ρ = ρ0。两个ρ0到A的提升ρ, ρ′称为等价，记为ρ ∼ ρ′，若存在一个GL2(A)中模mA同余

于单位阵的矩阵，使得ρ′通过此矩阵共轭之后等于ρ。

ρ0到A的一个(Galois)形变[(Galois) deformation]指的是ρ0到A的提升的一个等价类。

设ρ是ρ0到A的一个提升，则ρ所在的形变我们也记成ρ。

定义17 ([1, Chapter I, 2.3]) 设ρ : GQ → GL2(A)是A上的二维Galois表示，则ρ的行

列式(determinant)

det ρ : GQ → A×

指的是ρ与GL2(A)上行列式det : GL2(A)→ A×的复合。

在研究问题的时候，经常会限制考虑的Galois表示，使其的行列式等于指定的1维表

示。作为一个例子，我们先定义(p-adic)分圆特征[(p-adic) cyclotomic character]χp : GQ →Z×

p。我们考虑Q(ζp∞) :=∪∞

n=1 Q(ζpn)是Q添加所有pn次单位根所得的扩域。其Galois群为

Gal(Q(ζp∞)/Q) = lim←−n

Gal(Q(ζpn)/Q) ∼= lim←−n

(Z/pnZ)× ∼= Z×p ,

49

因此我们可由Galois群中元素的限制GQ → Gal(Q(ζp∞)/Q), σ 7→ σ|Q(ζp∞ )，定义χp如下：

对σ ∈ GQ，定义χp(σ) = (mn + Z/pnZ)∞n=1 ∈ Zp，其中mn ∈ Z使得σ(ζpn) = ζmn

pn 成立。

由于m1 = 0，因此这个定义得到的元素在Z×p中。现在我们看系数环上的Galois表示。我

们说ρ有行列式χp，若det(ρ)是χp与自然群同态Z×p → A×的复合。

对每个Q的素点l，Ql是Q在l处对应的局部域，对每个Ql我们取定它的一个代数闭

包Ql以及Q到Ql的一个嵌入。即若l是有限素点，则Ql是l-adic数域，即Q在l-adic范数|·|l下的完备化。若l = ∞是无限素点，则Q∞ := R是Q在通常Euclid范数| · |∞下的完备化，且Q∞ := C。在l处的局部Galois群GQl

:= Gal(Ql/Ql)。当l = ∞时GQ∞ := Gal(C/R) =

⟨c⟩是一个二阶循环群，其中c是复共轭。根据代数数论我们知道，对每个l，在Ql上的范数| · |l都可以唯一延拓到Ql上。因此我

们可以看到，GQl中的元素都是Ql的连续自同构。

使用我们取定的嵌入Q ⊂ Ql，Ql的自同构在Q上的限制可以得到Q的自同构，因此有群同态GQl

→ GQ。由于Q在Ql中稠密，因此群同态是单射，我们可以把GQl看成GQ的子

群，称为GQ的分解子群(decomposition subgroup)。严格地说，分解子群并不是良好定义

的，因为它依赖于嵌入Q ⊂ Ql的选取，不过取不同的嵌入，得到的分解子群是互相共轭

的。

对于l = ∞是Q的有限素点，GQl的群作用保持Ql的整数环Zl及其极大理想mZl

。因

此GQl在Fl = Zl/mZl

上自然作用，保持Fl中元素。因此有自然映射GQl→ Gal(Fl/Fl)，而

且可以知道此映射是满射。定义l处的惯性群(inertia group)Il := ker(GQl→ Gal(Fl/Fl)

)。

于是我们得到正合列

1→ Il → GQl→ Gal(Fl/Fl)→ 1.

群Gal(Fl/Fl)是射影循环(procyclic)群(即循环群的逆极限)，其有一个拓扑生成

元，为Frobenius元，是将Fl中每个元素映成它的l次幂。GQ中的Frobenius元Frobl就

是Gal(Fl/Fl)中Frobenius元的原象，在相差Il中元素乘积的意义下定义。

下面我们考虑Galois表示ρ : GQ → GL2(A)的局部性质，即将ρ限制在GQl上得到

ρ|GQl: GQl

→ GL2(A)

的性质。在数论中有很多重要的例子表明，如果l跑遍除去有限多个素点的Q的素点集合，则ρ在同构意义下唯一被这些ρ|GQl

所决定。

定义18 ([1, Chapter I, 2.7]) 我们说ρ是奇表示，若det ρ(c) = −1，其中c是GQ∞中的

复共轭。

定义19 ([1, Chapter I, 2.8]) 我们说ρ在l处非分歧，若Il ⊂ ker ρ|GQl。

50

如果ker ρ是GQ的指数有限的子群(例如当ρ是Artin表示和模p表示)，则ρ在l处非分歧，

当且仅当l在该群的固定域中非分歧。由于当K/Q是固定的有限代数扩张的时候，只有有限个素数在K中分歧，因此我们得到这时ρ在几乎所有素数处非分歧。

如果ρ在l处非分歧，则ρ|GQl: GQl

→ GL2(A)可以通过GQl/Il = Gal(Fl/Fl)分解。因

此这时我们可以谈论ρ在Frobl处的取值。

定义20 ([1, Chapter I, 2.9]) 我们说ρ在p处平坦，若对任意A的指数有限的理想I，

将ρ|GQl模掉I得到的表示GQp → GL2(A/I)都可以延拓为Zp上的有限平坦群概形。

3.2 模Galois表示, Serre猜想

固定一个Q中p之上的素理想p。设f =∑∞

n=1 anqn是一个权为2、导子为N、特征

为ε的(归一化的)新形式。设Kf是由ε的值以及Fourier系数an∞n=1生成的数域在p处的完

备化，以及Of是Kf的整数环。由Eichler和Shimura的构造(模Abel簇Af上的p-adic Tate模

所对应的Galois表示)，f对应一个二维奇Galois表示：

ρf : GQ → GL2(Of )

使得对素数l - pN，ρf在l处非分歧，而且Tr(ρf (Frobl)) = al,det(ρf (Frobl)) = ε(l)l。

对任意权为w的模形式，也可以类似构造Galois表示，只不过最后一个式子变

成det(ρf (Frobl)) = ε(l)lw−1。类比这个性质与Hecke算子在Fourier系数上作用的性质(命

题11)，我们可以得到模Galois表示的定义：

定义21 ([1, Chapter I, 4.3]) 系数环A上的二维Galois表示ρ : GQ → GL2(A)称

为模Galois表示，若存在整数N > 0以及同态π : T′(N) → A，其中T′(N) ⊂End(S2(Γ1(N)))是S2(Γ1(N))的由⟨d⟩和Tl, l - pN生成的Hecke子代数，使得ρ在Np之外非

分歧，以及对任意素数l - pN，都有Tr(ρ(Frobl)) = π(Tl),det(ρ(Frobl)) = π(⟨l⟩)l成立。

著名的Serre猜想说的是对任意ρ是有限域上的二维绝对不可约奇Galois表示，都存在

权为2的特征形式f，使得ρf ∼ ρ，而且对f的导子有较好的估计。特别地，ρ是模Galois表

示。

Serre猜想一直到2008年才被完全证明。不过在当时，Ribet已经证明了一个重要的特

殊情况，Serre将其称之为epsilon猜想：

定理22 (Ribet定理, [1, Chapter I, 4.5]) 设f是权为2，导子为Nl的新形式，其中l -N为素数。若ρf绝对不可约，而且如下两个条件有一个成立：

(1) ρf在l处非分歧；

(2) l = p且ρf在p处平坦，

则存在一个权为2，导子为N的新形式g，使得ρf ∼ ρg。

51

在Fermat大定理的证明中用此定理就足够了。

4 椭圆曲线

此章内容主要来自[1]和[4]。

4.1 定义和性质

定义23 椭圆曲线是指一个二元组(E,O)，其中E是一个亏格为1的光滑射影曲线，

O是E上的一个点(无穷远点)。我们说(E,O)是域K上的椭圆曲线，若E是K上的曲线，

O ∈ E(K)是E上的K-有理点。

任意椭圆曲线都可以看成P2中的光滑三次曲线，由如下的方程给出：

E : y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6 (3)

其中x = XZ , y = Y

Z。此方程称为E的Weierstrass方程。这时O = [0 : 1 : 0]是P2中的一个无

穷远点。如果E是K上的椭圆曲线，则参数a1, a2, a3, a4, a6可以取为K中的元素。

定义24 两个由Weierstrass方程定义的椭圆曲线

E : y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6

E′ : y2 + a′1xy + a′

3y = x3 + a′2x

2 + a′4x+ a′

6

称为同构，若E′可以由E经过如下的变量替换(然后两边除以u6)得到：

ψ :

(x

y

)7→(

u2x+ r

u3y + u2sx+ t

)=

(u2 0

u2s u3

)(x

y

)+

(r

t

)

其中u ∈ K×, r, s, t ∈ K。变量替换ψ称为容许的参数替换。

容易验证椭圆曲线的同构是个等价关系，因此我们把同构的椭圆曲线看作等同。容

易知道，如果K的特征char(K) = 2, 3，则通过容许的参数替换之后，a1, a2, a3可以取为0，

因此E的Weierstrass方程有如下形式：

E : y2 = x3 +Ax+B (4)

52

我们定义椭圆曲线(3)的一些量：

b2 = a21 + 4a2, b4 = 2a4 + a1a3, b6 = a2

3 + 4a6,

b8 = a21a6 + 4a2a6 − a1a3a4 + a2a

23 − a2

4, c4 = b22 − 24b4, c6 = −b32 + 36b2b4 − 216b6,

∆ = −b22b8 − 8b34 − 27b26 + 9b2b4b6,

j =c34∆, 若∆ = 0.

对于方程(4)，

b2 = 0, b4 = 2A, b6 = 4B,

b8 = −A2, c4 = −48A, c6 = −864B,

∆ = −16(4A3 + 27B2),

j = −172864A3

∆= 1728

4A3

4A3 + 27B2, 若∆ = 0.

∆称为椭圆曲线的判别式(discriminant)，j称为椭圆曲线的j-不变量(j-invariant)。容易

验证4b8 = b2b6 − b24, 1728∆ = c34 − c26，以及在容许的参数替换下，c′4 = u−4c4, c′6 =

u−6c6,∆′ = u−12∆, j′ = j。于是j在容许的参数替换下保持不变。这也是j-不变量名字的

由来。

命题25 设E,E′是代数封闭域K上的两个椭圆曲线。则两者同构当且仅当j(E) =

j(E′)。

当∆ = 0时E上没有奇点，是光滑曲线。当∆ = 0时E是退化的椭圆曲线，其上恰好

有一个奇点。这时如果c4 = 0，则奇点处有两个不同的切线方向，称为节点(node)；如

果c4 = 0，则奇点处有两个相同的切线方向，称为尖点(cusp)。

4.2 群律

椭圆曲线E上的点有一个加法群结构，称为群律(group law)，其上的加法的定义

可以有多种不同的看法。设P,Q是E上的点，则P + Q定义为唯一的点R使得除子(P ) +

(Q)和(R) + (O)线性等价，即(P ) + (Q)− (R)− (O)是主除子。几何上看，三个点相加为

零当且仅当三点共线(若有两个点重合，则此直线是经过此点的切线)。因此要计算两个点

相加，只需要把这两个点连起来，所得直线与椭圆曲线相交于第三个点，然后再将此点与

无穷远点连起来，所得直线与椭圆曲线相交于第三个点，即为所给两个点的相加。用此几

何直观很容易得到两点相加的坐标的表达式：设P = (x1, y1), Q = (x2, y2)是以方程(4)给

53

出的椭圆曲线E上的两个点，则有

x(P +Q) =

(y2 − y1

x2 − x1

)2

− x1 − x2, 若P和Q是不同点，

x41 − 2Ax2

1 − 8Bx1 +A2

4(x31 +Ax1 +B)

, 若P和Q是相同点。

类似地可以得到y坐标的表达式。这些式子是原来两点坐标的有理函数。由加法就可以得

到E上的乘m映射：[m] : E → E，其中m ∈ Z。如果E是退化的椭圆曲线，则前面描述的群律使得E上非奇异点的集合Ens上面有

群结构。当E有一个节点的时候，Ens ∼= Gm有乘法群结构；当E有一个尖点的时候，

Ens ∼= Ga有加法群结构。

定义26 椭圆曲线E1, E2间非平凡的保持无穷远点的Abel簇的态射ϕ : E1 → E2称为

同源(isogeny)。

一个同源ϕ : E1 → E2总是一个群同态，即保持加法：ϕ(P + Q) = ϕ(P ) + ϕ(Q)。同

源的核kerϕ总是E1的有限子群。ϕ的次数(degree)定义为其作为曲线间的有限态射的次数，

也等于kerϕ的元素个数。将E1的点都映成O的态射的次数定义为0。

设同源ϕ : E1 → E2的次数为n，则其对应一个对偶同源(dual isogeny)

ϕ : E2 → E1

使得ϕ ϕ = [n]E1 , ϕ ϕ = [n]E2。对偶同源有如下性质：ˆϕ = ϕ, ϕ+ ψ = ϕ + ψ, ϕ ψ =

ψ ϕ以及[m] = [m]。由对偶同源我们知道，椭圆曲线间的同源是一个等价关系。

我们考虑E的自同态全体End(E)，其中包含E到自己的所有同源，以及零态射。

End(E)在加法和映射复合下构成一个环，称为E的自同态环(endomorphism ring)。显然

对任意m ∈ Z，E中的乘m映射都在自同态环中。

命题27 ([1, §2.4]) 设E为域K上的椭圆曲线，则自同态环End(E)同构于如下三种情

况之一：(1) Z；(2)某个虚二次域的序(order)，即乘法封闭的格；(3)某个四元代数的极

大序。最后一种情况只在K的特征非零时出现，这时E称为超奇异(supersingular)椭圆曲

线。

如果K的特征不为2或3，则E的自同构群Aut(E)为如下三种情况之一： (1) µ2，

若j(E) = 0, 1728；(2) µ4，若j(E) = 1728；(3) µ6，若j(E) = 0。

当E的自同态环End(E)严格比Z大的时候，我们称椭圆曲线E有复乘(complex multi-

plication)。例如当j(E) = 0或1728的时候，E有复乘。

54

4.3 极小判别式, 导子, L-函数

设K是一个完备的局部域，v : K× → Z是其赋值。R是其整数环，p是唯一的

极大理想，k = R/p是剩余域。椭圆曲线E/K的极小Weierstrass方程是指方程(3)，其

中a1, a2, a3, a4, a6 ∈ R使得v(∆)最小。若char(k) = 2, 3，则极小Weierstrass方程的系数可

以选取使得a1, a2, a3为0。

我们有一些判定某Weierstrass方程是否是极小的充分条件[4, Chapter VII, Re-

mark 1.1]。根据容许的参数替换下∆变化的关系式，两个v(∆)只能相差12的倍数。因

此若某Weierstrass方程使得v(∆) < 12，则其为极小Weierstrass方程。同理，若v(c4) <

4或v(c6) < 6，则其也为极小Weierstrass方程。若剩余域的特征char(k) = 2, 3，则逆命题

成立：E的极小Weierstrass方程满足v(∆) < 12或v(c4) < 4。

设E/K有(3)形式的方程，E模p的约化(reduction)，记为E，是将E的Weierstrass方

程的系数模掉p得到的k上的椭圆曲线。即使原来的曲线E是非奇异的，E也有可能是奇

异的。如果E非奇异，则我们称E有一个好(good)[或稳定(stable)]约化。如果E有一个节

点，则称E有一个乘性(multiplicative)[或半稳定(semi-stable)]约化。这时若节点处的切

线方向在k中定义，则称为分裂(split)，否则成为非分裂(non-split)。如果E有一个尖点，

则称E有一个加性(additive)[或不稳定(unstable)]约化。有时我们把好约化和乘性约化统

称为半稳定约化。我们说数域K上的椭圆曲线E是半稳定的，若对K的任意有限素点p，

E在Kp上都是半稳定的，即有好约化或者乘性约化。

设K是数域，E/K是椭圆曲线，对每个K的素点p我们考虑E/Kp的一个极

小Weierstrass方程，设其判别式为∆p。E/K的极小判别式(minimal discriminant)是一

个整理想：

DE/K :=∏

p

pvp(∆p)

若K的类数为1(例如K = Q)，则可以找到E的Weierstrass方程使得它同时在K的所有素点

处极小。这个方程的判别式∆就等于E/K的极小判别式，在相差K中一个单位的12次幂的

意义下。

椭圆曲线的极小判别式反映了E的坏约化的多少。另一个反映此性质的量是E/K的

导子(conductor)，这是一个理想：

NE/K :=∏

p

pfp(E/K)

55

其中的指数fp(E/K)为[1, §2.14]：

fp(E/K) :=

0, 若E在p处有一个好约化，

1, 若E在p处有一个乘性约化，

2, 若E在p处有一个加性约化，且p - 6，

2 + δp, 若E在p处有一个加性约化，且p | 6。

最后一个情况比较复杂，包含一个整数δp，我们就不在这里介绍了。由此我们知道，对于

半稳定椭圆曲线，其导子就等于有乘性约化的理想p的乘积。

椭圆曲线E/K的L-函数L(E, s)定义为局部因子的乘积：

L(E, s) :=∏

p

Lp(q−sp )−1,

局部因子

Lp(T ) :=

1− apT + qpT2, 若E在p处有一个好约化，

1− T, 若E在p处有一个分裂乘性约化，

1 + T, 若E在p处有一个非分裂乘性约化，

1, 若E在p处有一个加性约化。

其中qp是p的范数，当E在p处有一个好约化时，定义ap = qp + 1− |E(kp)|，其中kp是Kp的

剩余域。

4.4 椭圆曲线与Galois表示

设E/K是椭圆曲线。E的m-torsion点是指E的几何点，乘m之后为O：

E[m] := ker[m] = P ∈ E(K)|[m]P = O

如果我们只考虑E的K-有理m-torsion点，则记为E(K)[m]。

命题28 设E/K为椭圆曲线。若char(K) = 0或者char(K) = p - m，则有E[m] ∼=Z/mZ× Z/mZ；若char(K) = p > 0，则有E[pr] ∼= Z/prZ或0。

对一个固定的素数l，通过映射[l] : E[ln+1] → E[ln]，可以给l的幂次的torsion点一个

逆向系统。其逆极限称为E的(l-adic)Tate模：

Tl(E) := lim←−n

E[ln]

56

如果char(K) = l，则Tl(E)是秩为2的自由Zl模：Tl(E) ∼= Zl × Zl。用此我们可以定

义2维Ql线性空间：Vl(E) := Tl(E)⊗Q ∼= Ql ×Ql。

设GK := Gal(K/K)是K的绝对Galois群。则由于E[m]中的点的坐标都是K上的

乘m映射的坐标表达式对应的代数方程的根，因此GK在E[m]上有群作用。而且此群

作用于E上的加法结构相容，因为加法对应的坐标表达式也是个K上的有理函数。因此

若char(K) = 0或者char(K) = p - m，则我们得到一个二维Galois表示

ρE,m : GK → Aut (E[m]) ∼= GL2(Z/mZ)

同样，对于Tate模Tl(E)，当char(K) = l时有二维Galois表示

ρE,l : GK → Aut (Tl(E)) ∼= GL2(Zl)

此表示的剩余表示正好是GK作用在E[l]上得到的Galois表示ρE,l。用ρE,l得到线性空

间Vl(E)上的Galois表示，即Ql上的二维Galois表示，也记为ρE,l。

命题29 ([1, §2.7]) Galois表示ρE,m的行列式det(ρE,m)等于分圆特征

χm : GK → Aut(µm) ∼= (Z/mZ)×.

现在设K = Q，l是素数。则有Galois表示ρE,l : GQ → GL2(Zl)，及其剩余表示ρE,l :

GQ → GL2(Fl)。这些表示有如下的性质：

命题30 ([1, Chapter I, 2.11]) 设E是Q上椭圆曲线，ρE,p是p-adic Tate模Tp(E)对应

的Galois表示，NE是E的导子。则ρE,p的行列式等于分圆特征χp，且ρE,p在pNE之外非分

歧。特别地，ρE,p是奇表示。

若E是极小判别式为∆E的半稳定椭圆曲线，则剩余表示ρE,p有局部性质：若素数l =p，则ρE,p在l处非分歧当且仅当p|ordl(∆E)；而且ρE,p在p处平坦当且仅当p|ordp(∆E)。

命题31 ([2, Proposition 2.8 (b), (c)]) 设E是Q上椭圆曲线，则ρE,p绝对不可约，

而ρE,p在几乎所有素数p处绝对不可约。如果E/Q没有复乘，则ρE,p在几乎所有素数p处都

为满射，从而ρE,p也是。

命题32 ([2, Theorem 2.9]) 设E是Q上椭圆曲线，则有：(a) 当p > 163时ρE,p不可

约。(b)若E半稳定，则当p > 7时ρE,p不可约。(c)若E半稳定，而且ρE,2平凡，则当p >

3时ρE,p不可约。

4.5 模椭圆曲线, Taniyama-Shimura-Weil猜想

Q上的椭圆曲线E称为模椭圆曲线(modular elliptic curve)，若存在一个权为2，导子

为NE，特征平凡的新形式f，使得两者的L-函数相等：

L(f, s) = L(E, s).

57

模椭圆曲线有多种等价定义：

定理33 ([1, Chapter I, 5.1]) 设E/Q是椭圆曲线。以下条件等价：(1) E是模椭圆曲线；

(2)对所有素数p，ρE,p是模Galois表示；

(3)对某个素数p，ρE,p是模Galois表示；

(4)存在一个的非平凡的Q上代数曲线的态射π : X0(NE)→ E；

(5) E同源于某个由权为2，导子为NE的新形式f对应的模Abel簇Af。

Taniyama-Shimura-Weil猜 想 是 说 每 个Q上 的 椭 圆 曲 线 都 是 模 椭 圆 曲 线。Wiles在1995年证明了此猜想在半稳定椭圆曲线上的情况，由此证明了Fermat大定理，

我们将在后面叙述。此猜想最终在1999年被证明，证明的想法基于Wiles的方法。现在这

称为模性定理(Modularity Theorem)。

5 Fermat大定理证明框架

5.1 椭圆曲线, Galois表示与Fermat大定理

对于任意互素的三个整数构成的三元组(A,B,C)使得A+B+C = 0，Gerhart Frey考

虑了如下形式的椭圆曲线：

EA,B,C : y2 = x(x−A)(x+B)

并阐述了EA,B,C的一些算术性质与整数三元组(A,B,C)的算术性质的关系。在我们的情

形中，只需要考虑(A,B,C) = (ap, bp, cp)是方程(1)的可能的非平凡解的情况。由于这三个

整数互素，不失一般性，可以设a ≡ −1 (mod 4)以及2 | b。

命题34 ([1, Chapter I, 1.1]) 设p > 5是素数，a, b, c为非零互素整数满足ap+bp+cp =

0。则Eap,bp,cp是半稳定椭圆曲线，极小判别式∆ap,bp,cp = 2−8(abc)2p，导子Nap,bp,cp =∏

l|abc l。

证明 Eap,bp,cp的形如(3)的Weierstrass方程是

y2 = x3 + (B −A)x2 −ABx (5)

经过容许的参数替换(u, r, s, t) = (2, 0, 1, 0)得到

y2 + xy = x3 +B −A− 1

4x2 − AB

16x (6)

58

由于A = ap ≡ −1 (mod 4)，以及p > 5，32|B = bp，因此方程的各项系数都是整数。

此方程的判别式∆ = 2−8A2B2(A + B)2 = 2−8(abc)2p，以及c4 = A2 + AB + B2, c6 =

(A−B)(A+ B2 )(A+2B)。由于A是奇数，B

2是偶数，因此c6是奇数，故此方程在2处极小。

对素数l - abc，则l - ∆，故此方程在l处极小。对素数l = 2使得l | abc，则l恰好整除a, b, c三者之一。若l | a则l - b，故c4 = A2 + AB + B2 ≡ B2 ≡ 0 (mod l)，同理若l | a则l - c4。若l | c则有c4 = (A + B)2 − AB = c2p − AB ≡ −AB ≡ 0 (mod l)。故此方程在l处极小。

故得(6)是全局极小Weierstrass方程，极小判别式为∆ap,bp,cp = 2−8(abc)2p。

若素数l - abc，则方程(5)已经是模l极小，且l - ∆ = 24(abc)2p，故Eap,bp,cp在l处有好

约化。若l = 2，则(6)在F2中约化为E : y2 +xy = x3或y2 +xy = x3 +x2，两者都以(0, 0)为

奇点，前者写为y(y + x) − x3 = 0，为分裂乘性约化，后者为(y2 + xy − x2) − x3 = 0，

为非分裂乘性约化。因此Eap,bp,cp在2处有乘性约化。若2 = l | abc，则方程(5)已经

是模l极小。若l | ab，则E的奇点为(0, 0)，方程写为[y2 − (B − A)x2] − x3 = 0，由

于l - 4(B − A)，因此奇点为节点。若l | c，则在Fl中A + B = 0，E的奇点为(A, 0)，方程

写为[y2 −A(x−A)2]− (x−A)3 = 0，由于l - 4A，因此奇点为节点。故Eap,bp,cp在l处有乘

性约化。故得到Eap,bp,cp是半稳定椭圆曲线，导子Nap,bp,cp =∏

l|abc l。

记E := Eap,bp,cp，则其诱导一个Galois表示：

ρap,bp,cp := ρE,p : GQ → GL2(Fp)

Gerhart Frey和Jean-Pierre Serre注意到此Galois表示有不寻常的局部性质[11]：

命题35 ([1, Chapter I, 3.1]) ρap,bp,cp是奇的绝对不可约Galois表示，在2p之外非分

歧，在p处平坦。

证明 由命题32(c)得到其为不可约表示，再由Serre的一个定理，由其不可约及E半

稳定得到其绝对不可约。剩下的断言由命题30得到。

人们怀疑根本不存在GQ → GL2(Fp)的Galois表示满足这些性质，但是这在当时并没

有被证明。不过根据Ribet的定理(定理22)，不存在模Galois表示满足这些性质：

证明 如果ρap,bp,cp ∼ ρfap,bp,cp是模Galois表示，即fap,bp,cp是权为2的新形式，导子

为Nap,bp,cp =∏

l|abc l。因为ρap,bp,cp是绝对不可约，在2p之外非分歧，在p处平坦，根据定

理22，存在权为2，导子为2的新形式g使得ρg = ρap,bp,cp。但是我们前面已经算得尖点形

式空间S2(Γ0(2))的维数是0，所以根本不存在这样的新形式g。

因此要证明Fermat大定理，只需要证明ρap,bp,cp是模Galois表示就够了。

定理36 (Wiles, [1, Chapter I, 5.3]) Q上任意半稳定椭圆曲线都是模椭圆曲线。

59

特别，Eap,bp,cp是模椭圆曲线，ρap,bp,cp是模Galois表示。

5.2 Wiles的工作

这一节我们简要介绍Wiles为了证明Fermat大定理所做的一些工作。

我们设素数p > 3，k是特征为p的有限域。ρ0 : GQ → GL2(k)是具有行列

式χp的Galois表示。

定义37 ([1, Chapter I, 7.1]) 我们说一个Galois表示ρ : GQ → GL2(A)在p处

是ordinary，若ρ在惯性群Ip的限制在一组适当的基下的表示为ρ|Ip =

(χp ∗0 1

)。我们

说ρ在素数l处半稳定(semistable)，若如下两个条件之一成立：

(1) l = p而且ρ在p处平坦或者ordinary(或者两者都成立)；

(2) l = p而且在一组适当的基下，ρ|Ip =

(1 ∗0 1

)。

我们说ρ是半稳定的，如果它在每个素数处都是半稳定的。

半稳定Galois表示的一个例子：设E/Q是半稳定椭圆曲线，则ρE,p : GQ → GL2(Zp)是

半稳定的。接下来我们假设ρ0是半稳定的。

定义38 ([1, Chapter I, 7.2]) 一个形变类型(deformation type)D指的是剩余表示ρ0的

形变所满足的一组性质。更确切地说，D是一个从以k为剩余域的系数环的范畴到集合范畴的一个函子，对于系数环A，D(A)是满足性质D的ρ0到A的形变的全体。

为了证明定理36，我们只考虑一类特殊的形变类型。设S = l = p|ρ0在l分歧，以及与S不交的素数的有限集合ΣD。我们定义形变类型D中的形变ρ满足：(1)具有行列式χp；

(2)在S∪p∪ΣD之外非分歧；(3)在ΣD之外半稳定；(4)若p ∈ ΣD且ρ0在p处平坦，则ρ也

在p处平坦。粗略地说，最后三个条件是说ρ在ΣD之外与ρ0有相同的局部性质。我们可以

知道，若ρ0在p处ordinary，则ρ也是。

接下来我们假设ρ0是绝对不可约的。使用Mazur的Galois形变理论，Wiles给每个形变

类型D对应了一个系数环RD，称为万有形变环(universal deformation ring)，以及ρ0的类

型D的万有形变(universal deformation)ρD : GQ → GL2(RD)。万有形变环和万有形变满

足如下的万有性质：对任意的ρ0的具有类型D的形变ρ : GQ → GL2(A)，都存在唯一的同

态πA : RD → A使得ρ经过ρD分解：

GQρD //

ρ$$I

IIIIIIII GL2(RD)

∃!πA

GL2(A)

60

关于Galois形变理论的介绍，以及万有形变环的构造，可以参阅[1, Chapter VIII, IX,

XIII]。

如果假设ρ0还是模Galois表示，而且ρ0|GQ(√−3)也是绝对不可约的，Wiles定义了另一

个系数环TD，称为万有模形变环(universal modular deformation ring)，以及万有模形

变(universal modular deformation)ρD,mod : GQ → GL2(TD)，满足如下的万有性质：对

任意的ρ0的具有类型D的形变ρ : GQ → GL2(A)，使得ρ是模Galois表示，都存在唯一的同

态πA : TD → A使得ρ经过ρD,mod分解：

GQρD,mod//

ρ$$I

IIIIIIII GL2(TD)

∃!πA

GL2(A)

Wiles花了很大的工夫才证明和TD和ρD,mod的存在性，并构造出来。可以参见[1, Chapter

VII, X, XII]。

根据万有形变环RD的万有性质，存在唯一的环同态φD : RD → TD使得ρD,mod =

φD ρD成立：

GQρD //

ρD,mod $$IIIIIIIII GL2(RD)

∃!φD

GL2(TD)

Wiles证明了一个“主定理”，其一个特殊情况描述了φD的性质：

定理39 ([1, Chapter I, 7.4]) 设ρ0 : GQ → GL2(k)是具有行列式χp的半稳定、绝对不

可约的模Galois表示，而且ρ0|GQ(√−3)也是绝对不可约的。则映射φD : RD → TD是完全相

交环(complete intersection ring)的同构。

完全相交环的定义比较难，我们就不在此叙述，可以参见[1, Chapter XI]。我们由此

定理可以知道的是，φD : RD → TD是完全相交环的同构，特别地，是环的同构。由φD是

同构，我们立即得到如下推论：

命题40 设ρ0满足定理39的条件，则ρ0的任意的具有类型D的形变都是模Galois表

示。

定理39的证明是用所谓的“Wiles数值判据”(Wiles’ numerical criterion)：

定理41 ([1, Chapter I, 7.5]) 设R, T是两个系数环，O是完备的离散赋值环，以及下

61

面交换图表：

Rφ //

πR @@@

@@@@

T

πT~~~~

~~~

O使得图表中的每个映射都是满射。设IR := kerπR, IT := kerπT , ηT := πT (AnnT (IT ))。则

以下结论等价：

(1) φ是完全相交环的同构；

(2) IR/I2R是有限集，而且|IR/I2

R| 6 |O/ηT |；(3) IR/I

2R是有限集，而且|IR/I2

R| = |O/ηT |。

用此定理，我们设f是权为2的新形式，ρf : GQ → GL2(Of )是ρ0的一个具有类型D的形变。由TD的万有性，存在唯一的环同态πTD : TD → Of，使得ρf = πTD ρD,mod。

设πRD := πTD φD，则有以下交换图表：

RDφD //

πRD !!BBB

BBBB

B TD

πTD||||

||||

O

为了证明φD是同构，Wiles要证明定理41的等价条件(2)。这样就变成了一个纯代数的问

题。此证明的框架和具体细节请参见[1]的相关章节。

5.3 半稳定椭圆曲线的模性定理

这一节我们将综述定理36的大体证明过程。

我们前面已经说过，Q上半稳定椭圆曲线E对应的Galois表示ρE,p是半稳定的。由命

题30，我们知道ρE,p以χp作为行列式。

Serre的一个定理说，当E半稳定时，对任意素数p > 3，剩余表示ρE,p为满射或者

可约。因此当p > 3时，由ρE,p不可约可以推出其绝对不可约，而且当p = 3时，此等价

于ρE,3|GQ(√−3)绝对不可约。因此由命题40和模椭圆曲线的等价定义，我们得到下面命题：

命题42 ([1, Chapter I, 7.8]) 设E/Q是半稳定椭圆曲线。若对于某个素数p > 3，

ρE,p为不可约模Galois表示，则E是模椭圆曲线。

Wiles的证法是证明对于p = 3或p = 5，以上命题的条件成立。此证明基于以下几个

定理，我们不加证明地陈述它们：

定理43 ([1, Chapter I, 7.9]) 对任意Q上椭圆曲线E，若ρE,3不可约，则ρE,3是

模Galois表示。

62

此定理也称为Wiles minus epsilon，是Langlands和Tunnell的一个深刻的定理的推论，

与GL2的Langlands纲领有关。关于GL2的Langlands纲领的介绍，以及此定理的更多内容，

可以参见[1, Chapter VI]。

定理44 ([1, Chapter I, 7.10]) 设E/Q是半稳定椭圆曲线，使得ρE,5不可约。则存在

另一个半稳定椭圆曲线E′/Q使得ρE′,3不可约，而且ρE′,5 ∼ ρE,5。

在[1, Chapter XVI]中，我们可以构造出一族椭圆曲线E′/Q使得ρE′,3不可约，而

且ρE′,5 ∼ ρE,5，而且这些椭圆曲线在5之外都是半稳定的。在这之中我们取E′在5-adic意

义下充分接近E，我们就找到了所求的椭圆曲线。

定理45 ([1, Chapter I, 7.11]) 设E/Q是半稳定椭圆曲线，则ρE,3和ρE,5至少有一个

不可约。

如果此定理的结论不成立，则E[15]会有一个15阶的Galois不变子群，与[1, Chapter

XVI]的Lemma 9 (iv)矛盾。

定理36的证明 设E/Q是半稳定椭圆曲线。若ρE,3不可约，则由定理43，其为

模Galois表示，再由命题42，E是模椭圆曲线。

若ρE,3可约，则由定理45，ρE,5不可约，由定理44，存在另一个半稳定椭圆曲

线E′/Q使得ρE′,3不可约，而且ρE′,5 ∼ ρE,5。因此由前面得到E′是模椭圆曲线，故ρE′,5是

模Galois表示，故ρE,5也是模Galois表示。由命题42，E是模椭圆曲线。

我们在这里提一下，其实只需要Wiles minus epsilon(定理43)以及命题42，利用方

程(1)的非平凡解所对应的半稳定椭圆曲线(5)的参数的特殊性，就能得到(5)是模椭圆曲

线，从而证明Fermat大定理。此证明的过程请参见[12]。

6 结论

到现在为止，我们已经综述了Galois表示、模形式和椭圆曲线的基本概念和性质，以

及怎样由模形式和椭圆曲线造出Galois表示，以及两者造出的Galois表示的联系。

为了证明Fermat大定理，我们假设方程(1)存在一个非平凡解，由此我们造出了一个

半稳定椭圆曲线，由椭圆曲线造出Galois表示，其有不寻常的局部性质，使得其不可能

是模Galois表示。但是由Wiles的半稳定情形的模性定理，此Galois表示一定是模Galois表

示，发生了矛盾。因此我们证明了(1)的非平凡解不存在，从而Fermat大定理成立。

为了证明Wiles的半稳定情形的模性定理，我们使用Galois形变的理论，构造了万有

形变环以及万有模形变环，并使用Wiles数值判据，通过一个纯代数的问题，证明了在某

些情况下万有形变环和万有模形变环的同构性，进而得到某些情况下剩余表示的Galois形

63

变的模性。我们利用GL2的Langlands纲领以及其它数学工具，证明了半稳定椭圆曲线一

定存在某个剩余表示满足我们需要的性质，因此其Galois形变一定是模Galois表示，进而

得到半稳定椭圆曲线的模性。

其它的形如A + B + C = 0的不定方程也可以用类似的方法研究，通过研究由其

构造的椭圆曲线的Galois表示的局部性质，可以得到某些方程的解的不存在性。可以参

见[11]里面所举的例子，以及[1, Chapter XX]。

在1999年椭圆曲线的模性定理被证明，以及在2008年Serre猜想被证明，使得我

们可以断言任意Q上椭圆曲线都是模椭圆曲线，以及其对应的Galois表示的剩余表

示的模Galois提升对应新形式的权和导子的更好的估计。因此我们可以对更多的形

如A+B + C = 0的不定方程的解做出结论。

Fermat大定理的陈述是如此简单和初等，使得任意一个中学生都能看懂，但是其证

明却异常艰难，困扰了无数的数学家。从Fermat大定理的提出，直到其最终被证明，花了

超过350年的时间，其间促进了很多数学学科和理论的发展，例如代数数论，包括理想理

论、ζ-函数及相关解析理论、类域论，Langlands纲领，以及模形式、椭圆曲线和Galois表

示理论，以及椭圆曲线模性定理和Serre猜想的提出和证明。难怪有数学家说Fermat大定

理是“下金蛋的母鸡”。

参考文献

[1] Gary Cornell, Joseph H. Silverman, Glenn Stevens, editors. Modular Forms and Fer-

mat’s Last Theorem. New York: Springer, 1997.

[2] Henri Darmon, Fred Diamond, Richard Taylor. Fermat’s Last Theorem. In Elliptic

Curves, Modular Forms & Fermat’s Last Theorem, 2nd ed. John Coates, S.T. Yau,

editors. Cambridge, MA: International Press, 1997.

[3] Harold M. Edwars. Fermat’s Last Theorem: A Genetic Introduction to Algebraic

Number Theory. Graduate Texts in Mathematics 50. New York: Springer-Verlag,

1977.

[4] Joseph H. Silverman. The Arithmetic of Elliptic Curves, 2nd ed. Graduate Texts in

Mathematics 106. New York: Springer, 2009.

[5] Fred Diamond, Jerry Shurman. A First Course in Modular Forms. Graduate Texts

in Mathematics 228. New York: Springer, 2005.

[6] 陆洪文, 李云峰. 模形式讲义. 北京: 北京大学出版社, 1999.

64

[7] 潘承洞, 潘承彪. 初等数论. 北京: 北京大学出版社, 1992.

[8] 潘承洞. 代数数论. 济南: 山东大学出版社, 2001.

[9] 冯克勤. 代数数论. 北京: 科学出版社, 2000.

[10] Tim Hsu. Identifying Congruence Subgroups of the Modular Group. Proceedings of

the American Mathematical Society. Volume 124, Number 5, May 1996.

[11] Gerhart Frey. Links between solutions of A−B = C and elliptic curves. In Number

Theory, proceedings of the Journees arithmetiques, held in Ulm, 1987. H.P. Schlick-

ewei, E. Wirsing, editors. Lecture Notes in Mathematics 1380. Berlin, New York:

Springer-Verlag, 1989.

[12] Noam Elkies. Wiles minus epsilon implies Fermat. In Elliptic Curves, Modular Forms

& Fermat’s Last Theorem, 2nd ed. John Coates, S.T. Yau, editors. Cambridge, MA:

International Press, 1997.

65

局部域与整体域的欧拉特征公式

卢伟∗

指导教师：印林生教授

摘要

局部域与整体域的欧拉特征公式是由John Tate分别在1962年与1965年给出的。

论文目的是关于局部域与整体域的欧拉特征公式的介绍与详细地证明。在本

文中，首先我们陈述Galois上同调的基本定义以及相关性质，然后我们将分别

介绍与证明局部域的欧拉特征公式与整体域的欧拉特征公式。其证明思路源

于Haruzo Hida书中[1,Ch4.4.4和4.4.5]以及J.S.Milne书中[2,Ch1.5]。在学习过程中，我

发现Haruzo Hida在证明局部欧拉特征公式中，有些打印错误；这些打印错误导致

后面证明中需要做大量的修改；但其大致思路与方向是没有任何问题的。然而在他

证明整体欧拉特征公式的时候，我发现后面在处理等式φ(M) = φ(M∗(1))时，他的

证明有误，并且这个错误是非平凡的。但在证明这个定理之前他声明整个证明来源

于J.S.Milne。于是我查阅了J.S.Milne在其Arithmetic duality theorems书中的相关章

节[2,Ch1.5]，发现他在处理上述等式的方法确实有很大的问题。随后沿着J.S.Milne的

证明思路,在印林生教授与李彬博士的指导与帮助下，我用完全不同于他的方法独

立证明了φ(M) = φ(M∗(1))。最后在3.3小节中给出了反例来说明J.S.Milne证明的错

误。

关键词：错误, “*”函子, φ(M) = φ(M∗(1)), 反例

1 预备知识

在这一节里，我们将不加证明地介绍一些Galois上同调的基本知识。这些证明细节可

以在参考书[4]和[5]中扎到。

1.1 上链与上同调

令G是一个有限群，M是一个G-模。从G-模范畴到阿贝尔群范畴的函子M 7→ MG 是

左正合的。这个函子的导出函子用Hn(G,−)来表示。其中Hn(G,M) 被称作群G的上同调

群，我们可以用链复形通过如下方法去对它进行计算：

∗基数72

66

首先定义Cn(G,M) := Map(Gn,M)，其中Cn(G,M)中的元素f是M的上链且有n个

变元在G中的函数。微分映射

dn : Cn(G,M) → Cn+1(G,M)

定义如下

dn(fn)(g1, · · · , gn) = g1 · f(g2, · · · , gn) +

n∑

i=1

(−1)if(g1, · · · , gigi+1, · · · , gn+1)

+(−1)n+1f(g1, · · · , gn),

我们可以很容易地直接验证dn+1 dn = 0。此时上同调群定义为

Hn(G,M) = Hn(C•(G,M)) = ker dn/Imdn−1.

当G是一个profinite群时，G在离散的阿贝尔群范畴CG上的作用是连续的，其

中CG是G-模范畴的子范畴。所以对任意一个（离散）G-模M，我们可以定义

Hnc (G,M) := lim−→

HG

Hn(G/H,MH),

其中H 跑遍G的所有正规开子群如果我们想要计算上同调群Hnc (G,M)，我们还是可

以采用上面相同的方法去计算，唯一需要修改的是上链必须是连续的（此时的连

续自然由profinite的定义给出），这时在不发生混淆的情况下，我们用Hn(G,M)去表

示Hnc (G,M)。

1.2 corG/U映射与resG/U映射

令U是G的一个闭子群，我们有限制映射

resG/U : Hn(G,M) → Hn(U,M).

如果U是G的一个有限指标开子群，则我们有余限制映射

corG/U : Hn(U,M) → Hn(G,M).

有了以上的定义，我们将给出下列性质：

命题1.1 有下列性质成立：

1. corG/U resG/U (x) = [G : U ]x，其中x ∈ Hn(G,M);

2. 如果G是一个有限群，M是一个有限G-模，且(|G|, |M |) = 1。则对所有的正整数q，

Hq(G,M) = 0。

67

1.3 Inflation和Restriction序列

命题1.2 令U是G的一个正规闭子群，假设对所有的p = 1, 2, · · · , q−1，Hp(U,M) =

0。则我们有如下正合序列:

0 → Hq(G/U,MU ) → Hq(G,M) → H0(G/U,Hq(U,M)) → Hq+1(G/U,MU ).

1.4 Shapiro引理

令U是G的一个闭子群，其诱导模由IndGUM = HomZ[U ](Z[G],M)给出。

命题1.3 则我们有如下同构：

Hn(G, IndGUM) = Hn(U,M)

对所有n ≥ 0。

1.5 Tate上同调

令G是一个有限群，M是一个G-模，则Tate上同调群定义如下：

HnT (G,M) =

Hn(G,M), n ≥ 1

H0(G,M), n = 0

kerNm/IGM, n = −1

Hi−1(G,M), n < −1

其中Nm : m 7→ ∑g∈G

gm，IG ⊂ Z[G]由g − 1所生成，g ∈ G。

命题1.4 假设G是一个有限循环群，且由g所生成，则

HnT (G,M) ∼= Hn+2

T (G,M).

推论1.5 同样地，假设G是一个有限循环群且由g所生成，则对所有的n ≥ 1，我们

有

H2n(G,M) ∼= H0T (G,M) = MG/Nm(M)

以及

H2n−1(G,M) ∼= kerNm/IGM.

命题1.6 如果G是一个循环群，M是一个有限模，则

|H0T (G,M)| = |H1

T (G,M)|.

68

2 局部欧拉特征公式

定理2.1 (局部情形) p是给定的一个有理素数，令K是Qp的一个有限扩张，令G =

Gal(Qp/K)。如果M是一个有限G-模。我们有如下的局部欧拉特征公式:

|H0(G,M)| · |H2(G,M)||H1(G,M)| =

|H0(G,M)| · |H0(G,M∗(1))||H1(G,M)| = ||M ||K ,

其中M∗(1) = Hom(M,K×)，|n|K = [OK : nOK ]−1其中n为正整数。

其中第一个等式来源于Tate duality。下面我们不加证明的给出此性质，如下：

命题2.2 (Tate duality) 令M是一个有限生成的离散Z[G]-模，则对所有的0 ≤ r ≤ 2，

有

Hr(G,M∗(1)) ∼= H2−r(G,M)∗

其中M∗(1) = Hom(M,K×)，N∗ = Hom(N,Q/Z)是阿贝尔群N的Pontryagin对偶模。

特别地，如果M是一个有限模，则所有的上同调群Hr(G,M)都是有限群，并且如

果r ≥ 3，则Hr(G,M) = 0。

2.1 局部情形的证明

为方便起见，我们用Hn(M)去代替Hn(G,M)。因为M =⊕l

M [l∞]，其中l为有

理素数。利用上同调群的加性特征，我们只需考虑M = M [l∞]这种情形即可。

现在我们假设M = M [l∞]则Hq(M)是一个有限长度的Zl-模。对任意的Zl-模N我们

有|N | = llengthZl(N)。这是因为任意一个非零单模它一定同构于Zl/l。这里length(N)是Zl-

模N的Jordan-Holder序列的长度。此时，欧拉特征定义如下：

χ(M) = χ(G,M) =

2∑

q=0

(−1)qlengthZlHq(M),

χ′(M) = χ′(G,M) = logl(||M ||K) =

−[K : Qp]lengthZp

M l = p,

0 l = p.

注意观察定理2.1的左右两边，实际上我们只需证明

χ(M) = χ′(M) =

−[K : Qp]lengthZp

M l = p,

0 l = p.

为了更好地理解这个定理，首先我们来看一种非常特殊的情况：M = Fl =

Z/lZ (G在Fl上是平凡作用)。由Tate Duality知， dimFlH0(G,Fl) = dimFl

FGl = 1,

69

dimFlH2(G,Fl) = dimFl

H0(G,µl)∗ = dimFl

µ∗l (K) = dimFl

µl(K), 其中µl(K) = z ∈K|zl = 1。另一方面，由Kummer理论，我们有dimFl

H1(G,Fl) = dimFlH1(G,µl)

∗ =

dimFlH1(G,µl) = dimFl

K×/(K×)l。换句话，其中最后一个等式原因如下：

1 −−−−→ µl −−−−→ K× x 7→xl

−−−−→ K× −−−−→ 1.

我们有

1 −−−−→ H0(G,µl) −−−−→ H0(G,K×) −−−−→ H0(G,K

×) −−−−→ H1(G,µl) −−−−→ H1(G,K

×)

∥∥∥∥∥∥

∥∥∥∥∥∥

∥∥∥

1 −−−−→ µl(K) −−−−→ K× x 7→xl

−−−−→ K× −−−−→ H1(G,µl) −−−−→ 1,

其中H1(G,K×) = 1由Hilbert 90定理给出。所以H1(G,µl) ∼= K×/(K×)l。因为K× ∼=

O×K × Z，O× ∼= OK × µ ∼= OK × µl∞(K) × ∏

q =l

µq∞(K)，其中OK是K的整数环，我们有：

K×/(K×)l ∼=

Z/lZ ⊕ µl(K) l = p,

Z/pZ ⊕OK/pOK ⊕ µp(K) l = p.

下面分两种情形讨论。

当l = p时，

χ(Fp) = dimFpH0(G,Fp) − dimFpH

1(G,Fp) + dimFpH2(G,Fp)

= 1 − dimFp(Fp ⊕OK/pOK ⊕ µp(K)) + dimFpµp(K)

= −dimFpOK/pOK = −[K : Qp]dimFpFp = χ′(Fp).

当l = p时，

χ(Fl) = 1 − dimFl(Fl ⊕ µl(K)) − dimFl

µl(K) = 0 = χ′(Fl).

由Tate duality知，这个定理同样对M = µl = F∗l (1)也成立，特殊情况验证完毕。下面我

们来考虑一般的情形。

首先，我们证明如果W0 → L → M → N → 0是一个有限Zl[G]-模的正合序列，则

有χ′(M) = χ′(N)+χ′(L)，χ(M) = χ(N)+χ(L)，χ′(M) = χ′(M ss)和χ(M) = χ(M ss)成

立。其中M ss =n⊕

q=1Mq/Mq−1是一个Zl[G]-模，0 = M0 ⊂ M1 ⊂ · · ·Mn = M是一

个Jordan-Holder序列，Mq均为Zl[G]-模。

70

事实上，χ′(M) = −[K : Qp]lengthZpM = −[K : Qp](lengthZp

L + lengthZpN) =

χ′(N) + χ′(L)。由性质1.2我们知道0 → H0(L) → H0(M) → H0(N) → H1(L) →H1(M) → H1(N) → H2(L) → H2(M) → H2(N) → 0是一个正合列。则我们有：

χ(M) =2∑

q=0

(−1)qlengthHq(M)

=

2∑

q=0

(−1)q(lengthHq(L) + lengthHq(N))

=2∑

q=0

(−1)qlengthHq(L) +2∑

q=0

(−1)qlengthHq(N)

= χ(L) + χ(N).

以及

χ′(M ss) = χ′(n⊕

q=1

Mq/Mq−1)

=

n∑

q=1

χ′(Mq/Mq−1)

=

n∑

q=1

(χ′(Mq) − χ′(Mq−1))

= χ′(Mn) = χ′(M).

对于χ(M) = χ(M ss)同理如上。

由于Mq/Mq−1是一个Fl[G]-模，所以M ss也是一个Fl[G]-模。因此不失一般性，我们

可以直接假设M本身是一个Fl[G]-模。此时dimFlM = lengthZl

M。

现在我们回顾下Grothendieck群的基本概念。令G是一个profinite群，E是一个给定

的域。我们考虑由如下数据所构成的范畴RepE(G)：

1. 范畴里的所有元素都是有限维E−线性空间并且在离散拓扑下有连续的G作用；

2. 所有的态射都是E[G]−线性映射。

RepE(G)的Grothendieck群RE(G)是一个阿贝尔群，它由如下生成元与关系所确定：

RE(G)是由形如符号[M ]的元素所生成，其中M ∈ RepE(G)为一个类。这唯一的关

系[M ] = [N ] + [L] 由短正合列0 → L → M → N → 0所确定。

现 在 我 们 回 到 原 证 明 中。 我 们 考 虑 由 所 有 的 有 限Fl[G]-模 所 构 成 的 模

范畴RepFl(G)，它的Grothendieck群是RFl

(G)。事实上，我们可以把χ 和χ′看作是

71

从Grothendieck群RFl(G)到Z的两个函数。我们只需对RFl

(G)的生成元验证即可。由

于Z是自由扭模，所以我们只需验证RFl(G) ⊗Z Q的生成元即可。而RFl

(G) ⊗Z Q的生成元我们可由如下性质得到：

命 题2.3 (参 考[2, lemma2.10])令G是 一 个 有 限 群，对G的 任 意 一 个 子 群H，

令IndGH是RFp(H) ⊗ Q → RFp(G) ⊗ Q 的一个同态，把H-模的类映射到对应的G-模的

类。则RFp(G) ⊗ Q是由IndGH的像所生成，其中H跑遍order与p互素的G的所有循环子群。

我们选取域K的一个有限扩张F，使得Gal(Qp/F )在M上作用平凡。为方便起见，

令G = Gal(F/K)。因此，对定理2.1我们只需要验证RFl(G)的生成元即可。由性质2.3，

我们知道RFl(G) ⊗Z Q是由IndG

Hρ所生成，其中循环子群的Horder与l互素，ρ : H → K×是

一个特征，其中K是Fl的一个有限扩张。因此我们可以假设M = IndGHρ = IndG

Hρ，其

中H = Gal(Qp/FH)。由Shapiro’s引理知Hq(G, IndG

Hρ) ∼= Hq(G, IndG

Hρ)∼= Hq(H, ρ)，因

此χ(G, IndGHρ) = χ(H, ρ)。所以我们只需验证ρ即可。（换句话，只需验证一维线性空

间V (ρ)，其中H是通过ρ在其上作用）

Qp

G′

G

F

G

H

FH

K

因此我们假设FH = K，则H = G，M = V (ρ)在K是一维的以及G = H是一个循环群，

且(|G|, l) = 1。由性质1.1(2)对所有的q ≥ 1，我们有Hq(G,M) = 0 for all q ≥ 1。因此，

我们有如下inflation和restriction 序列：

Hq(G/G′,MG′) −−−−→ Hq(G,M) −−−−→ H0(G/G′, Hq(G′,M)) −−−−→ Hq+1(G/G′,MG′

)∥∥∥

∥∥∥∥∥∥

∥∥∥0 = Hq(G,M) −−−−→ Hq(G,M) −−−−→ H0(G,Hq(G′,M)) −−−−→ Hq+1(G,M) = 0,

其中G′ = Gal(Qp/F )。从而对q = 0, 1, 2，我们可以得到Hq(G,M) ∼= H0(G,Hq(G′M))。

因此

Hq(G′,M) = Hq(G′,K) = Hq(G′,Fl) ⊗FlK =

K q = 0;

((F×/(F×)l)∗ ⊗FlK q = 1;

µ∗l (F ) ⊗Fl

K q = 2.

72

则

χ(G,M) = dimFlKG − dimFl

(((F×)/(F×)l)∗ ⊗FlK)G + dimFl

(µ∗l (F ) ⊗Fl

K)G.

由于我们已经验证了M = Fl和M = µl这两种情形，因此我们可以假设ρ既不是平凡作用

也不是分圆特征作用，从而可以得到KG = (µ∗l (F ) ⊗Fl

K)G = 0。这是因为Galois群在K上的作用是由平凡特征所决定以及ρ在µp上是由分圆特征作用所决定。因此，χ(G,M) =

−dimFl(((F×)/(F×)l)∗ ⊗Fl

K)G.

如同前面考虑特殊情况一样，我们还是分两种情形讨论。

当l = p时，我们只需证明如下等式：

χ(G,M) = −dimFl(((F×)/(F×)l)∗ ⊗Fl

K)G = −[K : Qp]dimFpM = χ′(G,M).

因为µ是F×的极大扭子群，则我们有：

1 −−−−→ µ −−−−→ F× −−−−→ F×/µ −−−−→ 1yp

yp

yp

1 −−−−→ µ −−−−→ F× −−−−→ F×/µ −−−−→ 1,

其中p : x 7→ xp。由蛇形引理我们知道：

1 → µ/µp → (F×)/(F×)p → (F×/µ)/(F×/µ)p → 1.

下面我们如下定义“∗”函子：“∗”=Hom(−,Q/Z) = Hom(−,Fp) (第二个等号仅在这种

情形下成立)，我们知道这是个反变左正合函子，因此

1 → ((F×/µ)/(F×/µ)p)∗ → ((F×)/(F×)p)∗ → (µ/µp)∗ → Ext1((F×/µ)/(F×/µ)p,Fp) = 1.

我们知道K是平坦的，则

1 → ((F×/µ)/(F×/µ)p)∗ ⊗Fp K → ((F×)/(F×)p)∗ ⊗Fp K → (µ/µp)∗ ⊗Fp K → 1.

因此

1 → H0(G, ((F×/µ)/(F×/µ)p)∗⊗FpK) → H0(G, ((F×)/(F×)p)∗⊗FpK) → H0(G, (µ/µp)∗⊗FpK).

而µ/µp ∼= µp(F ), 则

H0(G, (µ/µp)∗ ⊗Fp K) = (µ∗p(F ) ⊗Fp K)G = 0.

因此

H0(G, ((F×/µ)/(F×/µ)p)∗ ⊗Fp K) ∼= H0(G, ((F×)/(F×)p)∗ ⊗Fp K).

73

换句话说，我们有

dimFp(((F×)/(F×)p)∗⊗FpK)G = dimFp(((F

×/µ)/(F×/µ)p)∗⊗FpK)G = dimFp(((F×/µ)⊗ZFp)

∗⊗FpK)G.

我们把F加法赋值用v : F× → Z来表示，则我们有正合列：

1 → O×F /µ → F×/µ → Z → 0.

由于此正合列的项均为自由扭模，张量乘上Fp后，我们依然有正合序列：

1 → O×F /µ⊗Z Fp → F×/µ⊗Z Fp → Fp → 0.

再次对其用“∗”函子作用，我们有：

0 → Hom(Fp,Fp) = Fp → (F×/µ⊗Z Fp)∗ → (O×

F /µ⊗Z Fp)∗ → Ext1(Fp,Fp) = 0.

因此我们有：

0 → KG → ((F×/µ⊗Z Fp)∗ ⊗Fp K)G → ((O×

F /µ⊗Z Fp)∗ ⊗Fp K)G → H1(G,K) = 0.

所以，

dimFp((F×/µ⊗Z Fp)

∗ ⊗Fp K)G = dimFp((O×F /µ⊗Z Fp)

∗ ⊗Fp K)G.

现在我们利用下述性质将表示ρ提升到特征为0的域上去考虑，此时新的表示用ρ来

记。

命题2.4 (参考[1, 推论2.7]) 令域K是Qp的有限扩张，其整数环记为O。对于O的极

大理想mO，令E = O/mO。假设p不整除|G|以及所有G的所有不可约K-表示均为绝对不

可约的，则G的所有不可约E-表示均为绝对不可约的。同时约化映射ρ 7→ (ρ mod mO)诱

导出G的不可约K-表示等价类与G的不可约E-表示等价类之间的双射，且保维数。

所以我们选取Qp的唯一非分歧扩张L，其维数是dimFpK。由p-adic分析的性质，我们

有OL/(p) ∼= K，O×L

∼= (1 + pOL) × K×，其中OL是L的p-adic整数环。通过上述性质里的

同构，我们可以考虑ρ在O×L里取值。我们把这个新特征（或者一维表示）ρ : G → O×

L叫

做ρ的Teichmuller提升。因为O×F /µ是自由扭模以及(|G|, ρ) = 1，通过性质2.4，对ρ的唯一

的Teichmuller提升，我们有：

dimFp((O×F /µ⊗Z Fp)

∗ ⊗Fp K)G = dimFp((O×F /µ⊗Zp Fp)

∗ ⊗Fp K)G

= dimFp(Hom(O×F /µ⊗Zp Fp,Fp) ⊗Fp K)G

= RankZp(Hom(O×F /µ⊗Zp Zp,Zp) ⊗Zp OL)G

= dimQp(Hom(O×F /µ⊗Z Q,Qp) ⊗Qp L)G.

74

由p-adic理论，作为G-模，我们有O×F /µ⊗Z Q ∼= F。因此

dimQp(Hom((O×F /µ) ⊗Z Q,Qp) ⊗Qp L)G = dimQp(Hom(F,Qp) ⊗Qp L)G.

通过正规基定理，F ∼= K[G] ∼= Qp[G][K:Qp]。则

dimQp(Hom(F,Qp) ⊗Qp L)G = dimQp(Hom(Qp[G][K:Qp],Qp) ⊗Qp L)G

= [K : Qp]dimQp(Hom(Qp[G],Qp) ⊗Qp L)G.

作为G-模，我们有Hom(Qp[G],Qp) ∼= Qp[G]，通过ψ : f 7→ ∑σ∈G

aσσ，给出。其中aσ =

f(σ)，则

[K : Qp]dimQp(Hom(Qp[G],Qp) ⊗Qp L)G

= [K : Qp]dimQp(Qp[G] ⊗Qp L)G

= [K : Qp]dimQp(Qp[G] ⊗Qp L0)G · · · · · · (∗)

= [K : Qp]dimQp(Qp[G]G ⊗Qp L0)

= [K : Qp]dimQp(Qp ⊗Qp L0)

= [K : Qp]dimQpL0

= [K : Qp]dimQpL

= [K : Qp]dimFpK= [K : Qp]dimFpM,

其中Qp[G]G ∼= Qp以及(∗)来自如下同构：

Qp[G] ⊗Qp L∼= Qp[G] ⊗Qp L

0

由

σ ⊗m 7→ σ ⊗ σ−1m,

给出，其中L0是平凡G-模且作为Qp−线性空间有L ∼= L0。

当l = p时，我们只需验证χ(G,M) = −dim((F×/(F×)l)∗ ⊗FlK)G = 0。同理如上进

行讨论后，我们知道((F×/(F×)l)∗ ⊗FlK)G ∼= ((F×/µ⊗Z Fl)

∗ ⊗FlK)G。但F×/µ是Zp-模，

而l在Zp中不可逆。因此F×/µ⊗Z Fl = 0即χ(G,M) = 0。

注释2.5 事实上，dimFpMG = dimFp(M

∗)G，因为G是一个有限循环群，令σ是G的

75

生成元，则我们有

dimFp(M∗)G = dimFp(HomFp(M,Fp))

G

= dimFpHomFp[G](M,Fp)

= dimFpHomFp[G](M/(σ − 1)M,Fp)

= dimFpHomFp(M/(σ − 1)M,Fp)

= dimFpM/(σ − 1)M = dimFpMG

其中M是Fp[G]-模。因此，通过这个结论有：

dimFp((F×/(F×)p)∗ ⊗Fp K)G = dimFp(((F

×/(F×)p)∗ ⊗Fp K)∗)G

= dimFp((F×/(F×)p) ⊗Fp K∗)G.

所以对于证明定理2.1，我们只需验证

dimFp((F×/(F×)p) ⊗Fp K∗)G = [K : Qp]dimFpM.

Haruzo Hida在其证明中，一开始就把“*”函子漏掉（他说是typo错误）去计算。虽说

“*”函子最后是不起作用的，但这是完全不可取的，因为没有任何理由能够说明“*”函

子可以漏掉，且证明过程中“*”函子确实带来了相当多的障碍。鉴于此，在印老师的指

导以及李彬博士的帮助下，我对其证明做了大量的修正性工作。我们从上面的证明过程中

可看出，这些修正性的工作并不是非常容易。

3 整体欧拉特征公式

定理3.1 (整体情形) 给定一个有限模M，令K是Q的一个有限扩张，S是Q的有限素点集且包含阿基米德素点以及|M |的所有素因子。Σ是在S之上K的素点集，Ks 是K 的最

大代数扩张且在Σ之外都是非分歧的，我们令SS = Gal(Ks/K)。若M是一个有限SS-模，

则有：|H2(SS ,M)| · |H0(SS ,M)|

|H1(SS ,M)| =∏

v∈Σ∞

|H0(Gv,M)|||M ||Kv

,

其中|n|Kv = n[Kv:R]，v是阿基米德素点，Gv = Gal(Kv/Kv)。

在这个定理里，我们知道M是SS-模，由于Gv是SS的分解群，我们也可以把M看

作Gv-模。在证明这个定理前，我们将介绍一个非常重要的定理，这个定理由John

Tate和Georges Poitou证明，它是解决整个欧拉特征公式的关键。但我们在这里只对她进

行叙述而不加证明，其证明细节在参考文献[1]，[2]，[4]，[8]中均能找到。

76

命题3.2 令K是Q的一个有限扩张，S是Q的有限素点集且包含阿基米德素点，Σ是

在S之上K的素点集。固定S中的一个素点p，若M是一个有限SS-模，且其order为p的幂。

则有以下性质：

1.

Hr(SS ,M) ∼=∏

v∈Σ(R)

Hr(Gv,M) 对所有 r ≥ 3,

其中Σ(R)是K的阿基米德实素点集；

2. 我们有如下长正合列：

0 → H0(SS ,M) →∏

v∈Σ0

H0(Gv,M) ×∏

v∈Σ∞

H0T (Gv,M) → H2(SS ,M

∗(1))∗

→ H1(SS ,M) →∏

v∈Σ

H1(Gv,M) → H1(SS ,M∗(1))∗

→ H2(SS ,M) →∏

v∈Σ

H2(Gv,M) → H0(SS ,M∗(1))∗ → 0.

3.1 整体情形的证明

因为M =⊕

lM [l∞]，同样我们可以假设M = M [l∞]。现在我们对l > 2进行证明。令

φ(M) = χ(SS ,M) −∑

v∈Σ∞

(lengthZlH0(Gv,M) − [Kv : R]lengthZl

M)

则我们只需证明φ(M) = 0。

由性质3.2(1)，对所有的q ≥ 3，我们有Hq(SS ,M) =∏

v∈Σ(R)

Hq(Gv,M)。而|Gv| =

2以及l > 3，我们有gcd(|Gv|,M) = 1。通过性质1.1(2)，对所有的q ≥ 1，我们

有Hq(Gv,M) = 0，因此对所有的q ≥ 3，有Hq(SS ,M) = 0。

当Zl[SS ]-模序列0 → L → M → N → 0是短正合列时，我们有下列有限Zl-模长正合

列：

0 → H0(SS , L) → H0(SS ,M) → H0(SS , N) → · · · → H2(SS , N) → 0

所以如同局部欧拉特征，我们同样有加性等式χ(M) = χ(L) + χ(N)和φ(M) = φ(L) +

φ(N)。其中χ和φ同样可看作是从Grothendieck群RFl(G)到Z的两个函数。

77

然后由性质3.2(2)，我们有χ(M) + χ(M∗(1)) =∑v∈Σ

χv(Gv,M)。其中

χv(Gv,M) =

2∑q=0

(−1)qlengthZlHq(Gv,M), v 是有限素点;

2∑q=0

(−1)qlengthZlHq

T (Gv,M), v 是阿基米德素点.

则 ∑

v∈Σ

χv(Gv,M) =∑

v∈Σ∞

χv(Gv,M) +∑

v∈Σ0

χv(Gv,M),

其中Σ0是有限素点集。我们知道

∑

v∈Σ∞

χv(Gv,M) =∑

v∈Σ∞

2∑

q=0

(−1)qlengthZlHq

T (Gv,M).

因为Gv是一个循环群，由性质1.6知

lengthZlH2

T (Gv,M) = lengthZlH1

T (Gv,M)

lengthZlH0

T (Gv,M) = lengthZlH1

T (Gv,M) = lengthZlH1(Gv,M).

所以 ∑

v∈Σ∞

χv(Gv,M) =∑

v∈Σ∞

lengthZlH1(Gv,M).

另一方面，因为对v ∈ Σ，我们有||M ||Kv = 1。通过乘积公式∏v

||M ||Kv = 1以及前面

证明的局部欧拉特征公式，我们有：

∑

v∈Σ0

χv(Gv,M) = logl

∏

v∈Σ0

(||M ||Kv) = − logl

∏

v∈Σ∞

(||M ||Kv) = −∑

v∈Σ∞

[Kv : R]lengthZlM.

所以，

φ(M) + φ(M∗(1)) = χ(M) + χ(M∗(1))

−∑

v∈Σ∞

(lengthZlH0(Gv,M) + lengthZl

H0(Gv,M∗(1)) − 2[Kv : R]lengthZl

M)

=∑

v∈Σ∞

(lengthZlH1(Gv,M) − lengthZl

H0(Gv,M) − lengthZlH0(Gv,M

∗(1)) + [Kv : R]lengthZlM)

= 0.

最后一个等式源于下面性质：

78

命题3.3 (参考[2, 定理2.3(c)]) 令Kv = R或C，Gv = Gal(Kv/Kv)，|n|Kv = n[Kv :R]。

对任意有限Gv-模M，我们有：

|H0(Gv,M))| · |H0(Gv,M∗(1))|

|H1(Gv,M)| = ||M ||Kv .

因此我们只需证明φ(M) = φ(M∗(1))即可。

3.2 对φ(M) = φ(M∗(1))修改证明

由于J.S.Milne的证明有误，下面的证明是在印林生教授的指导与李彬博士的帮助下

独立用自己的方法对其进行详细的证明。如同局部欧拉特征公式，我们选取K的一个

有限扩张F，使得S′S = Gal(Ks/F )在M和µl上作用是平凡的。为方便起见，我们令G =

Gal(F/K)。同样用讨论局部的方法，我们可以假设G是一个循环群，它的degree与l互素；

且M是一个Fl[G]-模。则对所有的q ≥ 3，我们有Hq(S′S ,M) = 0。因为gcd(|G|, l) = 1，

则对所有的q ≥ 0，Hq(G,M) = 0。同理，再次运用inflation和restriction序列，我们

有Hq(SS ,M) ∼= H0(G,Hq(S′S ,M))。

由于我们把φ看作是从Grothendieck群RFl(G)到Z的函数（或同态）。令

χ′ : RFl(G) ⊗ Q → RFl

(G) ⊗ Q,

[M ] 7→2∑

i=0

(−1)i[H i(S′S ,M)]

及

θ : RFl(G) ⊗ Q → Q,

[M ] 7→ dimFlMG,

因此我们有

χ = θ χ′.

我们知道[H0(S′S , µl)] = [µl]，以及

(i) [H1(S′S , µl)] = [O×

F,S/l] + [ClS(F )[l]]，

(ii) [H2(S′S , µl)] = [ClS(F )/l] − [Fl] + [

⊕p∈S\S∞(F )

Fl] + [⊕

p∈S∞(F )

H0T (Gp,Fl)]，

其中O×F,S是S−单位群，ClS(F )[l]是S−理想类群ClS(F )的l−扭部分。其证明细节可以参

考[8,(8.7.4)]。

79

注意[O×F,S/l] = [

⊕p∈S(F )

Fl] + [µl] − [Fl]及[ClS(F )[l]] = [ClS(F )/l]，则我们有

χ′([µl]) = [⊕

p∈S∞(F )

H0T (Gp,Fl)] − [

⊕

p∈S∞(F )

Fl].

对于有限Fl[G]-模M，我们有：

(iii) χ′(M∗(1)) = [M∗] · χ′([µl]), 参考[2, 引理5.4],

(iv) [M ] · [Fl[G]] = dimFlM · [Fl[G]]

因此，

χ′([M∗(1)]) = [M∗ ⊗ (⊕

p∈S∞(F )

(H0T (Gp,Fl) − Fl))]

= [M∗ ⊗ (⊕

v∈Σ∞

⊕

p|v(H0

T (Gp,Fl) − Fl))].

同理，我们有

χ′([M ]) = [M(−1) ⊗ (⊕

p∈S∞(F )

(H0T (Gp,Fl) − Fl))]

= [M(−1) ⊗ (⊕

v∈Σ∞

⊕

p|v(H0

T (Gp,Fl) − Fl))].

其中M(−1) = (M∗(1))∗。因此，我们要证明φ(M) = φ(M∗(1))其实就是要证明：

−θ([M∗ ⊗ (⊕

v∈Σ∞(R)

⊕

p|v(H0

T (Gp,Fl) − Fl))]) −∑

v∈Σ∞

(dimFlH0(Gv,M

∗(1)) − [Kv : R]dimFlM∗(1))

= −θ([M(−1) ⊗ (⊕

v∈Σ∞(R)

⊕

p|v(H0

T (Gp,Fl) − Fl))]) −∑

v∈Σ∞

(dimFlH0(Gv,M) − [Kv : R]dimFl

M),

经整理后，只需证明：

∑v∈Σ∞(R)

(θ([M∗ ⊗ (⊕

p|v(H0T (Gp,Fl) − Fl))]) + dimFl

M∗(1)Gv)

=∑

v∈Σ∞(R)

(θ([M(−1) ⊗ (⊕

p|v(H0T (Gp,Fl) − Fl))]) + dimFl

MGv).

下面我们将上述素点v进行分类讨论。

当v = C，p = C时，有G-模同构：

⊕

p|vFl

∼= Fl[G]

80

以及

H0T (Gp,Fl) = 0.

这时我们可以很容易验证上述左右两边相等，与证明局部情形时完全相同；

当v = R，p = R时，有G-模同构：

⊕

p|vFl

∼= Fl[G]

以及 ⊕

p|vH0

T (Gp,Fl) ∼= H0T (Gp,Fl) ⊗Fl

Fl[G].

此刻处理方法同上。当v = R，p = C时，我们有

H0T (Gp,Fl) = 0,

这时只需独立的证明如下等式：

∑v∈Σ(R)

(θ([M∗ ⊗ (⊕

p|v,p is complex Fl)]) + dimFlM∗(1)Gv)

=∑

v∈Σ(R)

(θ([M(−1) ⊗ (⊕

p|v,p is complex Fl)]) + dimFlMGv).

下面我们将对剩下的每个v进行独立地证明。

事实上，

θ([M∗ ⊗ (⊕

p|v,p is complex

Fl)]) = dimFl(M∗ ⊗ (

⊕

p|v,p is complex

Fl))G.

显然Gv是G的一个子群，定义G′= G/Gv，则我们有

θ([M∗ ⊗ (⊕

p|v,p is complex

Fl)]) = dimFl((M∗ ⊗ (

⊕

p|v,p is complex

Fl))Gv)G

′

= dimFl((M∗)Gv ⊗ (

⊕

p|v,p is complex

Fl))G

′

= dimFl((M∗)Gv ⊗ Fl[G

′])G

′

= dimFl(M∗)Gv

= dimFlMGv .

其中最后两个等式分别由(iv)和注释2.5给出。

我们用同样的方法再次讨论有

θ([M(−1) ⊗ (⊕

p|v,p is complex

Fl)]) = dimFlM∗(1)Gv .

81

所以，我们完全证明了l > 2的情形。

当l=2时，我们有M∗(1) = M∗因此我们只需要对φ的加性证明稍微做点修改。这个修

改证明是非常容易的，具体可参考文献[1,Ch4.4.5]。

至此，我们完全证明了整体欧拉特征公式。

3.3 对J.S.Milne错误证明的说明及其反例

在 参 考 文 献[1,Ch4.4.5]与[2,Ch1.5]中， 他 们 直 接 去 证 明χ(M) =

χ(M∗(1))和dimFlMGv = dimFl

(M∗(1))Gv这两个等式。综合上面修正性证明与整体

欧拉特征公式结论来看，是根本不可能得到这两个等式的。换句话说这两个等式是完全不

等的。事实上，我们无法说清楚等式χ(M) = χ(M∗(1))是来源于χ′，同样我们也没办法说

明dimFlMGv = dimFl

(M∗(1))Gv。对后一个等式，我们有如下反例：

令K = Q(√

3)，L = K(√

−1) = Q(√

3,√

−1) ⊃ µ3；其中L是K的一个循环扩张，且

它的degree为2与3互素。易知域K里面在3之上的素数在域L里是非分歧的（利用分歧性质

的判别式法则）。所以令M = F3，则M∗(1) = µ3。但我们可以发现dimFl

MGv = 1 = 0 =

dimFl(M∗(1))Gv。

参考文献

[1] Haruzo Hida, Mordular forms and Galois Cohomology, Cambridge Studies in Ad-

vanced Mathematics 69, Cambridge University Press, 2000.

[2] J.S. Milne, Arithmetic Duality Theorems(2nd Edition), BookSurge, LLC, 2006.

[3] Jean-Pierre Serre, Linear Representations of Finite Groups, GTM 42, Springer-

Verlag, 1977.

[4] Jean-Pierre Serre, Galois Cohomology, Srpinger-Verlag, 1991.

[5] Jean-Pierre Serre, local fields, Srpinger-Verlag, 1979.

[6] Andre Weil, Basic Number Theory, Springer-Verlag, 1973.

[7] John Tate(noted by Helena Verrill and William Stein), Galois Cohomology,

http://wstein.org/Tables/Notes/tate-pcmi.html, 1999.

[8] J. Neukirch, A. Schmidt, K, Wingberg, Cohomology on Number Fields, Grundlehren

der mathematischen Wissenschaften 323, Springer, 2008.

82

Fully Nonlinear Equation

方汉隆∗

指导教师：Ben Andrews 教授

Abstract

We give an introduction to fully nonlinear elliptic equations. We focus on theproofs of Krylov-Safonov Estimate, Krylov-Evans Theorem and Caffarelli’s C2,α andW 2,p regularity results. We may assume the readers have knowledge of linear ellip-tic equations’ foundation. Also, we leave the Alexandroff-Bakelman-Pucci MaximumPrinciple’s proof to readers. We mainly based on Caffarelli and Cabre’s book. And werefer readers to Han and Lin’s book and Evans and Gariepy’s book for basic details.

Keywords: Fully nonlinear equation; viscosity solution; A-B-P maximal princi-ple; Harnack Inequality; C2,α; perturbation

1 Introduction

Compared with linear and quasilinear elliptic equations, the fully nonlinear elliptic equa-tions have not been fully developed. We are still unable to apply this kind of equations toGeometric Problems as successfully as linear and quasilinear. However, the fully nonlin-ear elliptic equations’ development maybe is in the centre of elliptic PDEs. Real Monge-Ampere Equations is the core of Optimal Transport. The C2,α Estimate of ComplexMonge-Ampere Equation is the core in Calabi-Yau’s Existence. But we still have not gotthe C2,α Estimate in non concave situation. We have counterexamples to that. We stillhope to find some proper assumption to enlarge the assumption of concave. One of thereasons for this is we have good regularity theory for linear and quasilinear equations, weof course want to get that for fully nonlinear. Second is if we want to do fully nonlinearequations on manifold, we first need a good theory on domain. In this article we haveno attempts to give any progress on this direction. We just follow Caffarelli and Cabre’sbook to show the mature results until 80s’about concave situation.

2 Viscosity Solutions of Elliptic Equations

We consider equations of the form

F (D2u(x), x) = f(x) (1)

∗基数83

83

where x ∈ Ω and u and f are functions defined in a bounded domain Ω of Rn. F (M,x)is a real valued function defined on S × Ω, where S is the space of real n × n symmetricmatrices. We assume that F is uniformly elliptic operator, i.e.,

Definition 2.1. F is a uniformly elliptic if there are two positive constants λ ≤ Λ (calledellipticity constants) such that for any M ∈ S and x ∈ Ω

λ∥N∥ ≤ F (M +N,x) − F (M,x) ≤ Λ∥N∥ ∀N ≥ 0

where ∥M∥ = sup|x|=1|Mx|. We recall that M ∈ S can be decomposed as M = M+ −M−

where M+,M− ≥ 0 and M+M− = 0. So we have the following

F (M +N, x) ≤ F (M,x) + Λ∥N+∥ − λ∥N−∥ ∀M,N ∈ S ∀x ∈ Ω

Definition 2.2. A continuous function u in Ω is a viscosity subsolution(resp. viscositysupersolution) of (1) inΩ, when the following condition holds if x0 ∈ Ω, ϕ ∈ C2(Ω) andu− ϕ has a local maximum at x0 then

F (D2ϕ(x0), x0) ≥ f(x0)

(resp. if u − ϕ has a local minimum at x0 then F (D2ϕ(x0), x0) ≤ f(x0)) we say thatF (D2u, x) ≥ (resp. ≤,=)f(x) in the viscosity sense in Ω whenever u is a subsolution(resp.supersolution, solution) of (1) in Ω

Definition 2.3. Let 0 < λ ≤ λ. For M ∈ S, we define Pucci’s extremal operators:

M−(M,λ,Λ) = M−(M) = λ∑

ei>0

ei + Λ∑

ei<0

ei

M+(M,λ,Λ) = M+(M) = Λ∑

ei>0

ei + λ∑

ei<0

ei

where ei = ei(M) are the eigenvalues of M

Definition 2.4. Let f be a continuous function in Ω and λ ≤ Λ two positive constants. Wedenote by S(λ,Λ, f) the space of continuous functions u in Ω such that M+(D2u, λ,Λ) ≥f(x) in the viscosity sense in Ωwe also define

S(λ,Λ, f) = S(λ,Λ, f) ∩ S(λ,Λ, f)

S∗(λ,Λ, f) = S(λ,Λ,−|f |) ∩ S(λ,Λ, |f |)

3 Alexandroff Estimate and Maximum Principle

We start defining the concept of convex envelope of a continuous function. A function Ldefined in Rn is said to be affine if

L(x) = l0 + l(x),

where l0 ∈ R and L is a linear function.

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Theorem 3.1. Let v be a continuous function in an open convex set A. The envelope ofv in A is defined by

Γ(v)(x) = supL

L(x) : L ≤ v in A, L is affine

for x ∈ A

We have that Γ(v) is a convex function in A. The set v = Γ(v) = x ∈ A : v(x) =Γ(v)(x) is called the (lower) contact set of v. The points in the contact set are calledcontact points.

Theorem 3.2. Let Ω be a bounded domain of Rn and f be a continuous and boundedfunction in Ω. Assume that u∈ S(λ,Λ, f) in Ω, u is continuous in Ω and u≥ 0 on ∂Ω.Then

supΩu ≤ C diam(Ω) ∥f+∥Ln(Ω∩u=Γu)

Here C is a universal constant, diam(Ω) is the diameter of Ω and Γu is the convex envelopein B2d of −u−, where Bd is a ball of radius d = diam(Ω) such that Ω ⊂ Bd and we haveextended u ≡ 0 outside Ω

We do not give the details of the proof, since it depends on the deep fact that Lipschitzfunctions is almost everywhere differentiable and convex function is almost everywheretwice punctional differentiable. We just need the results. If anyone have interests in thesetwo theorem, we strongly recommend them to [EG]. The following result is the maximumprinciple for viscosity solutions

Corollary 3.3. Assume that u∈ C(Ω). Then

(1)u ∈ S(λ,Λ, 0)and u ≤ 0 on ∂Ω imply u ≤ 0 in Ω.

(2)u ∈ S(λ,Λ, 0)and u ≥ 0 on ∂Ω imply u ≥ 0 in Ω.

4 Krylov-Safonov’s Harnack Inequality

Early in 60’s, De Giorgi, Nash and Moser gave the Harnack Inequality for divergenceelliptic equations. Until 80’s, Krylov and Safonov gave the Harnack Inequality for non-divergence elliptic equations.

If the readers are familiar with De Giorgi-Nash-Moser Iteration, they can find thedivergence elliptic equations can take test functions and use Energy Method to get thefinal result. The key point is that the derivatives can be controlled by solution itself andby Sobolev and Poincare Inequality we can get solution being controlled by derivatives.Then we can bootstraps and iterations.

When we focus on non divergence situation, we lost the important Energy Method.But Krylov and Safonov geniously use the Alexandroff(who is Perelman’s advisor) Maxi-mum Principal to get the estimate.

We have confidence that one day we will do further just as Krylov and Safonov didwhen people had divergence form but no non-divergence form.

We try to give a completed proof of this theorem.

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4.1 Two Tools

Sometimes the solution may not be directly use Alexandroff Maximum Principle. We needto adjust it to the right form. Then what we get is the set of solution which get big valuescan not be too large in the measure sense. I call it ”Pull down test function”.

Definition 4.1.

Ql(x0) =n∏

i=1

(xi0 − l

2, xi

0 +l

2)

and Ql = Ql(0)

Lemma 4.2. Given 0 < λ ≤ Λ constants, there exists a smooth function φ in Rn anduniversal positive constants C and M > 1 such that (recall Q1 ⊂ Q3 ⊂ B2

√n)

φ ≥ 0 in Rn\B2√

n (2)

φ ≤ −2 in Q3 (3)

M−(D2ϕ, λ,Λ) ≤ Cξ in Rn (4)

where 0 ≤ ξ ≤ 1 is a continuous function in Rn with suppξ ⊂ ΩMoreover, φ ≥ −M in Rn

Proof. Consider the universal constant α =max1, (n− 1)Λ/λ− 1. We have that B1/4 ⊂B1/2 ⊂ Q1 ⊂ Q3 ⊂ B3

√n/2 ⊂ B2

√n. Define

φ(x) = M1 −M2|x|−α in Rn\B1/4;

we choose M1 and M2 positive such that

φ|∂B2√

n≡ 0 and φ|∂B3

√n/2

≡ −2

(2) therefore holds; it is possible to extend φ smoothly to all Rn such that (3)holds. Thisextension depends only on n, λ,Λ We check(4). We first get

M(D2ϕ)(x) ≤ C(n, λ,Λ) for |x| ≤ 1/4

Because of the rotational symmetry of φ, for |x| ≥ 1/4, we only need to consider atthe point (r, 0, · · · , 0),

∂ijϕ = 0 if i = j,

∂11 = −M2α(1 + α)r−α−2,

∂iiφ = M2αr−α−2 for i > 1

so we have

M+(D2ϕ)(x) = M2[λ(n− 1)α|x|−α−2 − λα(1 + α)|x|−α−2]

= M2[α|x|−α−2(Λ(n − 1)) − λ(1 + α))] ≤ 0

We take 0 ≤ ξ ≤ 1 smooth such that ξ ≡ 1 in B1/4, ξ ≡ 0 outside B1/2 so that (4) holds

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Now we turn to a corollary of the Calderon-Zygmund cube decomposition. The keypoint of Calderon-Zygmund lemma is ”Critical Concentration”, which means the massfirst time concentrates, so it does not concentrate too much.

Definition 4.3. Let Q1 be the unit cube. We split it into 2ncubes of half side. We dothe same splitting with each one of these 2n cubes and we iterate this process. The cubesobtained in this way are called dyadic cubes.

If Q is a dyadic cube different from Q1, we say that Q is the predecessor of Q, if Qis one of the 2n cubes obtained from dividing Q

Lemma 4.4. Let A ⊂ B ⊂ Q1 be measurable sets and 0 < δ < 1 such that(a) |A| ≤ δ, and(b) If Q is a dyadic cube such that |A ∩Q| > δ|Q|, then Q ⊂ BThen |A| ≤ δ|B|

Proof. We have|Q1 ∩A|

|Q1|= |A| ≤ δ

We divide Q1 into 2n dyadic cubes. If Q is one of these 2n subcubes of Q1 and satisfies|Q ∩ A|/|Q| ≤ δ, we then split Q into 2n dyadic cubes. We iterate this process. In thisway we pick a family Q1, Q2, . . . of dyadic cubes(different from Q1) such that

|Qi ∩A||Qi| > δ, ∀i.

Use the fact that any point in a measurable set has density 1. We conclude A ⊂ ∪Qi,except for a set of measure zero.Also we have

|Qi ∩A||Qi|

≤ δ ∀i.

Since |Qi∩A||Qi| > δ and (b) holds. We have that Qi ⊂ B. Hence,

A ⊂∪

i≥1

Qi ⊂ B

Then|A| ≤ δ|B|

4.2 Harnack Inequality and Normalization

We introduce the Harnack Inequality for viscosity solutions and state the normalizationof the argument. In the following proofs we often scale the solution for simplification. Inorder to conserve the elliptic constants we need some standard process.

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Theorem 4.5. Let u ∈ S∗(λ,Λ, f) in Q1 satisfy u ≥ 0 in Q1, where f is continuous andbounded in Q1. Then

supQ1/2

u ≤ C( infQ1/2

u + ∥f∥Ln(Q1)),

where C is a universal constant.

Theorem 4.6. If u ∈ S(λ,Λ, f), then consider

x = x0 +1

Ky, y ∈ Q1, x ∈ Q = Q1/K(x0)

and the function

u(y) =u(x)

E

then u(y) ∈ S(λ,Λ, f(x)/(K2E))

Now we use this process to normalize the Harnack Inequality to a constant one

Lemma 4.7. Let u ∈ S∗(λ,Λ, f) in Q4

√n, u ∈ C(Q4

√n) satisfy u ≥ 0 in Q(4

√n), where

f is continuous and bounded in Q4√

n. Assume that infQ1/4≤ 1 and ∥f∥Ln(Q4

√n) ≤ ϵ0.

Then supQ1/4≤ C, where ϵ0 and C are universal constants.

We show that the Harnack inequality follows from the constant one: Applying (4.6)we take K = 1 and E = infQ1/4

+ δ + ∥f∥Ln/ϵ0 and then let δ −→ 0. Finally we use astandard covering procedure.

4.3 Main steps of proof

We first get a density decay result for subsolution. Then for solution we have two sidecontrols of decay and increase. From the density result to uniform bound we need abootstrap blowing up.

Lemma 4.8. There exist universal constants ϵ0 > 0, 0 < µ < 1 and M > 1, such that ifu ∈ S(|f |) in Q4

√n, u ∈ C(Q4

√n) and f satisfy

u > 0 in Q4√

n, (5)

infQ3

≤ 1 and (6)

∥f∥Ln(Q4√

n) ≤ ϵ0 (7)

then|u ≤ M ∩Q1| > µ. (8)

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Proof. Take φ as in (4.2) and define ω = u+ φ. Recall(2)(3) and (4) we have

ω ∈ S(|f | + Cξ) in B2√

n

We see easily we can apply the Alexandroff Maximum Principle to ω in B2√

n and get

1 ≤ C

(ˆ

ω=Γω∩B2√

n

(|f | + Cξ)n

)1/n

(9)

≤ C∥f∥Ln(Q4√

n) + C|ω = Γω ∩Q1|1/n (10)

Taking ϵ0 small enough, (10) and (7) imply

1

2≤ C|ω = Γω ∩Q1|1/n

≤ C|u ≤ M ∩Q1|1/n

Lemma 4.9. Let u be as in (4.8). Then

|u > Mk ∩Q1| ≤ (1 − µ)k

for k = 1, 2, 3, . . . , where M and µ are as in (4.8)As a consequence, we have that

|u ≥ t ∩Q1| ≤ dt−ϵ ∀t > 0,

where d and ϵ are positive universal constants

Proof. For k=1 (4.9) just (4.8). Suppose now that (4.9) holds for k-1, and let

A = u > Mk ∩Q1, B = u > Mk−1 ∩Q1.

(4.9) will be proved if we show that

|A| ≤ (1 − µ)|B|.

We want to apply (4.4). If we have show that a dyadic cube Q satisfy

|A ∩Q| > (1 − µ)|Q|

then Q ⊂ B. We can use the (4.4) to get

|A| ≤ (1 − µ)|B|.

If there is a dyadic cube Q satisfy

|A ∩Q| > (1 − µ)|Q|

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but Q ⊂ B.We denote Q = Q1/2i(x0) and x ∈ Q such that u(x) ≤ Mk−1. Now we use(4.6) take

K = 12i and E = Mk−1 then by (4.8) we have

µ < |u(y) ≤ M ∩Q1| = 2m|u(x) ≤ MK ∩Q|.

Hence |Q\A| > µ|Q|The following is easy.

If we turn to supersolution, we see that the set where supersolution takes small valuewill have a lower bound of measure. Hence if we combine we may get the bootstrapblowing up argument.

Lemma 4.10. Let u ∈ S(−|f |) in Q4√

n. Assume that f satisfies(7) and u satisfies(4.9)Then there exist universal constants M0 > 1, and σ > 0 such that, for ϵ as in (4.9)

and ν = M0/(M0 − 12) > 1, the following holds: if j > 1 is an integer and x0 satisfies

|x0|∞≤1/4 (11)

andu(x0) ≥ νj−1M0, (12)

thenQj := Qlj (x0) ⊂ Q1 and sup

Qj

≥ νjM0,

where lj = σM−ϵ/n0 ν−ϵj/n.

Proof. Take σ > 0 and M0 > 1 such that

1

2σn > d2ϵ(4

√n)n

and

σM−ϵ/n0 + dM−ϵ

0 ≤ 1

2,

with d and ϵ as in (4.9). Since lj = σM−ϵ/n0 ν−ϵj/n so lj ≤ 1/2 combined with (11) we have

Qlj/(4√

n)(x0) ⊂ Qlj (x0) = Qj ⊂ Q1. (13)

We suppose that supQju < νjM0 then we use the(4.9)and(13)we have the density decay

that

|u ≥ νjM0

2 ∩Qlj/(4

√n)(x0)| ≤ |u ≥ νjM0

2 ∩Q1| ≤ dν−jϵ

(M0

2

)−ϵ

. (14)

Now we turn the solution ”up to down” to get the density increase. We use a similar formof (4.6) that

x = x0 +lj

4√ny. y ∈ Q4

√n. x ∈ Qj = Qlj (x0)

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and the function

v(y) =νM0 − u

νj−1 (x0)

(ν − 1)M0.

then we figure out the definition fields’ transformation

x ∈ Qlj (x0)[resp. Q3lj/(4

√n)(x0),Qlj/(4

√n)(x0)

]⇔

⇔ y ∈ Q4√

n

[resp. Q3,Q1

]

We can check that v(y) ∈ S[(

lj4√

n)2(ν(j−1)(ν − 1)M0)

−1f(x) =: f(y)]

∩ C(Q4√

n). Also

v > 0 in Q4√n and infQ3v ≤ 1 and

∥f(y)∥Ln(Q4√

n) =σM

−ϵ/n0 ν−ϵ/n

4√nνj−1(ν − 1)M0

∥f(x)∥Ln(Qlj(x0)) ≤ ϵ0

then we use (4.8)and(4.9) get

|v(y) > M0 ∩Q1| < dM−ϵ0 .

so we have

|u(x) < νjM0

2 ∩Qlj/(4

√n)(x0)| ≤

( lj4√n

)ndM−ϵ

0 .

this inequality and the decay one gives

( lj4√n

)n ≤( lj4√n

)ndM−ϵ

0 + dν−jϵ(M0

2

)−ϵ

after simplify the inequality we get contradiction with 12σ

n > d2ϵ(4√n)n

Now we use the bootstrap blowing up argument to get the final result.

Proof of (4.7). Recall that lj = σM−ϵ00 ν−ϵj/n, j = 1, 2, 3, . . . Therefore, there exists a

universal integer j0 ≥ 1 such that ∑

j≥j0

lj ≤ 1/4

We claim that supQju ≤ νj0−1M0, then we get the ”constant” form of Harnack Inequality.Suppose the claim is not true. Then there exists xj0 such that

|xj0 |∞ ≤ 1/8 and u(xj0) ≥ νj0−1M0.

We can use the (4.10)to get the existence of a point xj0+1 such that |xj0+1 −xj0 |∞ ≤ lj0/2and u(xj0+1) ≥ νj0M0

We repeat this process and getting a sequence of points xjj≥j0 such that

|xj+1 − xj | ≤ lj/2 and u(xj+1) ≥ νjM0 ∀j ≥ j0,

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since

|xj |∞ ≤ |xj0 |∞ +

j−1∑

k=j0

|xk+1 − xk|∞

≤ 1

8+

∑

k≥j0

lk2

≤ 1

4,

Hence we can always apply the (4.10), which contradicts with the fact that u is continuousso is bounded in Q1/2

Now we state a corollary of the above result. We only assume u to be subsolution.

Corollary 4.11. Let u ∈ S(λ,Λ, f) in Q1, where f is continuous and bounded in Q1.Then, for any p > 0,

supQ1/2

u ≤ C(p)(∥u+∥LP (Q1/4) + ∥f∥Ln(Q1)),

where C(p) is a constant depending only on n, λ,Λ and p

Proof. we first assume that u ∈ S(f) ⊂ S(−|f |) in Q4√n, ∥f∥Ln(Q4

√n) ≤ ϵ0, u

+ ∈Lϵ(Q1) ≤ d1/ϵ, where ϵ is as in (4.8) and d and ϵ are as in (4.9). We then have

|u ≥ t ∩Q1| ≤ t−ϵ

ˆ

Q1

(u+)ϵ ≤ dt−ϵ ∀t > 0

and hence u satisfies (4.9). It follows that (4.10)holds and then the ”constant” Harnackinequality holds. We have

supQ1/4

≤ C.

Rescaling u, we get

supQ1/4

u ≤ C(∥u+∥L−ϵ(Q1) + ∥f∥Ln(Q4

√n)

)

We get the result when p = ϵ, by interpolation we can get result for any p.

Also we state a corollary for the supersolution.

Corollary 4.12. Let u ∈ S(λ,Λ, f) in Q1, satisfy u ≥ 0 in Q1, where f is continuousand bounded in Q1. Then

∥u∥Lp0 ≤ C( infQ1/2

+∥f∥Ln(Q1))

where p0 > 0 and C are universal constants.

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Proof. we use the formula

ˆ

Q1

up0 = p0

ˆ ∞

0tp0−1|u ≥ t ∩Q1|dt

then substitute the estimate for |u ≥ t ∩Q1|

After we getting the Harnack Inequality we can immediately get the Cα regularity ofthe solution. That is

Theorem 4.13. Let u ∈ S∗(λ,Λ, f) in Q1. Then(1) For a universal constant µ < 1

oscQ1/2

u ≤ oscQ1

u+ ∥f∥Ln(Q1).

(2) u ∈ Cα(Q1/2) and

∥u∥Cα(Q1/2) ≤ (∥u∥L∞(Q1) + ∥f∥Ln(Q1)),

where 0 < α < 1 and C > 0 are universal constants.

We do not give the proof since it’s just the same as in the linear elliptic theory.Nest we state some result of boundary estimates, where we use barrier function for proofjust the same as in the Schauder estimate. The only difference is we use AlexandroffMaximum Principle instead of Maximum Principle.

Lemma 4.14. Let u ∈ S(λ,Λ, 0) in B1. Assume that 0 < β < 1, u ∈ C(B1) andu|∂B1 = φ, where φ ∈ Cβ(∂B1).Then, for any x0 ∈ ∂B1, u is Cβ/2 −Holder continuous at x0, and

supx∈B1

|u(x0) − u(x)||x− x0|β/2

≤ 2β/2 supx∈∂B1

|φ(x) − φ(x0)||x− x0|β

.

We refer the readers to [HL] or just [CC]. I want to point out that for linear ellipticequation we may improve global Cβ/2 to global Cβ by potential integral. I don’t know ifit’s true for this problem.We combine with the interior estimate getting following results.

Lemma 4.15. Let u ∈ S(λ,Λ, 0) in B1. Assume that 0 < β < 1, u ∈ C(B1) andu|∂B1 = φ, where φ ∈ Cβ(∂B1). Then u ∈ Cγ(B1) and

∥u∥Cγ(B1) ≤ C∥φ∥Cβ(∂B1)

Where C is a universal constant and γ = min(α, β/2), with α(universal) as in (4.13)

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Theorem 4.16. Let u be continuous in B1 satisfy u ∈ S(λ,Λ, f) in B1, where f is acontinuous function. Let φ := u|∂B1 and let ρ(|x − y|) be a modulas of continuity of φ;that is ρ is a nondecreasing function in (0,∞) with limδ→0ρ(δ) = 0 such that

|φ(x) − φ(y)| ≤ ρ(|x− y|) ∀x, y ∈ ∂B1.

Assume finally that K is a positive constant such that ∥φ∥Ln(∂B1) ≤ K and ∥f∥Ln(B1) ≤ K.

Then there exists a modulus of continuity ρ∗ of u in B1, i.e., ρ∗ is nondecreasing,

limδ→0ρ∗(|x − y|) ∀x, y ∈ B1,

and ρ∗ depends only on n, λ, λ,K and ρ.

5 Negative Combination of Solutions and Regularity Im-provement

Before going on towards Evans-Krylov Theorem, we introduce two theorems. They allbased on Jesen’s Approximate Solutions.

5.1 Jesen’s Approximate Solutions

Let u be a continuous function in Ω and let H be an open set such that H ⊂ Ω. We define,for ϵ ≥ 0, the upper ϵ− envelop of u (with respect to H):

uϵ(x0) = supx∈H

u(x) + ϵ− 1

ϵ|x− x0|2, forx0 ∈ H

Theorem 5.1. (a) uϵ ∈ C(H) and uϵ ↓ u uniformly in H as ϵ → 0(b) For any x0 ∈ H there is a concave paraboloid of opening 2/ϵ (the coefficient of quadraticterm) that touches uϵ by below at x0 in H. Hence uϵ is C1,1 by below in H. In particular,uϵ is punctually second order differentiable at almost every point of H(c) Suppose that u is a viscosity subsolution of F (D2u) = 0 in Ω and that H1 is an openset such that H1 ⊂ H. We then have that for ϵ ≤ ϵ0 (where ϵ0 depends only on u, H andH1) u

ϵ is a viscosity subsolution of F (D2u) = 0 in H1, in particular, F (D2uϵ(x0)) ≥ 0a.e x ∈ H1

Remark: We say that a continuous function u in Ω is punctually second order differ-entiable at x) ∈ Ω if there exists a paraboloid P such that

u(x) = P (x) + o(|x− x0|2) as x → x0

Remark: We also can define uϵ the lower ϵ− envelop of uBefore we prove this theorem we need some concept which help analysis a function locally.Let us say that P is a paraboloid of opening M whenever

P (x) = l0 + l(x) ± M

2|x|2,

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where M is a positive constant, l0 is a constant and l is a linear function. P is convexwhen we have + and concave when we have −.Given two continuous functions u and v defined in an open set A and a point x0 ∈ A, wesay that v touches u by above at x0 in A whenever

u(x) ≤ v(x) ∀x ∈ A

u(x0) = v(x0).

Let u be a continuous function defined in Ω and A be an open subset of Ω. For x0 ∈ A,we define

Θ(u,A)(x0)

to be the infimum of all positive constants M for which there is a convex paraboloid ofopening M that touches u by above at x0 n A. If no such constant M exists, we define itto be ∞. Hence Θ(u,A) is a measurable function in A.Similarly we define

Θ(u,A)(x0) ∈ [0,∞]

andΘ(u,A)(x0) = sup

Θ(u,A)(x0),Θ(u,A)(x0)

≤ ∞

Given x0 ∈ Ω, we say that u is C1,1 by above at x0 if Θ(u,A)(x0) < ∞ for some neigh-borhood A of x0.Let us consider the second differential quotients of u at x0

∆2hu(x0) =

u(x0 + h) + u(x0 − h) − 2u(x0)

|h|2 ;

here h ∈ Rn and we assume that x0 + h and x0 − h belong to Ω. Note that ∆2hP ≡ M

when P is convex paraboloid of opening M. It follows that, for any x0 ∈ Ω,

−Θ(u,B|h|(x0))(x0) ≤ ∆2hu(x0) ≤ Θ(u,B|h|(x0))(x0),

if B|h|(x0) ⊂ Ω.

Lemma 5.2. Let 1 < p ≤ ∞ and u be a continuous function in Ω. Let ϵ be a positiveconstant and define

Θ(u, ϵ)(x) := Θ(u,Ω ∩Bϵ(x))(x)), x ∈ Ω

Assume that Θ(u, ϵ) ∈ Lp(Ω). Then D2u ∈ Lp(Ω) and

∥D2u∥Lp(Ω) ≤ 2∥Θ(u, ϵ)∥Lp(Ω).

95

Proof. Since 1 < p ≤ ∞ we only need to prove that

|ˆ

Ω

uφij | ≤ 2∥Θ(u, ϵ)∥Lp(Ω)∥φ∥Lp′ (Ω)

for any C∞ function φ with compact support in Ω and any indices i, j Here p′ denotes theconjugate exponent of p.To show it, we have

∂ijφ =1

2(∂ei+ej ,ei+ejφ− ∂iiφ− ∂jjφ)

=1

2(2∂vvφ− ∂iiφ− ∂jjφ),

where v = (ei + ej)/√

2 and ei is the canonical basis of Rn. Hence it suffices to prove

|ˆ

Ω

uφii| ≤ ∥Θ(u, ϵ)∥Lp(Ω)∥φ∥Lp′(Ω)

Let K ⊂⊂ Ω be the support of φ. We have thatˆ

Ω

uφii = limδ→0

ˆ

K

u∆2δeiφ = lim

δ→0

ˆ

K

(∆2δeiu)φ

Take δ < ϵ and δ <dist(K,Rn\Ω) we have

|∆2δeiu| ≤ ∆(u, ϵ)

Lemma 5.3. Let u be continuous in Ω and B be a convex domain such that B ⊂ Ω. Letϵ > 0 and define

Θ(u, ϵ)(x) := Θ(u,Ω ∩Bϵ(x))(x), .x ∈ B

Assume that, for some constant K, Θ(u, ϵ)(x) ≤ K for any x ∈ B. Then u ∈ C1,1(B) and

|Du(x) −Du(y)| ≤ 2n∥Θ(u, ϵ)∥L∞(B)|x− y| ∀x, y ∈ B.

Proof. We know(since Θ(u, ϵ) < ∞ for any x ∈ B) that u is differentiable at any point of B.By the above lemma we know that D2u ∈ L∞(B) and ∥D2u∥L∞(B) ≤ 2∥Θ(u, ϵ)∥L∞(B).Since ui ∈ W 1,∞ and B is convex, we have that ui is continuous an d

ui(x) − ui(y) =

1ˆ

0

d

dtui(tx+ (1 − t)y)dt

=∑

i

1ˆ

0

∂iju(tx+ (1 − t)y)dt(xj − yj),

for any x, y ∈ B. Using that ∥D2u∥L∞(B) ≤ 2∥Θ(u, ϵ)∥L∞(B),we get the result.

96

Remark: we also can prove directly.Using the Alexandroff-Buselman-Feller theorem we get the following theorem:

Lemma 5.4. Let u be a continuous function in a convex domain Ω such that

Θ(u,Ω)(x) ≤ K ∀x ∈ Ω,

for some positive constant K. Then

u(x) +K

2|x|2

is convex in Ω. In particular, u is punctually second order differentiable at almost everypoint x ∈ Ω.

We turn back to Jesen’s Approximate Solutions.

Lemma 5.5. Let x0, x1 ∈ H. Then(1) ∃x∗

0 ∈ H such that uϵ(x0) = u(x∗0) + ϵ− |x∗

0 − x0|2/ϵ(2) uϵ(x0) ≥ u(x0) + ϵ(3) |uϵ(x0) − uϵ(x1)| ≤ (3/ϵ)diam(H)|x0 − x1|.(4) 0 < ϵ < ϵ′ ⇒ uϵ(x0) ≤ uϵ′

(x0)(5) |x0 − x∗

0|2 ≤ ϵ oscHu.

(6) 0 < uϵ(x0) − u(x0) ≤ u(x∗0) − u(x0) + ϵ

Proof. (1)(2)(4)(6)are easy. For (3), letx ∈ H and note that

uϵ(x0) ≥ u(x) + ϵ− 1

ϵ|x− x0|2

≥ u(x) + ϵ− 1

ϵ|x− x0|2 − 1

ϵ|x1 − x0|2 − 2

ϵ|x− x1||x1 − x0|

≥ u(x) + ϵ− 1

ϵ|x− x1|2 − 3

ϵdiam(H)|x1 − x0|.

taking the supremum over x ∈ H, we get (3)For (5), we note by (1)(2)

1

ϵ|x∗

0 − x0|2 = u(x∗0) + ϵ− uϵ(x0) ≤ u(x∗

0) − u(x0).

We can give the proof of properties of Jesen’s Approximate Solutions:

97

Proof. (3)in (5.5) gives the continuity of uϵ. (4)(5)(6)imply the other assertion in(a). For(b),

P0(x) = u(x∗0) + ϵ− 1

ϵ|x− x∗

0|2 ≤ uϵ(x) ∀x ∈ H

and that equality holds at x0. That is, P0 touches uϵ by below at x0 in H. Then we usethe (5.4) we get the (b).For (c). Let x0 ∈ H1 and let P (x) be a paraboloid that touches uϵ by above at x0.Consider the paraboloid

Q(x) = P (x+ x0 − x∗0) +

1

ϵ|x0 − x∗

0|2 − ϵ

We have

u(x) ≤ uϵ(x+ x0 − x∗0) +

1

ϵ|x0 − x∗

0|2 − ϵ

Therefore

u(x) ≤ P (x+ x0 − x∗0) +

1

ϵ|x0 − x∗

0|2 − ϵ = Q(x)

and u(x0) = Q(x∗0). Hence Q touches u by above at x∗

0 we have

0 ≤ F (D2Q) = F (D2P )

Now we prove a theorem allowing a negative combination of solutions in some sense.

Theorem 5.6. Let u be a viscosity subsolution of F (D2ω) = 0 in Ω and v be a viscositysupersolution of F (D2ω) = 0 in Ω. Then

u− v ∈ S(λ

n,Λ) in Ω

Proof. We fix H and H1 such that H1 ⊂ HH ⊂ Ω; we will prove that for ϵ small enough,uϵ − vϵ ∈ S(λ/n,Λ) in H1. It follows that u− v ∈ S(λ/n,Λ) in Ω.Let P be a paraboloid such that uϵ − vϵ ≤ P in Br(x0) ⊂ H1 and uϵ(x0)− vϵ(x0) = P (x0).We need to see that M+(D2P, λ/n,Λ) ≥ 0. We may assume that B2r(x0) ⊂ H. Takeδ ≥ 0 and define

ω(x) = vϵ(x) − uϵ(x) + P (x) + δ|x− x0|2 − δr2

Using (5.5) (b), we know that for any x ∈ Br(x0) there exists a convex paraboloid P x ofopening K (independent of x) that touches ω by above at x in Br(x). And we have ω > 0on ∂Br(x0) and ω(x0) ≤ 0, we have

0 <

ˆ

Br(x0)∩ω=ΓωdetD2Γω

By (5.4) we know that almost everywhere in Brx0 we have uϵ and vϵ are punctually secondorder differentiable in A. Hence by (5.5)(c), we have

F (D2vϵ(x)) ≤ 0 and F (D2uϵ(x)) ≥ 0 forx ∈ A

98

Since Γω is convex and Γω ≤ ω, we have that

D2ω(x)isnonnegativedefinite, forx ∈ A ∩ ω = Γω

We choose one point x1 ∈ ω = Γω ∩A. At this point we have

0 ≤ F (D2uϵ(x1)) = F (D2vϵ(x1) −D2ω(x1) +D2P + 2δI)

≤ F (D2vϵ(x1) +D2P + 2δI) ≤ F (D2vϵ(x1) +D2P ) + 2Λδ

≤ F (D2vϵ(x1)) + Λ∥(D2P )+∥ − λ∥(D2P−∥ + 2Λδ

≤ Λ∥(D2P )+∥ − λ∥(D2P )−∥ + 2Λδ ≤ M+(D2P, λ/n,Λ) + 2Λδ.

Letting δ → 0, we get M+(D2P, λ/n,Λ) ≥ 0

Now we turn to regularity improvement and get C1,α regularity for F (D2u) = 0. Wedenote, for h > 0, Ωh = x ∈ Ω : d(x, ∂Ω) > h.

Lemma 5.7. Let u be a viscosity solution of F (D2u) = 0 in Ω. Let h > 0 and e ∈ Rn

with |e| = 1. Then

u(x+ he) − u(x) ∈ S(λ

n,Λ) in Ωh

Lemma 5.8. Let 0 < α < 1, 0 < β ≤ 1 and K > 0 be constants. Let u ∈ L∞([−1, 1])satisfy ∥u∥L∞([−1.1]) ≤ K. Define, for h ∈ R with 0 < |h| ≤ 1,

vβ,h(x) =u(x+ h) − u(x)

|h|β , x ∈ Ih,

where Ih = [−1, 1 − h] if h > 0 and Ih = [−1 − h, 1] if h < 0. Assume that vβ,h ∈ Cα(Ih)and ∥vβ,h∥Cα(Ih) ≤ K, for any 0 < |h| ≤ 1. We then have

1. If α+ β < 1 then u ∈ Cα+β([−1, 1]) and ∥u∥Cα+β([−1,1]) ≤ CK;

2. If α+ β ≥ 1 then u ∈ C0,1([−1, 1]) and ∥u∥C0,1([−1,1]) ≤ CK;

where the constants C in (1)(2) depends only on α+ β

Proof. By symmetry of the problem with respect to the change x → −x, it is enough tobound |u(x+ ϵ) − u(x)| for

−1 ≤ x ≤ 0 and x+ ϵ ≤ 1

Let i ≤ 0 be the integer such that x + 2iϵ ≤ 1 < x + 2i+1ϵ, and define τ0 = 2iϵ. Then−1 ≤ x < x+ τ0 ≤ 1 and

1/2 ≤ τ0 ≤ 2

Defineω(τ) = u(x+ τ) − u(x), 0 < τ ≤ τ0

99

We have that

|ω(τ)−2ω(τ/2)| = |u(x+τ)−2u(x+τ/2)+u(x)| =(τ2

)β |vβ,τ/2(x+τ/2)−vβ,τ/2(x)| ≤ K(τ2

)α+β,

since∥vβ,τ/2∥Cα([−1,1−τ/2]) ≤ K. Hence

|ω(τ0) − 2ω(τ0/2)| ≤ CKτα+β0 , |2ω(τ0/2) − 22ω(τ0/2

2)| ≤ CK21−(α+β)τα+β0 , . . . ,

|2i−1ω(τ0/2i−1) − 2iω(τ0/2

i)| ≤ CK2(i−1)(1−(α+β))τα+β0 ,

For some constant C which depends (as all C’s in the rest of this proof) only on α + β.Adding all the inequalities, we get

|ω(τ0) − 2iω(ϵ)| = |ω(τ0) − 2iω(τ0/2i)| ≤ CKτα+β

0

i−1∑

j=0

2j(1−(α+β)).

Since 2−i = τ−10 ϵ ≤ 2ϵ and ∥u∥L∞([−1, 1]) ≤ K, we have

|ω(ϵ)| ≤ 2−i|ω(τ0)| + CK2−iτα+β0

i−1∑

j=0

2j(1−(α+β))

≤ 4Kϵ+ CKϵτα+β0

i−1∑

j=0

2j(1−(α+β))

If α + β < 1 we get |ω(ϵ)| ≤ 4Kϵ + CKϵτα+β−10 2i(1−(α+β)) = 4Kϵ + CKϵα+β ≤ CKα+β .

If α+ β ≥ 1, we get |ω(ϵ)| ≤ 4Kϵ+ CKϵτα+β+10 ≤ CKϵ

Lemma 5.9. Let u be a viscosity solution of F (D2u) = 0 in B1. Then u ∈ C1,α(B1/2)and

∥u∥C1,α(B1/2) ≤ C(∥u∥L∞(B1) + |F (0)|),

where 0 < α < 1 and C are universal constants.

Proof. We fix e ∈ Rn with |e| = 1 and 0 < h < 1/8. By the above lemma, we know thatfor 0 < β < 1

vβ(x) =1

hβ(u(x+ he) − u(x)) ∈ S(λ/n,Λ) ∈ B7/8 (15)

Hence by Cα interior estimates we have that

∥vβ∥Cα(Br) ≤ C(r, s)∥vβ∥L∞(B(r+s)/2) ≤ C(r, s)∥u∥C0,β(Bs),

where 0 < r < s ≤ 7/8, 0 < h < (s−r)/2, α is universal and C(r, s) depends on n, λ,Λ, r, s.By making α slightly smaller, we can assume that there is a universal integer i such thatiα < 1 and (i+ 1)α > 1. We have u ∈ S(λ/n,Λ,−F (0)) in B1. Hence

∥u∥L∞(B7/8) ≤ C(∥u∥L∞(B1) + |F (0)|) =: CK

100

We define K = ∥u∥L∞ + |F (0)|. We apply (15) with β = α and r = r1 < s = 7/8 to get

∥vα∥Cα(Br1 ) ≤ C(r1)∥u∥Cα(B7/8) ≤ C(r1)K

where 0 < h < (7/8 − r1)/2, and C(r1) depends only on n, λ,Λ, r1. We can apply abovelemma get that

∥u∥C2,α(Br2 ) ≤ C(r1, r2)K for r2 < r1

We now apply (15) and above lemma with β = 2α, after finite steps we get

∥u∥C0,1(B3/4) ≤ CK

We finally apply(15) with β = 1 and get

∥v1∥Cα(B1/2) ≤ C∥u∥C0,1(B3/4) ≤ CK ∀|e| = 1 ∀0 < h < 1/8

We conclude that u ∈ C1,α(B1/2) and ∥u∥C1,α(B1/2) ≤ CK

We now state some easy application of the above lemma.

Theorem 5.10. Let F be concave and let u and v be viscosity subsolutions of F (D2ω) = 0in ω. Then 1

2(u+ v) is a viscosity subsolution of F (D2ω) = 0 in ω

Corollary 5.11. Let f be concave and suppose that F (D2u) = 0 in Ω in the viscositysense. Let e ∈ Rn with |e| = 1 and h > 0. Then

u(x+ he) + u(x− he) − 2u(x)

h2∈ S(λ/n,Λ) in Ωh

Corollary 5.12. Let F be concave and u ∈ C2Ω be a solution of F (D2u) = 0 in Ω. Then,for any e ∈ Rn with |e| = 1,

uee =∂2u

∂e∂e∈ S(

λ

n,Λ) in Ω

Proof of (5.10). It’s enough to see that 12(uϵ + vϵ) is viscosity subsolution of F (D2ω) = 0.

Let P be a paraboloid that touches 12(uϵ +vϵ) by above at x0. We need to show F (D2P ) ≥

0.Define

ω(x) = P (x) + δ|x− x0|2 − δr2 − 1

2(uϵ + vϵ).

We have x1 ∈ Ω such that uϵ, vϵ and ω are punctually second order differentiable at x1,and

D2P + δ|x− x0|2 − 1

2(uϵ + vϵ)(x1) ≥ 0

We also have that F (D2uϵ) ≥ 0 and F (D2vϵ(x1)) ≥ 0Since F is concave we get F (D2 1

2(uϵ + vϵ(x1))) ≥ 0, then

F (D2P + 2δI) ≥ 0

Letting δ → 0, we get F (D2P ) ≥ 0

101

6 Evans-Krylov Theorem

Now we turn to prove Evans-Krylov Theorem. This section is concerned with equations

F (D2u) = 0 (16)

for a concave operator F. We prove C2,α interior regularity for viscosity solutions of concaveequations of the form (16), for some universal α ∈ (0, 1)

Theorem 6.1. Let F be concave and u be a viscosity solution of F (D2u) = 0 in B1. Thenu ∈ C2,α(B1/2) and

∥u∥C2,α(B1/2) ≤ C(∥u∥L∞(B1) + |F (0)|

), (17)

where 0 < α < 1 and C are universal constants.

Remark: using the ellipticity condition for F, it is easy to see that there exists onet ∈ R such that F (tI) = 0 and t ≤ |F (0)|/λ. Define for any x0 ∈ Rn, the paraboloid

P (x) =t

2|x− x0|2.

ThenF (D2u) = F (D2(u− P ) + tI) =: G(D2(u− P )).

We have now that G(0) = 0; G is uniformly elliptic.

6.1 C1,1 Regularity

Theorem 6.2. Let F be concave and u be a viscosity solution of F (D2u) = 0 in B1. Thenu ∈ C1,1(B1/11) and

∥u∥C1,1(B1/11) ≤ C(∥u∥L∞(B1) + |F (0)|

), (18)

where C are universal constants.

Proof. We assume that F (0) = 0. By considering the functional t−1F (tD2ω), with t =∥u∥L∞(B1), we may assume that∥u∥L∞(B1) ≤ 1. We need to prove that ∥u∥C1,1 ≤ C.Where C denotes a universal constant.

Recall that F is a concave function on the space S of symmetric matrices; it followsthat there is a supporting hyperplane(by above) to the graph of F at 0 ∈ S. That isthere is a linear functional L on the space of symmetric matrices such that L(0) = 0 andL(M) ≥ F (M) for any M ∈ S.

Therefore, L is of the form L(M) =∑aijmij = tr(AM) for some A ∈ S. Then L is

uniformly linear operator with eigenvalues of A belong to [λ,Λ]. That is because for anyξ ∈ Rn, we have

aijξiξj = L(ξξt) ≥ F (ξξt) ≥ F (0) + λ∥ξξt∥ = λ|ξ|2 and,

aijξiξj = −L(−ξξt) ≤ −F (−ξξt) ≤ −F (0) + Λ∥ξξt∥ = Λ|ξ|2.

102

Hence, we can make a linear change of space variables such that in the new variables wehave L(D2u) = ∆u. Since L’s ellipticity constants is in [λ,Λ], we just need to prove thesituation L(D2u) = ∆u.

Since F (D2u) = 0 in the viscosity sense. and ∆φ = L(D2φ) ≤ F (D2φ) for anyφ ∈ C2, we immediately see that u also satisfies ∆u ≥ 0 in the viscosity sense in B1. Fromthis, we get that

u(x0) ≤

Sh(x0)

u

for any x0 ∈ B1/2 and 0 < h < 1/2; here Sh(x0) = ∂Bh(x0) andffl

denote average. To seethis we considers the harmonic function ω in Bh(x0) equal to u on Sh(x0). Then applythe maximum principle.We therefore have that the function

u∗h(x) =

1

h2

(

Sh(x)u− u(x)

)

is continuous and nonnegative in B1/2, for any 0 < h < 1/2. Note that u∗h is an approx-

imation of 12n∆u, in the following sense. Using Taylor’s formula, we see that if φ ∈ C∞

then φ∗h converges as h → 0 uniformly in compact sets to 1

2n∆φ; more over, for a constantC(n) depending only on n, ∥φ∗

h∥L∞(B1) ≤ C(n)∥D2φ∥L∞(B2).Using that u∗ ≤ 0, we now bound the L1− norm of u∗

h as follows. Let φ ≥ 0 be a C∞

function with compact support in B1/2 and φ ≡ 1 on B1/3. Then

ˆ

B1/3

|u∗h| =

ˆ

B1/3

u∗h ≤

ˆ

B1/2

uφ∗h ≤ C(n)∥D2φ∥L∞ ≤ C(n).

Recall that

Sh(x)

u =

Sh(0)

u(x+ y)dy

Because any positive linear combination of viscosity subsolution of F (D2u) = 0 is asubsolution, viscosity subsolutions is closed under uniform limits and u is a supersolution,we have u∗

h ∈ S(λ/n,Λ) in B1/2. Please notice the definition of u∗h

u∗h(x) =

1

h2

(

Sh(x)

u− u(x))

We now apply (4.11)(stated in balls and applied with p=1)to u∗h. We have that 0 < u∗

h ∈S(λ/n,Λ) and ∥u∗

h∥L1(B1/3) ≤ C(n). It follows that

∥u∗h∥L∞(B1/4) ≤ C,

for a universal constant C(independent of h).Let ψ be any C∞ function with compact support in B1/4. Since 2nψ∗

h → ∆ψ as h → 0

103

uniformly in B1/4, we have that

ˆ

B1/4

u∆ψ = 2n limh→0

ˆ

B1/4

uψ∗h = 2n lim

h→0

ˆ

B1/4

u∗hψ

and combined with ∥u∗h∥L∞(B1/4) ≤ C, we have

|ˆ

B1/4

u∆ψ| ≤ C∥ψ∥L1(B1/4)

There for ∆u in the distributional sense belongs to L∞(B(1/4)) and ∥∆u∥L∞(B1/4) ≤ C.

In particular, ∥∆u∥L2(B(1/4)) ≤ C and then by L2 regularity theory, u ∈ W 2,2(B1/5) and

∥D2u∥L2(B(1/5)) ≤ C (19)

By (5.11) we know that the pure second order incremental quotients of u,

∆2heu(x) =

1

h2[u(x+ he) + u(x− he) − 2u(x)],

belong to S(λ/n,Λ) inB1/2; here e is any vector with |e| = 1. (19)implies that ∥∆2heu∥L∞(B1/10) ≤

C, for a universal constant C independent of h ∈ (0, 1/10).(4.11) gives

supB1/11

≤ C ∀0 < h < 1/10, ∀|e| = 1 (20)

Then v(x) := u(x) − C2 |x|2 is a concave function in B1/11. The Alexandroff-Buselman-

Feller theorem implies that v and therefore u, is punctually second order differentiable atevery point in a set A with |B1/11\A| = 0. (20)implies that

uee(x) ≤ C ∀x ∈ A ∀|e| = 1

We also know that the weak derivative D2u of Du equals the punctually derivative a.e.,so we can see weak derivative D2u as the punctually derivative. Hence uee(x) is a linearcombination of D2u, then uee is an L2 function in B1/11. Because u is second punctuallydifferentiable, we have F (D2u) = 0 for any x ∈ A. Then

∥D2u∥ ≤ C sup|e|=1

uee(x)+ ∀x ∈ A

Hence D2u ∈ L∞(B1/11) and ∥D2u∥L∞(B1/11) ≤ C. We also know that W 2,∞(B1/11) =

C1,1(B1/11), so∥u∥C1,1(B1/11)

≤ C

Remark: We have C1,α regularity of u for free. Now we assume F is concave, thenget the C1,1. Without the concave condition, we still have a long way to go.

104

6.2 From C1,1 to C2,α

We have proved the C1,1 regularity for u, but next we need to assume u is C2. It seemsthere is a gap, however we can do the same argument for C1,1 solution u except we havethe almost everywhere C2,α regularity: there exists A ⊂ B1/2 and |B\A| = 0satisfies

1. D2u exists ∀x ∈ A;

2. |D2u(x) −D2u(y)| ≤ C|x− y|α ∀x, y ∈ A;

Where C is a constant independent of x, y. Then we can find a Cα function D2u, suchthat D2u = D2u in A. We integrate D2u back to get Du. Since Du is Lipschitz, we canhave Du = Du that is D(Du) = D2uHence we just need to prove the almost everywhere C2,α, without lost of generality, weassume u is C2.

Lemma 6.3. Under the assumption that F is concave, u ∈ C2(B1) satisfy F (D2u) = 0in B1, there exists a universal constant 0 ≤ δ0 ≤ 1 such that diam D2u(B1) = 2 impliesdiam D2u(Bδ0) ≤ 1

Remark: we can state a version for u ∈ C1,1, then you may believe there will be nobig difference between the proofs.

Lemma 6.4. Under the assumption that F is concave, u ∈ C1,1(B1) (thus D2u exists a.e.in B1) satisfy F (D2u) = 0 in B1, there exists a universal constant 0 ≤ δ0 ≤ 1 such thatdiam D2u(B1\A) = 2 implies diam D2u(Bδ0\A) ≤ 1, where A has measure zero.

Before going on towards the proof, we state several lemmas.

Lemma 6.5. Let v ∈ S(λ,Λ, 0) in B1 satisfy v ≥ 0 in B1. Then

infB1/2

v ≥ C|v ≥ 1 ∩B1/4|δ

where C and δ are positive universal constants.

Now we prove the main lemma towards Evans-Krylov Theorem

Lemma 6.6. Let F be concave and let

u ∈ C2(B1) satisfy F (D2u) = 0 in B1

Assume that1 ≤ diam D2u(B1) ≤ 2

and that D2u(B1) is covered by N balls B1, . . . , BN of radius ϵ (in the space S of symmetricmatrices) with N ≥ 1 and ϵ ≤ ϵ0, where ϵ0 > 0 is a universal constants.Then D2(B1/2) is covered by N-1 balls among B1, . . . , BN .

105

Proof. Take c0 universal constants such that

If F (M1) = F (M2) = 0 then c0∥M2 −M1∥ ≤ ∥(M2 −M1)+∥

For i = 1, 2, . . . , N we take xi ∈ B1 such that Bi ⊂ B2ϵ(Mi), where

Mi = D2u(xi)

Hence, taking ϵ0 such that 2ϵ ≤ 2ϵ0 ≤ c0/16, we have that

Bc0/16(M1), . . . , Bc0/16(MN )

cover D2u(B1). Since D2u(B1) has diameter at most 2, every Mi belongs to one closedball B of radius 2 in S. Let N ′ be the maximum umber of points in the ball B such thatthe distance between any two of them is at least c0/16. We have that N ′ depends onlyon n and c0. Therefore, we can assume that Bc0/8(Mi)N ′

i=1 cover B1 and therefore, thereexists one Mi, say M1, such that

|(D2u)−1(Bc0/8(M1)) ∩B1/4| ≥ η > 0

where η is a universal. Since diamD2u(B1) > 1 and take 2ϵ ≤ ϵ0 ≤ 1/4. Hence we haveM2 such that ∥M2 −M1∥ ≥ 1/4. Hence there is an e ∈ Rn and |e| = 1 such that

uee(x2) ≥ uee(x1) + c0/4

DefineK = sup

B1

uee and v = K − u

We have that 0 ≤ v ∈ S(λ/n,Λ, 0) in B1, by (5.12).Also we from above have |v(≥ c0/8) ∩ B(1/4)| ≥ η. We can apply (6.5) and get for auniversal constant C1

infB1/2

(K − uee) ≥ C1 > 0

By definition of K, there is one j, 1 ≤ j ≤ N , such that

K − uee(xj) < 3ϵ

If we finally take 5ϵ ≤ 5ϵ0 ≤ C1, then D2u(B1/2) ∩B2ϵ(M2ϵ(Mj))

Proof of (6.3). We remember the transformation

ω(x) = 4u(x/2) for x ∈ B1

We can normalize u in B1/2 to ω in B1, then continue the above process and get N − 2balls. When we get only one ball, we get the final result.

106

7 C2,α Perturbation

We skip the W 2,p estimate for the perturbation of fully nonlinear equation. The reasonis that this part maybe not so important as the C2,α perturbation in the application tothe geometry. Recently, Krylov has some new result maybe enlarge the W 2,p estimate, werefer the reader to those papers. While C2,α is more complicated, since we cannot hopeto get that right for all functions. We don’t know if the concave condition is the best orcan we beyond that. Anyway, Caffarelli’s work says if we have that large class of functionswhich has ”good solution”, then the small perturbation is also good.

7.1 Approximation lemma

Before we going on directly to the C2,α estimate we prove an approximation lemma whichis also a key point in the proof of W 2,p estimate.

Theorem 7.1. Let 0 < ϵ < 1 and u be a viscosity solution of F (D2u, x) = f(x) in B8√

n

such that ∥u∥L∞(B8√

n) ≤ 1. Assume that ∥β∥Ln(B7√

n) ≤ ϵ and that F (D2ω, 0) = 0 has C1,1

interior estimates (with constants ce). Then there exists h ∈ C2(B6√

n) and φ ∈ C(B6√

n)such that ∥h∥C1,1(B6

√n) ≤ c(n)ce (for a constant c(n) depending only on n),

u− h ∈ S(λ

n,Λ, φ) in B6

√n

and∥u− h∥L∞(B6

√n) + ∥φ∥Ln(B6

√n) ≤ C(ϵγ + ∥f∥Ln(B8

√n)),

where C is a positive constant depending only on n,λ,Λ and ce, and 0 < γ < 1 is universal.

Proof. Let h ∈ C2(B7√

n) be the solution of

F (D2h, 0) = 0 in B7

√n

h = u on ∂B7√

n

(21)

Using the C1,1 interior estimates and a covering argument, we have that

∥h∥C1,1(B6√

n) ≤ c(n)ce∥u∥L∞(B7√

n) ≤ c(n)ce.

By interior Holder estimates and a covering argument, we have that u ∈ Cα(B7√

n) and

∥u∥Cα(B7√

n) ≤ C(1 + ∥f∥Ln(B8√

n)),

with 0 < α < 1 and C > 0 universal constants. We use the boundary estimate by loweringthe Holder index.

∥h∥Cα/2(B7√

n) ≤ C∥u∥Cα(∂B7√

n) ≤ C(1 + ∥f∥Ln(B8√

n)).

107

Let 0 < δ < 1 and x0 ∈ B7√

n−δ. Then Bδ(x0) ⊂ B7√

n and we can apply interior C1,1

estimates in Bδ(x0) to h− h(x1) where x1 ∈ ∂Bδ(x0), we get

∥D2h(x0)∥ ≤ ceδ−2δα/2C(1 + ∥f∥Ln(B8

√n))

These implies that

|F (D2h(x0), x0)| ≤ Cδα/2−2β(x0)(1 + ∥f∥Ln(B8√

n)) forx0 ∈ B7√

n−δ

Since u− h = 0 on ∂B7√

n we have,

∥u− h∥L∞(∂B7√

n−δ) ≤ δα/2(1 + ∥f∥Ln(B8√

n)).

Since h is C2 we have u− h ∈ S(λ/n,Λ, φ) in B7√

n, with φ(x) = f(x) − F (D2h(x), x) ∈C(B7

√n). By the maximum principle, we have

∥u− h∥L∞(B7√

n−δ) + ∥φ∥Ln(B7√

n−δ) (22)

≤ ∥u− h∥L∞(∂B7√

n−δ) + C∥φ∥Ln(B7√

n−δ) (23)

C[δα/2 + δα/2−2∥β∥Ln(B7√

n)][1 + ∥f∥Ln(B8√

n)] + C∥f∥Ln(B8√

n) (24)

Recall that ∥β∥Ln(B7√

n) ≤ ϵ < 1. Take δ = ϵ1/2. We have γ = α/4

7.2 C2,α Estimates

In this part we study the C2,α regularity of viscosity solutions of

F (D2u, x) = f(x), x ∈ B1

We first define the oscillation of F in x near the origin

β(x) = βF (x) = supM∈S

|F (M,x) − F (M, 0)|∥M∥ + 1

,

Since we consider the regularity we can add a paraboloid such that we can assumeF(0,0)=f(0)=0. And we denotes the adimensional C2,α(Bd) norm:

∥ω∥∗C2,α(Bd) = ∥ω∥L∞(Bd)+d∥Dω∥L∞(Bd)+d

2∥D2ω∥L∞(Bd)+d2+α sup

x,y∈Bd

∥D2ω(x) −D2ω(y)∥|x− y|α .

Theorem 7.2. Assume that F is uniformly elliptic with ellipticity constants λ and Λ,that F and f are continuous in x and that F(0,0)=f(0)=0.Suppose that there are constants 0 < α < 1 and ce > 0 such that for any symmetric matrixM with F (M, 0) = 0 and any ω0 ∈ C(∂B1), there exists a omega ∈ C2(B1) ∩ C(B1) ∩C2,α(B1/2) which satisfies

F (D2ω(x) +M, 0) = 0 in B1,

ω = ω0 on ∂B1

(25)

108

And∥ω∥C2,α(B1/2) ≤ ce∥ω∥L∞(B1).

Assume that 0 < α < α, r0 > 0, C1 > 0,C2 > 0,

(

Br(0)

βn)1/n ≤ C1r

−α0 rα,

(

Br(0)

|f |n)1/n ≤ C2r

−α0 rα, ∀r ≤ r0 (26)

and that u is a viscosity solution of F (D2u, x) = f(x) in Br0(0). Then u is C2,α at theorigin; that is there is a polynomial P of degree 2 such that

∥u− P∥L∞(Br(0)) ≤ C3r−(2+α)0 r2+α ∀r ≤ r1 (27)

r0|DP (0)| + r20∥D2P∥ ≤ C3 (28)

C3 ≤ C(∥u∥L∞(Br0 (0)) + r20(C2 + 1)) (29)

andr1 = C−1r0 (30)

where C > 1 depends only on n,λ,Λ, ce, α, α and C1

Proof. We will prove that there is a constant δ > 0 (small enough) depending only on nλ, Λ, ce, α and α such that if u is viscosity solution of F (D2u, x) = f(x) in B1 = B1(0)with ∥u∥L∞(B1) ≤ 1,

(

Br(0)

βn)1/n ≤ δrα and

(

Br(0)

|f|n)1/n ≤ δrα ∀r ≤ 1, (31)

then there is a polynomial P of degree 2 such that

∥u− P∥L∞(Br) ≤ Cr2+α ∀r ≤ 1 (32)

and|DP (0)| + ∥D2P∥ ≤ C (33)

for some constant C > 0 depending only on n λ,Λ,ce,α and α.Then the theorem follows from the scaling. Consider

u(y) =r−21 u(r1y)

r−21 ∥u∥L∞(Br0 ) + δ−1C2

=:r−21 u(r1y)

K, y ∈ B1

if K ≥ 1(otherwise consider u(y) = r−21 u(r1y)).

Claim: There exists 0 < µ < 1 depending only on n,λ,Λ,ce,α and α and a sequenceof polynomials

Pk(x) = ak+ < bk, x > +1

2xtCkx

such that

109

F (Ck, 0) = 0 ∀k ≥ 0 (34)

∥u− Pk∥L∞(Bµk ) ≤ µk(2+α) ∀k ≥ 0 (35)

and

|ak − ak−1| + µk−1|bk − bk−1| + µ2(k−1)∥Ck − Ck−1∥ ≤ 13ceµ(k−1)(2+α) ∀k ≥ 0, (36)

where P0 ≡ P−1 ≡ 0. Then since Pk converges uniformly in B1 to a polynomial P ofdegree 2 which clearly satisfies (33) If we have proved the claim is true then we as followsto show the (32)

∥u− P∥L∞(Bµk) ≤ ∥u− Pk∥L∞(Bµk) +∞∑

i=k+1

∥Pi − Pi−1∥L∞(Bµk) (37)

≤ µk(2+α) +

∞∑

i=k+1

[|ai − ai−1| + µk|bi − bi−1| +

1

2µ2k∥Ci − Ci−1∥

](38)

≤ µk(2+α) + 13ce

∞∑

i=k+1

[µ(i−1)(2+α) + µkµ(i−1)(1+α) + µ2kµ(i−1)α

](39)

≤ Cµk(2+α) (40)

We turn to the claim now. We need take the µ carefully such that we can have thequantities related to µ is small enough to allow the solution have some property(rememberthat we are doing perturbation). We take µ small enough (depending on ce, α, α),such that

µα ≤ 1

2, µ ≤ 7

16, 28ceµ

α ≤ µα.

We take ϵ ∈ (0, 1) such that2Cϵγ ≤ ceµ

2+α

where C and γ is the constants in the (7.1). And we take δ(for which(31)) such that

2(1 + 26ce)c(n)δ = ϵ

(34)(35)(36)hold for k = 0, since P0 ≡ P−1 ≡ 0, F (0, 0) = 0 and ∥u∥L∞(B1) ≤ 1. We nowassume that they hold for k ≤ i and we prove them for k = i+ 1, where i ≥ 0. Consider

v(y) =(u− Pi)(µ

iy)

µi(2+α), y ∈ B1

Which satisfies µ−iαF (µiαD2v(y) + Ci, µiy) = µ−iαf(µiy), and hence Fi(D

2v, y) = fi(y)in B1, where

Fi(M,y) =1

µiα[F (µiαM + Ci) − F (Ci, µ

iy)], (41)

fi(y) =1

µiα[f(µiy) − F (Ci, µ

iy)]. (42)

110

We apply the rescaled (7.1)to v. We point out that Fi(D2ω, 0) = 0 a C2,α interior esti-

mates(with constant ce).Also we have

βFi(y) = supM

|F (µiαM + Ci, µiy) − F (µiαM+Ci,0)

µiα(∥M∥ + 1)− F (Ci, µ

iy) − F (Ci, 0)

µiα(∥M∥ + 1)|

≤ supM

(∥µiαM + Ci∥ + 1 + ∥Ci∥ + 1

∥M∥ + 1

) β(µiy)

µiα

≤ 2(1 + 26ce)µ−iαβ(µiy),

Since ∥Ci∥ ≤i∑

k=1

∥Ck − C(k − 1)∥ ≤ 13ce∞∑

k=1

µ(k−1)α = 13ce(1 − µα)−1 ≤ 26ce.

∥βFi∥Ln(B1) ≤ 2(1 + 26ce)µ−iαµ−i∥β∥Ln(Bµi ) ≤ 2(1 + 26ce)c(n)δ = ϵ;

∥fi∥Ln(B1) ≤ µ−iαµ−i(∥f∥Ln(Bµi )) + (1 + 26ce)∥βLn(Bµi )∥ ≤ c(n)δ(1 + 1 + 26ce) ≤ ϵ

The (7.1) gives the existence of h ∈ C2 ¯B3/4 such that

∥v − h∥L∞(B1/2) ≤ C(ϵγ + ϵ) ≤ 2Cϵγ

And h satisfies Fi(D

2h, 0) = 0 in B7/8,

h = v on ∂B7/8

(43)

HenceF (µiαD2h+ Ci, 0) in B7/8.

∥h∥∗C2,α(B7/16) ≤ ce∥v∥L∞(∂B7/8) ≤ ce.

Hence, since µ < 7/16,

∥h− P∥L∞(Bµ) ≤ ce(16

7)2+α ≤ 27ceµ

2+α,

where

P (y) = h(0)+ < ∇h(0), y > +1

2ytD2h(0)y

It follows that∥v − P∥L∞(Bµ) ≤ µ2+α

rescaling back

|u(x) − Pi(x) − µi(2+α)P (µ−ix)| ≤ µ(i+1)(2+α) ∀x ∈ Bµi+1

also Ci+1 = Ci + µiαD2h(0) and hence F (Ci+1, 0) = 0. Finally

|ai+1−ai| + µi|bi+1 − bi| + µ2i∥Ci+1 − Ci∥= µi(2+α)(|h(0)| + |∇h(0)| + ∥D2h(0)∥)

≤ 13ceµi(2+α)

111

8 Conclusion

This thesis is mainly based on [CC], even can be viewed as a copy of that book. What iselliptic theory? In my view just one sentence: knowing the boundary and try to interpretthe interior information. What is the regularity theory? In my view is that the oscillationis being controlled. What is the ABP maximal principle’s advantage? In my view is thatthe oscillation is controlled by integral quantity instead of the pointwise quantity.

Why we focus on elliptic theory? I don’t know. In a short talk with Caffarelli I knowthat elliptic has some sense stability as the critical point is a minimum. That is to say wecan try to get regularity. This property isn’t being shared by other types.

How to develop the elliptic theory? I don’t know. But I think we still need to findnew problem and observe new things. The fully nonlinear and the degenerate maybe arethe finally fields in elliptic theory. But after elliptic we may want to know the saddlecritical’s PDE aspects.

References

[CC] Luis A. Caffarelli and Xavier Cabre, Fully nonlinear elliptic equations, AmericanMathematical Society Colloquium Publications, vol. 43, American Mathematical So-ciety, Providence, RI, 1995.

[E] Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order el-liptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363.

[EG] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties offunctions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[HL] Qing Han and Fanghua Lin, Elliptic partial differential equations, 2nd ed., CourantLecture Notes in Mathematics, vol. 1, Courant Institute of Mathematical Sciences,New York, 2011.

[K] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad.Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian).

[KS] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equationswith measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian).

[S] Elias M. Stein, Singular integrals and differentiability properties of functions, Prince-ton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

112

Mirzakhani’s Volume Recursion and Witten-Kontsevich

Theorem

朱艺航∗

指导教师：Eduard Looijenga 教授

Abstract

This note is an exposition of M. Mirzakhani’s work [11] and [10] which gave a newproof of Witten-Kontsevich theorem on the intersection numbers of the ψ-classes onthe moduli spaces of stable curves.

Keywords: moduli space of curves; Weil-Petersson Form; Witten’s Conjecture

1 Introduction

In his famous 1991 paper [12] E. Witten conjectured that a generating function of theintersection numbers of the ψ-classes on the Deligne-Mumford compactification of modulispace of curves should be related to the Korteweg-de Vries hierarchy (KdV hierarchy),namely, it is the τ -function of the KdV hierarchy with initial value (d/dx)2 + 2x. Thefirst proof was given by M. Kontsevich in [6], using the interpretation of the modulispace concerned, in rough terms, as a moduli space of ribbon graphs (or fat graphs). Anexposition of these concepts and Kontsevich’s proof can be found in [7]. In this note wewill follow [10] to state the Witten conjecture in such a form that the concepts of KdVhierarchy and τ -function will not be involved. See Sect. 3.1.

The theme of this note is M. Mirzakhani’s work [11, 10], which gave a new proof ofWitten-Kontsevich Theorem. The key idea of Mirzakhani was to cut open a hyperbolicsurface along a closed geodesic to decompose the surface into simpler ones. Thus it is nat-ural to consider the moduli space Mg,n(L) of hyperbolic surfaces with geodesic boundarycomponents and prescribed boundary lengths, which generalizes the (open) moduli spaceof nonsingular curves Mg,n. The definitions of these concepts will be given in Sect. 2.By a classification of simple geodesics on a hyperbolic surface with geodesic boundaries,Mirzakhani obtained a generalization of MacShane’s identity [9] satisfied by the lengthsof certain geodesics on such a surface, which will be discussed in Sect. 5. Using thisidentity Mirzakhani managed to get recursion formulae for the Weil-Petersson volumesVg,n(L) := VolMg,n(L) of the various moduli spaces. See Sect. 3.2. On the other hand,by an application of Duistermaat-Heckman Theorem Mirzakhani showed that Vg,n(L) isin fact a polynomial in L with coefficients equal to the intersection numbers of certain

∗基数82

113

tautological classes over Mg,n. Thus the recursion formulae obtained give rise to relationssatisfied by these intersection numbers, and thus imply Witten-Kontsevich Theorem.

2 Moduli Spaces of Bordered Riemann Surfaces and Weil-Petersson Geometry

2.1 Mg,n and its Weil-Petersson Form

Let Mg,n be the moduli space of nonsingular complex algebraic curves. Mg,n is endowedwith a natural symplectic structure, given by the Weil-Petersson form ωWP [5]. As themoduli space Mg,n can be obtained as a quotient Tg,n/Modg,n of the Teichmuller spaceTg,n by the mapping class group Modg,n, ωWP can be expressed in the Fenchel-Nielsencoordinates of Tg,n. The following simple formula was proved by Wolpert in [13].

ωWP =∑

dlj ∧ dτj ,

where (lj , τj) are the length and twist parameters with respect to a pair-of-pants decom-position of the marking surface. These concepts will be explained in detail in the nextsubsection. Note that it is not obvious that the expression on the right-hand-side shouldbe invariant under Modg,n, which is generated by all the Dehn twists, not necessarilyalong the circles in the chosen pair-of-pants decomposition .This formula is fundamentalin studying the Hamiltonian geometry of Mg,n, as it shows that the length of a closedgeodesic is a Hamiltonian potential for the vector field induced by the infinitesimal Fenchel-Nielsen twist along that geodesic. These terms will be made more precise in what follows.Moreover, in [14] Wolpert showed that ωWP can be extended to Mg,n so that ωWP /2π

2

represents a cohomology class in H2(Mg,n,Q).(This is indeed the first kappa class κ1.)

2.2 Generalization to the Bordered Case

Now we generalize the setting to include hyperbolic surfaces with geodesic boundaries andgive precise definitions. Let S be a smooth manifold that is a finite disjoint union of closedsurfaces, and P = pi a finite set of points on S, such that each connected componentof S − P has negative Euler characteristic. Define a bordered Riemann surface of type(S, P ) to be a complete hyperbolic surface with n = #P boundary components, each ofwhich is a closed geodesic, such that the interior of the surface is diffeomorphic to S − P .Here by abuse of language we count a cusp of a hyperbolic surface as a geodesic boundarycomponent of length 0. Let X be a bordered Riemann surface of type (S, P ), define amarking of X to be a map ϕ : S − P → X that is a diffeomorphism onto the interiorX − ∂X. By abuse of notation we denote by ϕ(pi) the boundary component (or cusp) ofX which ϕ(q) approaches as q approaches pi on S. For L = (Lp1 , . . . , Lpn) ∈ RP , considerthe set

S(S, P ;L) = [ϕ : S − P → X] |X is a bordered Riemann surface of type (S, P ),

ϕ is a marking for X, such that ϕ(pi) has length Li. ,

114

where we define the equivalence relation [ϕ : S − P → X] = [ψ : S − P → Y ] if and onlyif there exists an isometry f : X → Y such that ψ = f ϕ. The group Diff+(S, P ) oforientation-preserving auto-diffeomorphisms of S that fix each pi ∈ P acts on S(S, P ;L)by precomposing with the markings. Define the Teichuller space of bordered Riemannsurfaces of type (S, P ) with prescribed boundary lengths L ∈ Rn

≥0 to be

T(S, P ;L) := S(S, P ;L)/Diff0(S, P ),

where Diff0(S, P ) ⊂ Diff+(S(S, P ) is the component of identity. Define the moduli spaceof bordered Riemann surfaces of type (S, P ) with prescribed boundary lengths L ∈ Rn

≥0 tobe

M(S, P ;L) := S(S, P ;L)/Diff+(S, P ).

Thus we have M(S, P ;L) = T(S, P ;L)/Mod(S, P ) where

Mod(S, P ) := Diff+(S, P )/Diff0(S, P ) = π0(Diff+(S, P ))

is the mapping class group.Obviously if (S, P ) =

⨿kj=1(S

(j), P (j)), and L = (L(j))1≤j≤k ∈ RP =⨿k

j=1 RP (j), then

T(S, P ;L) =k∏

j=1T(S(j), P (j);L(j)) (2.1)

M(S, P ;L) =k∏

j=1M(S(j), P (j);L(j)). (2.2)

We observe that the spaces T(S, P ;L) and M(S, P ;L) only depend on the diffeo-morphic class of S − P , thus it makes sense to introduce the following notations. For2g − 2 + n > 0 ,S a genus g closed surface, and P a set of n distinct points on S, let

Tg,n(L) := T(S, P ;L),

Mg,n(L) := M(S, P ;L),

and alsoModg,n := Mod(S, P ).

In such a case we will also use the term bordered Riemann surface of type (g,n) in placeof bordered Riemann surface of type (S,P) and use the symbol Sg,n to denote S − P .

Remark 2.1. In view of (2.1) and (2.2) it suffices to study Tg,n(L) and Mg,n(L), howeverit is convenient to treat more arbitrary pairs (S, P ).

Remark 2.2. When L = 0 ∈ Rn , Tg,n(0) = Tg,n,Mg,n(0) = Mg,n, by the equivalence ofcomplex structure and hyperbolic structure on surfaces of negative Euler characteristic.

115

2.3 Weil-Petersson Geometry

The Teichmuller space T(S, P ;L) also has Fenchel-Nielsen coordinates. Let P = γi1≤i≤m

be a pair-of-pants decomposition of S−P , that is, γi are mutually disjoint embedded circlesin S−P such that S−P−P has each connected component a pair of pants (i.e. a 2-sphereminus three points). For any embedded circle γ in S − P , we define the function

lγ : T(S, P, L) −→ R>0 (2.3)

[ϕ : S − P → X] 7→ the length of the unique geodesic in

the isotopy class of ϕ(γ) on X .

Thus we have a map

lP = (lγ1 , . . . , lγm) : T(S, P ;L) → Rm>0. (2.4)

A standard result is that this map is an Rm-principal bundle, where the action of Rm oneach fibre is given by the Fenchel-Nielsen shearing along the γi’s. In particular we have anon-canonical isomorphism of real analytic manifolds

(li, τi)1≤i≤m : Tg(L) −→ Rm>0 × Rm. (2.5)

(li, τi)1≤i≤m are called the Fenchel-Nielsen coordinates of T(S, P ;L) with respect to thepair-of-pants decomposition P, although τi cannot be chosen in a canonical way. Thus τmeasures the shearing in terms of arc length.

Remark 2.3. By counting the Euler characteristic it is easy to compute the cardinal mof a pair-of-pants decomposition P of S − P : Let n = #P . The number of connectedcomponents of S − (P ∪ P ) is (2m+ n)/3. Thus the Euler characteristic of S-P is

χ(S − P ) = −2m+ n

3= χ(S) − n.

Thus m = n− 32χ(S). In particular if S is a genus g closed surface then m = 3g − 3 + n

Now over T(S, P ;L) is defined a symplectic form ωWP invariant under Mod(S, P ),called the Weil-Petersson form, which has the form [11]

ωWP =∑

dlj ∧ dτj (2.6)

under the Fenchel-Nielsen coordinates (lj , τj). By the invariance under Mod(S, P ), ωWP

descends to a symplectic form in the orbifold sense over M(S, P ;L), which will still bedenoted by ωWP and called the Weil-Petersson Form. Note that (2.6) implies that theproducts (2.1) and (2.2) are actually compatible with the Weil-Petersson forms defined onthe various spaces, since a pair-of-pants decomposition for S always comes from those forthe connected components of S. We define the Weil-Petersson volume form to be ωm

WP /m!and the Weil-Petersson volume of M(S, P ;L) to be

V (S, P ;L) = Vol(M(S, P ;L)) :=1

m!

∫

M(S,P ;L)

ωWPm.

116

We also use the notations Vg,n(L) := Vol(Mg,n(L)), and Vg,n := Vol(Mg,n). By what hasbeen said and the following remark, the volume is multiplicative under the product (2.2).Here we set by convention V0,3(L) = 1 since M0,3(L) consists of only one point.

Remark 2.4. The factor 1/m! is a convention in symplectic geometry. It has the effect

that if a symplectic form ω =m∑

i=1dpi ∧dqi on a 2m-dimensional manifold is in its Darboux

form, then ωm/m! =m∏

i=1dpi ∧ dqi.

Remark 2.5. A remark on the finiteness of Vg,n(L). Bers observed that there exists a con-stant bg,n > 0 for each g, n ∈ Z≥0, 2g−2+n > 0, such that every bordered Riemann surfaceof type (g, n) has a pair of pants decomposition each constituent of which is a geodesic oflength ≤ bg,n. Consider the action of Modg,n on P|P is an isotopy class of pair of pantsdecompositions of Sg,n. The orbits Oi are indexed by the isomorphism classes of trivalentgraphs with n half edges of genus g, hence are finite in number. Pick Pi ∈ Oi for each i

and let (l(i)j , τ

(i)j ) be the Fenchel-Nielsen coordinate of Tg,n(L) with respect to Pi. Then

each Modg,n-orbit in Tg,n(L) intersects with the set ∆ =∪

i

l(i)j ≤ bg,n, 0 ≤ τ

(i)j < l

(i)j ∀j

for a positive number of times. This observation together with formula (2.6) implies thatVg,n < +∞.

3 Mirzakhani’s Main Results and Witten-Kontsevich The-orem

In this section we will first state Witten-Kontsevich Theorem. We will also state Mirza-khani’s main results (Theorem 1.1 and Section 5 in [11], Corollary 3.3 in [10])and showthat they imply Witten-Kontsevich, while the proof of these results will be presented inthe subsequent sections.

3.1 Statement of Witten-Kontsevich Theorem

Let Mg,n be the Deligne-Mumford compactification of Mg,n, i.e. it is the moduli spaceof n-pointed genus g stable curves [4]. Let Li, 1 ≤ i ≤ n be the tautological line bundle

over Mg,n, whose fibre at X ∈ Mg,n is the cotangent space to X at the ith marked point. Let ψi be its first Chern class. For any set d1, . . . , dn of integers define the symbol

⟨τd1 , . . . , τdn⟩g :=

∫

Mg,n

ψd11 · · ·ψdn

n . (3.1)

This is understood to be zero if∑di = 3g − 3 + n. Define a generating function of

these intersection numbers

F(λ, t0, t1, . . .) :=∞∑

g=0

λ2g−2∑

(d1,d2,··· )∈Z∞0

⟨∏τdj

⟩g

∏

r≥0

tnrr /nr!

117

where Z∞0 is the set of sequences of integers with all but finitely many coefficients zero, and

nr := # j : dj = r. Define a sequence of differential operators L−1, L0, L1, . . . , Ln, . . . asfollows,

L−1 =∂

∂t0+λ−2

2t0

2 +

∞∑

i=1

ti+1∂

∂ti

L0 =3

2

∂

∂t1+

∞∑

i=1

2i+ 1

2

∂

∂t1+

1

16

and for n ≥ 1,

Ln = −(2n+ 3)!!

2n+1

∂

∂tn+1+

∞∑

i=0

(2i+ 2n+ 1)!!

(2i− 1)!!2n+1ti

∂

∂ti+n

+λ2

2

n−1∑

i=0

(2i+ 1)!!(2n− 2i− 1)!!

2n+1

∂2

∂ti∂tn−i−1.

Now we can state Witten-Kontsevich Theorem as follows.

Theorem 3.1 (Witten-Kontsevich [10]). For n ≥ −1, Ln(eF) = 0, i.e.

0 = −(2n+ 3)!!

2n+1

∂F

∂tn+1+

∞∑

i=0

(2i+ 2n+ 1)!!

(2i− 1)!!2n+1ti∂F

∂ti+n

+λ2

2

n−1∑

i=0

(2i+ 1)!!(2n− 2i− 1)!!

2n+1

(∂2F

∂ti∂tn−i−1+∂F

∂ti

∂F

∂tn−i−1

). (3.2)

Remark 3.2. The original version of Witten’s conjecture involves KdV hierarchy and its τfunction, and Theorem 3.1 is a consequence of that. However, as it can be shown that therelations in Theorem 3.1 actually determine all the intersection numbers ⟨τd1 , . . . , τdn⟩g,Theorem 3.1 is actually equivalent to the original version.

3.2 Mirzakhani’s Main Results

Theorem 3.3. For 2g − 2 + n > 0, L = (L1, . . . , Ln) ∈ Rn≥0, Vg,n(L) has the form

Vg,n(L) =∑

|α|≤3g−3+n

CgαL

2α, (3.3)

where α = (α1, . . . , αn) is a multi index and 0 < Cα ∈ π6g−6+2n−2|α|Q. Moreover,

Cgα =

2m(g,n)

2|α|α!(3g − 3 + n− |α|)!

∫

Mg,n

ψαω3g−3+n−|α|, (3.4)

where ω = ωWP is the Weil-Pertersson form, ψ = (ψ1, . . . , ψn) are the ψ-classes asintroduced in Section 3.1 and m(i, j) := δ1,iδ1,j is equal to 1 if and only if i = j = 1 and 0otherwise.

118

Remark 3.4. Making use of the transcendence of π over Q it is convenient to express thefact 0 < Cα ∈ π6g−6+2n−2|α|Q in saying that Vg,n(L) is a homogeneous polynomial ofdegree 3g − 3 + n with positive rational coefficients in the variables L2

i and π2, where weview π as having the same homogeneous degree as Li in Q[L, π].

Theorem 3.5 (Mirzakhani’s Recursion Formulae, Section 5 in [11]). V0,3(L) = 1 by

convention, and V1,1(L) = L2

24 + π2

6 . Let g ∈ Z≥0, n ∈ N, 2g − 2 + n > 0, (g, n) =(0, 3) or (1, 1), L = (L1, . . . , Ln) ∈ Rn

≥0, L := (L2, . . . , Ln). Then Vg,n(L) satisfies thefollowing relation.

∂

∂L1L1Vg,n(L) = Acon

g,n (L1, L) + Adcong,n (L1, L) + Bg,n(L1, L), (3.5)

where

Acong,n (L1, L) :=

1

2

+∞∫

0

+∞∫

0

xyAcong,n (x, y, L1, L)dxdy,

Adcong,n (L1, L) :=

1

2

+∞∫

0

+∞∫

0

xyAdcong,n (x, y, L1, L)dxdy,

Bg,n(L1, L) :=

+∞∫

0

+∞∫

0

xBg,n(x, L1, L)dx,

Acong,n (x, y, L1, L) := Vg−1,n+1(x, y, L)H(x+ y, L1),

Adcong,n (x, y, L1, L) :=

∑

a∈Ig,n

V (a, x, y, L)H(x+ y, L1),

Ig,n := ((g1, I1), (g2, I2))|g1 + g2 = g,

2, . . . , n = I1 ⊔ I2, 2gj − 2 + #Ij ≥ 0,

V (a, x, y, L) :=Vg1,n1+1(x,LI1

)

2m(g1,n1+1) × Vg2,n2+1(x,LI2)

2m(g2,n2+1)

for a = ((g1, I1), (g2, I2)) ∈ Ig,n,

Li1,...,ik := (Li1 , . . . , Lik)

Bg,n(x, L1, L) :=

12m(g,n−1)

n∑i=2

12 (H(x, L1 + Lj) +H(x, L1 − Lj))Vg,n−1(x, L2 . . . , Lj , . . . , Ln)

H(x, y) :=1

1 + ex+y

2

+1

1 + ex−y

2

.

119

Remark 3.6. The formulae (3.5) actually determine Vg,n(L) for all permitted (g, n). Sinceall the Vg′,n′(L′) appearing on the right hand side of (3.5) have 2g′ − 2 + n′ < 2g − 2 + n,and only (g, n) = (0, 3) or (1, 1) will produce 2g − 2 + n = 1.

3.3 Computational Aspects of the Recursion Formulae

We develop some formulae to facilitate computations in the recursion formulae in Theorem3.5. In particular we show that Theorem 3.5 already implies the polynomial nature ofVg,n(L) (3.3) by induction on 2g − 2 + n. Note that V0,3(L) and V1,1(L) are of the form(3.3). For the induction step, we are interested in integrals of the forms

Fi(t) =+∞∫0

xiH(x, t)dx (3.6)

and Fi,j(t) =+∞∫0

+∞∫0

xiyjH(x+ y, t)dxdy, (3.7)

for i, j ≥ 0. We compute

Fi,j(t) =

+∞∫

0

+∞∫

0

xiyjH(x+ y, t)dxdy

=

+∞∫

0

du

u∫

0

dy(u− y)iyjH(u, t)

= B(i+ 1, j + 1)

+∞∫

0

u(i+ j + 1)H(u, t)du

= B(i+ 1, j + 1)Fi+j+1(t), (3.8)

where B(x, y) is Euler’s beta function and

B(i+ 1, j + 1) =i!j!

(i+ j + 1)!.

120

Hence it suffices to compute Fi(t).

Fi(t) =

+∞∫

0

xiH(x, t)dx

=

+∞∫

0

xi

(1

1 + ex+t+

1

1 + ex−t

)dx

=

+∞∫

0

(x+ t)i + (x− t)i

1 + exdx+

t∫

0

(t− x)i

1 + ex+

(t− x)i

1 + e−xdx

=

+∞∫

0

(x+ t)i + (x− t)i

1 + exdx+

t∫

0

(t− x)idx

=

+∞∫

0

(x+ t)i + (x− t)i

1 + exdx+

ti+1

i+ 1

=ti+1

i+ 1+

∑

0≤k≤i,i−k even2

(ik

)ti−k

+∞∫

0

xk

1 + exdx. (3.9)

We have

+∞∫

0

xk

1 + exdx =

+∞∫

0

∞∑

l=1

(−1)l+1xke−lxdx

=∞∑

l=1

(−1)l+1k!l−k−1

= (1 − 2−k)k!ζ(k + 1). (3.10)

Hence

Fi(t) =ti+1

i+ 1+

∑

0≤k≤i,i−k even2

(ik

)ti−k(1 − 2−k)k!ζ(k + 1). (3.11)

Note that for k ≥ 1 odd, ζ(k + 1) ∈ πk+1Q. Therefore for i ≥ 1 odd, Fi(t) is a rationalcoefficient polynomial homogeneous in t and π of degree i + 1. In the induction step ofproving (3.3), actually only the Fi(t) with odd t are involved. By Theorem 3.5,(3.11) and(3.8), we have proved (3.3) as well as the assertion that 0 < Cα ∈ π6g−6+2n−2|α|Q.

121

3.4 Mirzakhani’s Main Results Imply Witten-Kontsevich Theorem

In this subsection we use Theorem 3.3 and Theorem 3.5 to prove Theorem 3.1. Forα = (α1, . . . , αn) ∈ Zn

≥0, define

∆gα :=

1, if 3g − 3 + n = |α|0, otherwise

Thus the intersection numbers (3.1) and the coefficients in the volume polynomials arerelated by (3.4) in the following form.

⟨τα⟩g := ⟨τα1 , . . . ταn⟩g = ∆gαC

gα

2|α|α!(3g − 3 + n− |α|)!2m(g,n)

(3.12)

By virtue of (3.8) and (3.11), we know that for each Vg′,n′(L′) appearing on the right hand

side of (3.5), only its top coefficients (i.e. Cg′α′ with ∆g′

α′ = 1 ) will contribute to the topcoefficients of Vg,n(L). Explicitly,

(2α1 + 1)!! ⟨τα⟩ =1

2

∑

i+j=α1−2

(2i+ 1)!!(2j + 1)!!∑

I⊂2,...,n⟨ τiταI ⟩⟨ τjταIc ⟩

+1

2

∑

i+j=α1−2

(2i+ 1)!!(2j + 1)!!⟨τiτjτα2 · · · ταn⟩

+

n∑

j=2

(2α1 + 2αj − 1)!!

(2αj − 1)!!⟨τα2 · · · ταj−1τα1+αj−1ταj+1 · · · ταn⟩,(3.13)

whereταI

:=∏

i∈I

ταi ,

and

⟨· · · ⟩ :=+∞∑

g=0

⟨· · · ⟩g.

Now (3.13) implies (3.2).

4 A Computation for V1,1(L) as a Motivation for the Ideas

In this section we compute V1,1(L). The result is used as the starting point of the recursionof volume and the computation also illustrates the ideas. The computation is based onthe following lemma, which is a special case of Theorem 5.3.

Lemma 4.1. Let X be a bordered Riemann surface of type (1,1), L ≥ 0 be the length ofthe boundary geodesic. The following holds.

∑

γ∈FD(L, l(γ), l(γ)) = L (4.1)

122

where F is the set of all simple closed geodesics γ on X (countably many), l(γ) means thelength of γ, and D(x, y, z) : R3 → R≥0 is an explicitly defined function satisfying

∂

∂xD(L, x, x) =

1

1 + ex− L2

+1

1 + ex+L2

. (4.2)

Obviously for any simple closed geodesic γ on X, X − γ is of the same diffeomorphicclass. Take any marking ϕ : S1,1 → X, as γ runs through F , the isotopy class of ϕ−1(γ)on S1,1 exhausts an Mod1,1-orbit without repetition, where Mod1,1 permutes the isotopyclasses of closed curves on S1,1 naturally. Denote this orbit by O,(4.1) can be interpretedas an equality between two functions defined on T1,1(L):

∑

[γ]∈OD(L, lγ , lγ) = L, (4.3)

where lγ : T1,1(L) → R>0 is defined as in (2.3), which only depends on the isotopy class ofγ.

Take any embedded circle γ in S1,1 such that S1,1 − γ is connected, then [γ] ∈ O.Denote the stabilizer of [γ] in Mod1,1 by Stab(γ), then (4.3) becomes

∑

σ∈Mod1,1 / Stab(γ)

σ∗D(L, lγ , lγ) = L. (4.4)

Now letMγ

1,1(L) := T1,1(L)/Stab γ,

then

M1,1(L) =Mγ

1,1(L)

Mod1,1 /Stab(γ).

Denote by πγ the projection Mγ1,1(L) → M1,1(L). For any function g : Mγ

1,1(L) → R,define its pushforward

πγ∗g :=

∑

σ∈Mod1,1 / Stab γ

σ∗g, (4.5)

which is a well-defined function on M1,1(L). We have

∫

M1,1(L)

π∗g =

∫

Mγ1,1(L)

g, (4.6)

where on both sides we are integrating with respect to the Weil-Petersson volumes ωmWP /m!

inherited from T1,1(L), where m = 3g−3+n = 1. In particular, let g be the left-hand-sideof (4.4), which is well defined on Mγ

1,1(L), combining (4.4), (4.5) and (4.6) we have

LV1,1(L) =

∫

Mγ1,1(L)

D(L, lγ , lγ) (4.7)

123

and hence by (4.2)

∂

∂LLV1,1(L) =

∫

Mγ1,1(L)

1

1 + elγ− L2

+1

1 + elγ+L2

. (4.8)

Note that P = γ is a pair-of-pants decomposition of S1,1. Let (l, τ) = (lγ , τγ) be theFenchel-Nielsen coordinates as in (2.5). A fundamental domain of the action of Stab(γ)on T1,1(L) is (l, τ) ∈ R>0 × R | 0 ≤ τ < l. Using the dl ∧ dτ formula (2.6), we compute

∂

∂LLV1,1(L) =

+∞∫

0

dl

l∫

0

dτ1

1 + el−L2

+1

1 + el+L2

=

+∞∫

0

l

1 + el−L2

+l

1 + el+L2

dl

=

+∞∫

− L2

x+ L2

1 + exdx+

+∞∫

L2

x− L2

1 + exdx

= 2

+∞∫

0

x

1 + exdx+

0∫

− L2

x+ L2

1 + exdx−

L2∫

0

x− L2

1 + exdx

= 2

+∞∫

0

x

1 + exdx+

L2∫

0

(L

2− x)

[1

1 + ex+

1

1 + e−x

]dx

= 2

+∞∫

0

x

1 + exdx+

L2∫

0

(L

2− x)dx

= 2

+∞∫

0

xe−x

1 + e−xdx+

L2

8= 2

+∞∫

0

x∑

k≥1

(−1)k+1e−kxdx+L2

8

= 2∑

k≥1

(−1)k+1

k2Γ(2) +

L2

8= 2(ζ(2) − 1

2ζ(2)) +

L2

8

=L2

8+π2

6

Hence

V1,1(L) =L2

24+π2

6. (4.9)

In the next section we will prove Mirzakhani’s identity Theorem 5.9 (Theorem 4.2 in[11]), which will play the same role as (4.1) when computing general Vg,n(L). However, un-like the case here, for large (g, n) it is not easy to find fundamental domains for subgroups

124

of Modg,n. A new idea, that is, symplectic reduction, is used to avoid such difficulties as itallows us to reduce the computation for Vg,n to the computation of Vg′,n′ for some (g′, n′)with 2g′ − 2 + n′ < 2g − 2 + n, thus the recursion.

5 Mirzakhani’s Identity

In what follows a geodesic means a complete geodesic.

5.1 Geodesics on a Hyperbolic Pair of Pants

Our object of study is a hyperbolic pair of pants X (a bordered Riemann surface oftype (0, 3)) with geodesic boundary components a1,a2,a3. There is a unique geodesicb1 on X joining a2 and a3, which is orthogonal at the intersections. Similarly we haveb2 joining a1 and a3 and b3 joining a1 and a2. X − ∪bi is the union of two hyperbolicrectangular hexagonsX1 andX2. By an elementary theorem we know that the a hyperbolicrectangular hexagon is uniquely determined up to isometry by the lengths of its threemutually nonadjacent edges, hence X1 and X2 are isometric, having each bi as a commonedge. In particular each ai is divided into two arcs of equal length by its intersection pointswith bj and X has a canonical isometric involution j, mapping X1 isometrically onto X2.

Now we construct the universal cover of X. First we embed X1 into the upper halfplane H. Denote the image of X1 by Y1, and the images of bi by Bi, images of a1 ∩ X1

by Ai. Consider the group G = ⟨g1, g2, g3|g21 = g2

2 = g23 = 1⟩ = (Z/2Z) ∗ (Z/2Z) ∗ (Z/2Z).

For each g ∈ G, the length of g ,denoted by l(g), is the minimal integer k for whichg = gj1gj2 · · · gjk

, for some jl ∈ 1, 2, 3 , 1 ≤ l ≤ k. By induction on l(g), we define foreach g ∈ G a hexagon Yg in H with tree marked edges Bg

i , 1 ≤ i ≤ 3, and τg ∈ Aut(H) .For l(g) = 0, we have g = 1, and we define Yg = Y1, B

gi = Bi, and an element τg = id.

For general g ∈ G with l(g) > 0, let g = gig′ where l(g′) < l(g). Let σ be the Mobius

inversion with respect to the circle arc Bg′i . We define Yg := σYg′ , Bg

j = σBg′j , 1 ≤ j ≤ 3,

and τg = σ τg′ . Thus we have Yg = τg(Y1) and Bgi = τg(Bi). Define Ag

i = τg(Ai).Let X = ∪g∈GYg. We define a map π : X → X. Define π|Y1 to be the inverse of the

embedding X1 → Y1 and define π|Yg:= jl(g) π|Y1 τ−1

g , where j is the involution of X.

Thus we have defined π : X → X, which is easily seen to be a covering map. SinceG is a free product, X has the homotopy type of a trivalent tree, and is therefore simplyconnected. Hence (X, π) is the universal cover of X.

Remark 5.1. For each g ∈ G with even length we can define a deck transformation tgby defining tg|Y ′

g:= τgg′ τ−1

g′ for each g′ ∈ G. Since these transformations are al-

ready transitive on the fibres of π, we know that Deck(X/X) = tg|l(g) is even ∼=g ∈ G|l(g) is even. The fundamental group π1(X) of X is a free group of two gen-erators h1, h2, which is indeed isomorphic to Deck(X/X) by the map sending h1 tog2g3 and sending h2 to g3g1.

By the transitive property of Aut(H), we may arrange that

A1 =y√

−1|1 ≤ y ≤ el(a1)/2,

125

intersecting with B3 and B2 at p3 =√

−1 and p2 = el(a1)/2√

−1 respectively, and that Y1

lies to the left of the imaginary axis. We will assume this is the case in what follows.For any x ∈ a1 ∩X1 (x ∈ A1), let γx (γx) be the geodesic ray emanating from x (from

x and to the left), orthogonal to a1 (A1). We study its behavior.Let y2 ∈ A1 be such that γy2 is asymptotic to the half circle supporting A2. Let

y2 = π(y2). We claim that for any x in the open interval ]p3, y2[, x = π(x), γx is simpleand will finally touch a2 nonorthogonally, and γy2 is simple and asymptotic to a2. Similarly,we find y3 ∈ A1 such that γy3 is asymptotic to the half circle supporting A3. Let pi =π(pi), yi = π(yi), zi = j(yi), i = 2, 3, then we know that for any x in the open arc y2p3z2 andx = p3, γx is simple and will touch a2 nonorthogonally, while γy2 and γz2 are asymptoticto a2, spiraling around a2 in opposite directions, and γp3 is simple and orthogonal to a2.Similarly for the arc y3p2z3.

Proof of Claim.Denote by C(A1) the half circle in H circle supporting A1, etc. We mayapply a transformation of H to arrange that C(Bi) = z ∈ H| |z| = ri , i = 2, 3, r2 < r3and C(A2) =

√−1R>0. We may also arrange that A1 lies in the region r2 < |z| < r3,

that γy2 = x0 +√

−1R>0,−r3 < x0 < −r2 and that the direction of γy2 points upward.Define a new curve γ in the region r2 ≤ |z| ≤ r3 by reflecting γx along C(Bi) every timeit touches it ,i = 2, 3. It is obvious that γ has consecutive portions as a portion of a verticalline and a portion of a circle arc whose supporting circle passes through the origin. Inparticular γ is simple and is asymptotic to C(A2). This shows the desired property of γy2 .The behavior of γx for x = π(x), x ∈]p3, y2[ is similar.

We next find w1 ∈ A1 such that γw1 is orthogonal to τg1A1, and let w1 = πw1, thenγw1 is simple and touches a1 orthogonally at j(w1). We claim that for any x in theopen interval ]y2, y3[ and x = w1, x = π(x), γx is either non-simple or finally touches a1

nonorthogonally.Proof of Claim. Suppose γx is simple, and does not touch a1 nonorthogonally (i.e.

either does not touch a1, or touches a1 orthogonally). γx touches B1 before leaving Y1.When it goes into Yg1 , it either touches some Bg1

i , i = 2, 3 or touches Ag11 . The latter case

would imply that γx touches a1 nonorthogonally. Suppose γx touches Bg12 (the case for

Bg13 is the same) at y, then it enters Yg2g1 . Suppose it touches Ag2g1

1 . Then the intersectionmust be nonorthogonal, since otherwise we would have on X two different geodesic raysemanating from π(y) that are orthogonal to a1, namely a portion of γx and a portion ofb2, a contradiction. But this means that γx touches a1 nonorthogonally. Hence γx musttouch Bg2g1

1 or Bg2g13 . So we know γx first touches b1, then b2, then b1 or b3. Since γx

is simple, the alternative must be b2, and we know that after these three crossings γx

can never come close to a1, a2 or b3 again. But this means that γx will stay in Yg1(g2g1)n

and Y(g2g1)n , n ≥ 0 before its possible intersection with some Ag1(g2g1)n

2 or A(g2g1)n

2 , n ≥ 0.

Each Ag1(g2g1)n

2 and A(g2g1)n

2 , n ≥ 0 is supported on the half circle C supporting A2, not

intersecting with γx, hence γx will touch B(g2g1)n

1 and Bg1(g2g1)n

2 for each n ≥ 0. But

limn→∞

sups∈B

(g2g1)n

1

de(s, C) = limn→∞

sups∈B

g1(g2g1)n

2

de(s, C) = 0,

where de means the Euclidean distance in H, implying that γx is asymptotic to the circle

126

C, a contradiction.Let w1 = π(w1), w2 = jw1, from the above we know that for any x in the two open

arcs y2w1y3 and z2w2z3 , γx is either nonsimple or finally touches a1 again, and the secondintersection with a1 is orthogonal if and only if x = w1 or w2. We will call the open arcsy2w1y3 and z2w2z3 the main gaps on a1 for X and call the closed arcs y2p3z2 and y3p2z3the complementary arcs on a1 for X near a2 and a3 respectively.

We can actually compute the lengths of the complementary arcs as follows. Let li =l(ai), 1 ≤ i ≤ 3. First note that [p3, y2], B3, a portion of the half circle supporting A2,and γx are the four edges of a hyperbolic quadrangle with one angle zero and two infiniteedges. Hence we have

sinh d(p3, y2) =1

sinh l(B3),

or equivalently

sinh d(p3, y2) =1

sinh l(b3).

Since the hexagon X1 is determined by li/2, 1 ≤ i ≤ 3, so is l(b3), and we can write downan explicit formula

cosh l(b3) =cosh l3

2 + cosh l12 cosh l2

2

sinh l12 sinh l2

2

.

Therefore the length of the complementary arc y2p3z2 is

l(y2p3z2) = 2d(p3, y2) = logcosh l3

2 + cosh l1+l22

cosh l32 + cosh l1−l2

2

.

For later convenience let R(l1, l3, l2) be the length of the arc y2p2z2 and D(l1, l2, l3) be thetotal length of the main gaps on a1 for X. We have

R(l1, l3, l2) = l1 − l(y2p3z2) = l1 − logcosh l3

2 + cosh l1+l22

cosh l32 + cosh l1−l2

2

(5.1)

D(l1, l2, l3) = R(l1, l3, l2) + R(l1, l2, l3) − l1 = 2 loge

l12 + e

l2+l32

e−l12 + e

l2+l32

. (5.2)

We will view R and D as functions defined on R3 by the explicit formulae above. Notethat D(x, y, z) is a function of y + z. By explicit computation, we have

D(x, y, z) + D(x,−y, z) = 2R(x, y, z) (5.3)∂∂xD(x, y, z) = H(y + z, x) (5.4)

∂∂xR(x, y, z) = 1

2 (H(z, x+ y) +H(z, x− y)) (5.5)

where

H(x, y) =1

1 + ex+y

2

+1

1 + ex−y

2

. (5.6)

127

As a consequence of (5.4) and (5.5), we have, as x → 0,

D(x, y, z) ∼ xH(y + z, x) = 2x

1+ey+z2

(5.7)

R(x, y, z) ∼ x12 (H(z, y) +H(z,−y)) = x( 1

1+ez+y2

+ 1

1+ez−y2

) (5.8)

5.2 Geodesics on a Bordered Riemann Surface

Let X be a bordered Riemann surface, with geodesic boundary components βi, 1 ≤ i ≤ n.Let B = γ|γ is a simple geodesic on X with any possible intersection with βi orthogonal.For any x ∈ β1 let γx be the geodesic ray emanating from x orthogonal to β1. Weinvestigate cases when γx /∈ B.

Case 1. γx is simple and intersects with β1 nonorthogonally at y. x and y divide β1

into two portions a and b. In the homotopy class of a∪ γx (b∪ γx) on X there is a uniquesimple closed geodesic A (B). β1,A and B are the boundary components of a uniqueembedded hyperbolic pair of pants Y containing γx. Moreover, if Y ′ is an embedded pairof pants with geodesic boundaries such that β1 is one of its boundaries and γx is containedin Y ′, then Y ′ = Y . x is in the main gaps on β1 for Y .

Case 2. γx is simple and intersects with some βi, i > 1 nonorthogonally at y. Considerthe boundary a of a small neighborhood of β1 ∪βi ∪γx in X. In the homotopy class of a onX there is a unique simple closed geodesic A. β1, βi and A are the boundary componentsof a unique embedded hyperbolic pair of pants Y . Moreover, if Y ′ is an embedded pairof pants with geodesic boundaries such that β1 and βi are two of its boundaries and γx iscontained in Y ′, then Y ′ = Y . x is in the complementary arc on β1 for Y near βi.

Case 3. γx is nonsimple. Let y be the first self intersection of γx. Let γ be the portionof γx from x to y. Thus γ has the shape of a lasso. The boundary of a small neighborhoodof γ in X has two connected components, in which one is an embedded circle in X − ∂Xand the other is a portion of β1 union an embedded segment a in X with two end pointson β1. Similar to Case 1, a and β1 determines a unique embedded pair pants Y withgeodesic boundaries including β1, and Y contains γ. Hence x is in the main gap on β1 forY .

5.3 Mirzakhani’s Identity

Lemma 5.2. The set of x ∈ β1 for which γx ∈ B is of measure zero (with respect to the1-dimensional measure on β1).

Proof. By a theorem of Birman-Series [2], the union of all the simple geodesics ona complete hyperbolic surface without boundary is of Hausdorff measure 1. We doubleX to get such a complete hyperbolic surface without boundary Y , and every geodesic inB is a portion of a simple geodesic on Y . Hence we know that the union E = ∪γ∈Bγis of Hausdorff dimension 1 and is of measure zero in particular . Consider a half collarneighborhood U of β1 in X, we have µ2(E∩U) = µ1(E∩β1) sinh r = 0, where µi means thei-dimensional measure and r is the width of the collar neighborhood. Hence µ1(E∩β1) = 0.

By the previous lemma and the analysis in the previous subsection, we have provedthe following theorem.

128

Theorem 5.3 (Mirzakhani’s Identity). Let X be a bordered Riemann surface with geodesicboundaries βi, 1 ≤ i ≤ n. Let Li be the length of βi, then the following holds

∑

γ1,γ2∈F1

D(L1, l(γ1), l(γ2)) +

n∑

i=2

∑

γ∈F1,i

R(L1, Li, l(γ)) = L1 (5.9)

where the functions R and D are defined by (5.1) and (5.2), F1 is the set of all unorderedpairs γ1, γ2 such that there is an embedded pair of pants in X with β1, γ1 and γ2 asits geodesic boundaries, and F1,i is the set of all γ such that there is an embedded pair ofpants in X with β1, β2 and γ as its geodesic boundaries.

Remark 5.4. Lemma 4.1 is a consequence of Theorem 5.3 and (5.4) (5.6).

Remark 5.5. In what follows we will interpret (5.9) as an identity between functions definedon Tg,n(L). Then F1 is the set of all unordered pairs γ1, γ2 of non-peripheral isotopyclasses of embedded circles on Sg,n such that there is an embedded topological pair of pantsT in Sg,n ,with γ1, γ2 as its boundaries, and such that T is a punctured neighborhood of p1

; and F1,i is the set of all non-peripheral isotopy classes γ such that there is an embeddedtopological pair of pants T in Sg,n, with γ as its boundary, and such that T is a puncturedneighborhood of p1 and pi. Here an embedded circle in Sg,n is called nonperipheral if itdoes not bound a disc or a once punctured disc. We interpret l(γ) = lγ and l(γi) = lγi asin (2.3).

6 Integration over the Moduli Spaces and Symplectic Re-duction

6.1 Basic Definitions

We generalize the constructions in Sect. 4. Let γ1, . . . , γk be mutually disjoint non-peripheral isotopic classes of embedded circles in Sg,n, 2g − 2 + n > 0. Let

Γ := (γ1, . . . , γk) (6.1)

|Γ| := γ1, . . . , γk (6.2)

Stab(Γ) :=

k∩

i=1

Stab(γi) ⊂ Modg,n, (6.3)

where Modg,n permutes the isotopy classes of embedded circles in Sg,n naturally. Let

MΓg,n(L) := Tg,n(L)/Stab(Γ).

Define the function

lΓ = (lγ1 , . . . , lγk) : Tg,n(L) −→ Rk

>0

where lγi is defined by (2.3). lΓ descends to a function MΓg,n(L) → Rk

>0, which will be

denoted by LΓ. Denote by πΓ the projection MΓg,n(L) → Mg,n(L).

129

For any measurable function F : Rk>0 → R>0, let FΓ = F LΓ. Integration of FΓ

against the volume form ω3g−3+nWP /(3g − 3 + n)! over MΓ

g,n(L) is meaningful. We omitthe volume form in the notation when convenient. We introduce a general principle ofintegration.

Lemma 6.1. Let π : X → Y be a locally trivial fibration of manifolds. Let H be a groupthat acts on X and Y properly discontinuously, such that the actions are equivariant withrespect to π, i.e. π(hx) = gπ(x),∀x ∈ X,h ∈ H. Suppose dV is a volume form on Yinvariant under H, ω is a differential form on X such that dW = ω ∧ π∗dV is a volumeform on X invariant under H. Let π : X/H → Y/H be the induced fibration. Then forany y ∈ Y , ω is H-invariant on Hπ−1(y), and we have

∫

X/H

dW =

∫

y∈Y/H

dV

∫

π−1(y)=Hπ−1(y)/H

ω. (6.4)

Proof Since locally Hπ−1(y0) is a disjoint union of open sets of π−1(y), y ∈ Hy0, itsuffices to prove that

i∗y(h∗ω − ω) = 0, (6.5)

where iy : π−1(y) → X is inclusion. Now both dV and dW are invariant, hence (h∗ω−ω)∧π∗dV = 0. For x0 ∈ π−1(y0), we find local coordinates (x, y) : U → RM+N near x0 for X,local coordinates y : V → RN near y0 for Y such that π(x, y) = y, dV = dy = dy1 · · · dyn

locally. Now locally (h∗ω − ω) ∧ dy = 0, hence h∗ω − ω has the form∑

i dyi ∧ ωi. Hencei∗y0

(h∗ω − ω) = 0. (6.4) is just integration along fibres.

Remark 6.2. Since the mapping class group Modg,n has a normal subgroup of finite indexthat acts on Tg,n(L) without fixed point, any quotient of Tg,n(L) by a subgroup of Modg,n

has a finite Galois covering that is a manifold. Since any integral over a space is 1/r of thecorresponding integral over its r-fold covering, we may proceed as if the various quotientsof Tg,n(L) were manifolds obtained by a good group action when studying integrals onthem.

We extend |Γ| to a pair of pants decomposition P = γ1, . . . , γm of Sg,n, wherem = 3g − 3 + n ≥ k. Let (li, τi)1≤i≤m be the Fenchel-Nielsen coordinates. ApplyingLemma 6.1 to the fibration lΓ : Tg,n(L) → Rk

>0 with H = Stab(Γ) acting on Tg,n(L)naturally and acting on Rk

>0 trivially, we have

∫

MΓg,n(L)

FΓ =

∫

Rk>0

F (t)Vol(L−1Γ (t))dt, (6.6)

where Vol(L−1Γ (t)) =

∫

L−1Γ (t)

dτ1 · · · dτkdlk+1dτk+1 · · · dlmdτm.

We will compute Vol(L−1Γ (t)) in the next subsection.

For convenience we introduce the following definitions. Let S be a disjoint union ofclosed surfaces and P = pi|1 ≤ i ≤ n+ 2k a set of n+ 2k distinct points on S such that

130

S−P ∼= Sg,n −|Γ|, with pi corresponding to the ith marked point for Sg,n, 1 ≤ i ≤ n, andpn+j and pn+k+j corresponding to γj , 1 ≤ j ≤ k. We define

T (Sg,n − |Γ| ;L, l(Γ) = t) := T (S, P ;L, t, t),

M(Sg,n − |Γ| ;L, l(Γ) = t) := M(S, P ;L, t, t),

andMod(Sg,n − |Γ|) := Mod(S, P ).

6.2 Symplectic Reduction

We recall the theory of symplectic reduction. Suppose (M,ω) is a symplectic manifold ,and G is a Lie group acting smoothly on M preserving ω. Let g be the Lie algebra of Gand g∗ its dual space. A moment map is a map Φ : M → g∗ such that for any X ∈ g,the map ΦX : M → R, p 7→ Φ(p)(X) is a Hamilton potential for the vector field X#, i.e.ω(X#, ·) = dΦX , where X# is the vector field on M induced by X through the action.Such (M,ω,G,Φ) is called a Hamiltonian G-space. Assume Φ is ad∗-equivariant, that is

Φ(gp) = ad∗g−1Φ(p), for any g ∈ G, p ∈ M . For a ∈ g∗ let Ga =

g ∈ G|ad∗

g−1a = a

.Ga is

a closed subgroup of G, hence a Lie subgroup. By the ad∗-equivariancy of Φ, Ga acts onΦ−1(a). We have the following theorem.

Theorem 6.3 (Symplectic Reduction). Let Φ be an ad∗-equivariant moment map for(M,ω,G), a ∈ g∗ a regular value of Φ, and suppose that Ga acts on Φ−1(a) properlyand without fixed points, so that Ma := Φ−1(a)/Ga is a smooth manifold. On Ma thereis a unique symplectic form ωa such that π∗

aωa = i∗aω, where πa : Φ−1(a) → Ma is theprojection and ia : Φ−1(a) → M is the inclusion.

For a proof see [1] Chapter 4.Now let M = Tg,n(L), ω = ωWP , G = Rk. Let Γ = (γ1, . . . , γk) as in the previous

subsection. We define a G-action on (M,ω). We extend |Γ| to a pair of pants decom-position P = γ1, . . . , γm and let (li, τi)1≤i≤m be the Fenchel-Nielsen coordinates. Forg = (t1, . . . , tk) ∈ G, let

g(lP , τP) = (lP , τ1 − l1t1, . . . τk − lktk, τk+1, . . . τm).

Thus for example t = (1, 0, . . . , 0) gives a Dehn twist along γ1. We will call this action thenormalized shearing along Γ. By (2.6) this action preserves ω. Define

Φ =l2Γ2

= (l2γ1

2, . . . ,

l2γk

2).

By (2.6) Φ is an ad∗-equivariant moment map for the action. Moreover any a ∈ Rk>0 ⊂ g∗

is a regular value of Φ. The action of Ga = G on Φ−1(a) is proper and without fixedpoints. Let a = t2/2 = (t21/2, . . . , t

2k/2) ∈ Rk

>0, t ∈ Rk>0, then Φ−1(a) = l−1

Γ (t). The space

131

l−1Γ (t)/G has coordinates (li, τi)k+1≤i≤m and the symplectic form on it defined by Theorem6.3 is

∑mi=k+1 dli ∧ dτi. We recognize that

l−1Γ (t)/G ∼= T (Sg,n − |Γ| , L, l(Γ) = t) (6.7)

as symplectic manifolds.Our aim is to compute the volume of

L−1Γ (t) ⊂ MΓ

g,n(L) = Tg,n(L)/ Stab(Γ), t ∈ Rk>0.

The action of G on M and the moment map Φ are both equivariant with respect to theaction of Stab(Γ) on M . In particular we know that

(M ′, ω′, G,Φ′) = (MΓg,n(L), ωWP , G,

LΓ2

2)

satisfy the hypotheses of Theorem 6.3 (where we view M ′ as if it were a manifold byRemark 6.2). By (6.7)

LΓ−1(t)/G ∼= T (Sg,n − |Γ| ;L, l(Γ) = t)/ Stab(Γ) = M(Sg,n − |Γ| ;L, l(Γ) = t)

as symplectic manifolds, where Stab(Γ) acts on T (Sg,n −|Γ| ;L, l(Γ) = t) via the surjectivemap

Stab(Γ) −→ Mod(Sg,n − |Γ|). (6.8)

Let π : l−1Γ (t) → l−1

Γ (t)/G and π : LΓ−1(t) → LΓ

−1(t)/G be the projections. ApplyingLemma 6.1 again to π and H = Stab(Γ) we have

Vol(LΓ−1(t)) =

∫

X∈M(Sg,n−|Γ|)

∫

π−1(X)

dτ1 · · · dτk

dV (6.9)

where dV is the volume form on M(Sg,n − |Γ|) and we omit the prescribed lengths inthe notation. For a generic X ∈ M(Sg,n − |Γ|), we compute

∫π−1(X) dτ1 · · · dτk. Let

Y ∈ Tg,n(L) represent X. π−1(X) = Stab(Γ)π−1(Y )/Stab(Γ). We find a fundamental

domain for the Stab(Γ)-action on Stab(Γ)π−1(Y ) to be∏k

i=i[0,miti] × Y , where mi isequal to 1/2 if at least one connected component of Sg,n − γi is of type (1, 1) and equal to1 otherwise. Let M(Γ) = # i|mi = 1/2, we have

∫

π−1(X)

dτ1 · · · dτk = 2−M(Γ)t1 · · · tk. (6.10)

By (6.9)(6.10) we have

VolL−1Γ (t) = 2−MΓt1 · · · tkV (Sg,n − |Γ| , L, l(Γ) = t) (6.11)

where V (Sg,n − |Γ| , L, l(Γ) = t) = Vol(M(Sg,n − |Γ| , L, l(Γ) = t)). Combining (6.6) and(6.11) we have proved the following.

132

Theorem 6.4.∫

MΓg,n(L)

FΓ = 2−M(Γ)

∫

t∈Rk>0

F (t)t1 · · · tkdt1 · · · dtk (6.12)

6.3 The Covolume Formula

We use Theorem 6.4 to derive a fundamental formula. Let Γ = (γ1, . . . , γk), γi mutuallydisjoint non-peripheral isotopic classes of embedded circles in Sg,n, 2−2g+n > 0. Considera formal linear combination of the γi,

γ = c1γ1 + · · · + ckγk, ci ∈ R

called a multicurve. The mapping class group Modg,n acts on such multicurves linearly.Let Stab(γ) be the stabilizer of γ in Modg,n. Thus Stab(Γ) is a normal subgroup of Stab(γ)and these are equal when all the ci are distinct. Define

Sym(γ) := Stab(γ)/Stab(Γ),

lγ := c1lγ1 + · · · + cklγk: Tg,n(L) → R.

Letf : R → R>0

be a measurable function. Define

F (t1, . . . , tk) := f(c1t1 + · · · + cktk), for (t1, . . . , tk) ∈ Rk

FΓ = F LΓ is a function on MΓg,n(L). We use the projection πΓ : MΓ

g,n(L) → Mg,n(L)

to define a function πΓ∗FΓ on Mg,n(L) by pushing forward FΓ

πΓ∗FΓ(X) :=∑

Y ∈πΓ−1(X)

FΓ(Y ), for X ∈ Mg,n(L).

We have∫

Mg,n(L)

πΓ∗FΓ =

∫

MΓg,n(L)

FΓ. (6.13)

Let Y ∈ πΓ−1(X) , we compute

πΓ∗FΓ(X) =∑

σ∈Modg,n / Stab(Γ)

FΓ(σY )

=∑

σ∈Modg,n / Stab(Γ)

f(c1lσγ1(Y ) + · · · cklσγk(Y ))

= |Sym(γ)|∑

σ∈Modg,n

f(lσγ(Y ))

133

The function Y 7→ ∑σ∈Modg,n

f(lσγ(Y )) on Tg,n(L) is Modg,n-invariant and descends to a

function fγ : Mg,n(L) → R>0. Hence

πΓ∗FΓ = |Sym(γ)|−1 fγ (6.14)

By Theorem 6.4, (6.13) and (6.14), we have

Theorem 6.5 (The Covolume Formula, Theorem 7.1 in [11] ).

∫

Mg,n(L)

fγ =2−M(Γ)

|Sym(γ)|

∫

t∈Rk>0

f(c1t1 + · · · + cktk)V (Sg,n − |Γ| , L, l(Γ) = t) t1 · · · tkdt

7 Proof of Theorem 3.5

We use the interpretation in Remark 5.5 of Mirzakhani’s identity (5.9). For 2 ≤ i ≤ n,F1,i is an Modg,n-orbit of (multi)curves on Sg,n, since for each γ ∈ F1,i, Sg,n − γ is of thesame diffeomorphic type. Define Ri(x) := R(L1, Li, x), and choose any γi ∈ F1,i. Keepingthe notation of Section 6.3, we have

∑

γ∈F1,i

R(L1, Li, lγ) = Riγi. (7.1)

The first sum on the left hand side of (5.9) can be rewritten as∑

γ∈F ′1D(L1, lγ , 0),

where F ′ := γ1 + γ2| γ1, γ2 ∈ F1. We have used the fact that D(x, y, z) = D(x, y +z, 0). F ′ is the union of several Modg,n-orbits of multicurves. First there is an orbitF ′con := γ1 + γ2 ∈ F ′

1|Sg,n − γ1 ∪ γ2 has two connected components contained in F ′.Let F ′dcon = F ′ − F ′con. For any γ = γ1 + γ2 ∈ F ′dcon, Sg,n − γ1 ∪ γ2 is a disjointunion of a pair of pants and two surfaces S1 and S2 diffeomorphic to Sg1,n1+1 and Sg2,n2+1

respectively, such that g1 + g2 = g, n1 + n2 = n − 1, 2gj − 2 + nj ≥ 0, j = 1, 2. LetP = pi be the set of marked points for Sg,n, Pj = P ∩ Sj , Ij = i|pi ∈ Pj , j = 1, 2.Then 2, . . . , n = I1 ⊔ I2. Let Jg,n := (g1, I1), (g2, I2) |g1 + g2 = g, 2, . . . , n =

I1 ⊔ I2, 2gj − 2 + #Ij ≥ 0, j = 1, 2. F ′dcon is a disjoint union of Modg,n-orbits indexed byJg,n,

F ′dcon=

⨿

b∈Jg,n

Ob.

For each b ∈ Jg,n choose any γb ∈ Ob. Define D(x) := D(L1, x, 0), we have

∑

γ1,γ2∈F1

D(L1, lγ1 , lγ2) =∑

b∈Jg,n

Dγb. (7.2)

Combining (5.9) ( 7.1) (7.2), we have proved the following identity of functions definedon Mg,n(L).

∑

b∈Jg,n

Dγb+

n∑

i=2

Riγi

= L1. (7.3)

134

We have already computed V1,1(L) in Section 4. Integrate each side of (7.3) overMg,n(L) and apply Theorem 6.5 and properties (5.4) (5.5) , we have proved Theorem 3.5,where the only thing to notice is that for b = (g1, I1), (g2, I2) ∈ Jg,n, |Sym(γb)| = 2 ifand only if g1 = g2,#I1 = #I2 = ∅. In that case, the fibre over b of the forgetful mapϕ : Ig,n → Jg,n consists of one point. Otherwise |Sym(γb)| = 1 and ϕ−1(b) consists of twopoints.

8 Proof of Theorem 3.3

8.1 Duistermaat-Heckman Theorem

We will use the following version of Duistermaat-Heckman Theorem. Let (M,ω,G,Φ) bea Hamiltonian G-space, where G = Tn is a torus. Suppose Φ is ad∗-equivariant, proper,and has regular value 0 ∈ g∗ = Rn. Thus G0 = G acts on Φ−1(0). We suppose this actionis free, so that G will also act freely on Φ−1(a) for a near 0. Ehresmann’s fibration theoremdetermines for each a near 0 , an isotopy class of diffeomorphisms [fa : Φ−1(0) → Φ−1(a)].We will call such an fa natural in the sense of Ehresmann.

Theorem 8.1 (Duistermaat Heckman Theorem). Under the assumptions above, thereexists an ϵ > 0 such that ∀a ∈] − ϵ0, ϵ0[

n, G acts freely on Φ−1(a) and Φ−1(a) is naturallydiffeomorphic to Φ−1(0) in the sense of Ehresmann . Then Φ−1(a)/G is also naturallyidentified with Φ−1(0)/G . Denote by ωa the symplectic form on Φ−1(a)/G defined bysymplectic reduction and let fa : Φ−1(0) → Φ−1(a) be a natural diffeomorphism, a ∈] − ϵ0, ϵ0[

n. We have

[f∗aωa] = [ω0 +

k∑

j=0

ajc1(Lj)] ∈ H1dR(Φ−1(a)/G,R),

where Lj is the jth principal circle bundle in the principal Tn bundle Φ−1(0) → Φ−1(0)/G,and c1(Lj) is its first Chern class.

For a proof of this theorem see Chapter 30 in [3].

8.2 Teichmuller Space Obtained by Symplectic Reduction

Let 2g − 2 + n > 0. We can double Sg,n along the boundaries to get a closed surfaceSg′ , where g′ = 2g − 1 + n. More precisely, there exist mutually disjoint embedded circles

Γ = (γ1, . . . , γn) in Sg′ such that Sg′ − |Γ| = S(1)g,n ⊔ S

(2)g,n, where S

(i)g,n are two copies of

Sg,n. Take a pair of pants decomposition P for Sg,n. This gives rise to a pair of pants

decomposition P(i) =γ

(i)j

1≤j≤m

for S(i)g,n, i = 1, 2. Then P(1) ∪ P(2) ∪ |Γ| is a pair of

pants decomposition for Sg′ . Consider the locus L ⊂ Tg′ defined by the equations

lγ(1)j

= lγ(2)j

, τγ(1)j

= τγ(2)j

, 1 ≤ j ≤ m.

135

As in Section 6.2, the normalized shearing along Γ give rise to an RΓ action on Tg′ andΦ = lΓ

2/2 is an ad∗-equivariant, proper moment map. This action preserves the definingequations for L, hence (l−1

Γ (L) ∩ L)/RΓ is a locus in

l−1Γ (L)/RΓ = Tg,n(L) × Tg,n(L),

where L ∈ Rn≥0. This locus is just

(x, x) ∈ Tg,n(L) × Tg,n(L),

hence we have(l−1

Γ (L) ∩ L)/RΓ = Tg,n(L).

Consider the subgroup of ZΓ ⊂ RΓ generated by the Dehn twists along Γ. Define

Tg,n(L) := (l−1Γ (L) ∩ L)/ZΓ.

We know that the natural projection

Tg,n(L) −→ (l−1Γ (L) ∩ L)/RΓ = Tg,n(L)

is a principal Tn bundle.On the other hand, let i : L → Tg′ be the inclusion map. By (2.6) i∗ωWP is a symplectic

form on L. This form is invariant under ZΓ, hence it induces a symplectic structure onM := L/ZΓ. Now we have Tn acting on M symplectically, induced by the RΓ-action on L.The function lΓ descends to be a function l′Γ on M , and l′Γ

2/2 is a moment map for the

Tn action on M . For L ∈ Rn≥0, we have l′Γ

−1(L) = Tg,n(L) and l′Γ−1(L)/T k ∼= Tg,n(L) as

symplectic manifolds, where the symplectic structure on the left hand side is defined bysymplectic reduction.

8.3 Extension to the Case L = 0

We want to extend the principal Tn bundle

Tg,n(L) −→ (l−1Γ (L) ∩ L)/RΓ = Tg,n(L)

to the case L = 0. Recall the collar and cusp region theorem.

Lemma 8.2. Let X be a hyperbolic surface with cusps or geodesic boundary components.For every cusp on X, it has a unique neighborhood called the cusp region, which is isometricto

z ∈ H|Imz ≥ 1

2

/z ∼ z + 1 ∼=

z ∈ D∗| |z| ≤ e−π

,

where H is the upper half plane and D∗ is the unit open disc minus origin. For everygeodesic boundary of length l, it has a unique neighborhood, called a half collar region,isometric to

z ∈ H|d(z,√

−1R) ≤ w,Rez ≥ 0/z ∼ elz,

where d is the distance in H and w > 0 satisfies sinhw sinh l2 = 1. All the cusp regions

and half collar regions are disjoint.

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As can be seen from the metric structure, the cusp region of a cusp is foliated bygeodesics orthogonal to its boundary, and the half collar region of a geodesic boundary isfoliated by geodesics orthogonal to both of the two boundary components of this region.As the length of the geodesic boundary tends to zero, the half collar region Gromov-Hausdorff tends to a cusp region ,(i.e. convergence is uniform in any neighborhood with abounded distance from the geodesic boundary), and its boundaries and geodesic foliationalso converge.

Intuitively speaking, the space Tg,n(L), L ∈ Rn≥0 consists of pairs (X, (qi)1≤i≤n) where

X ∈ Tg,n(L) and each qi lies in the ith boundary component of X. This is equivalent tothe space of all pairs (X, (qi)1≤i≤n) where X ∈ Tg,n(L) and qi lies in boundary component

of the half collar region of the ith geodesic boundary other than the ith geodesic boundaryitself. This interpretation still makes sense if we let L = 0, where we replace the half collar

region of the ith geodesic boundary by the cusp region of the ith cusp.To be precise, we consider the Harvey bordification of Teichmuller spaces. We repro-

duce the exposition in Chapter 9 of [8]. First consider the map

ϕ :] − 1, 1[×S1 → C, (t, u) 7→ tu.

Both the restrictions ϕ|]−1,0[×S1 and ϕ|]0,1[×S1 parameterize the punctured unit disc D∗,so ϕ pulls back the metric

ds2 =|dz|2

(|z| log |z|)2

(induced from the covering H → H∗, w 7→ exp(2π√

−1w)) to (] − 1, 1[−0) × S1, calledthe standard degenerate hyperbolic metric on ] − 1, 1[×S1.

Let S be a closed surface of genus g, P = p1, . . . , pn a set of n points on S, 2g −2 + n > 0. Let Γ = (γ1, . . . , γk) be mutually disjoint non-peripheral isotopy classes ofembedded circles in S − P . A marking for a quasi-hyperbolic surface of type (S, P,Γ) isa triple (X, f, ds), where X is a surface with n boundary components whose interior isdiffeomorphic to S − P , f : S − P → X is a diffeomorphism onto the interior of X, andds is a hyperbolic metric on each component of X − P − f(|Γ|), such that each boundarycomponent of X is a geodesic and each f(γi) has a neighborhood U in X parameterizableby some orientation preserving ϕ :] − r, r[×S1 → U, 0 < r < 1 such that ϕ(0 × S1) = γi

and ϕ|(]−r,r[−0)×S1 is an isometry onto U − γi. Define an equivalence relationship [· · · ]between such objects by setting [X, f, ds] = [X ′, f ′, ds′] if and only if there exists a mapg : X → X ′ such that f ′ = g f and such that g induces an isometry from (X−f(|Γ|), ds)to (X ′ − f ′(|Γ|), ds′). As before, for such an object (X, f, ds) we abuse notation to denoteby f(pi) the boundary component of X which f(q) approaches as q approaches pi in S.For L = (L1, . . . , Ln) ∈ Rn

≥0 define the space

SΓ(S, P ;L) = [X, f, ds]|(X, f, ds) is a marking for a quasi-hyperbolic

surface of type (S, P,Γ) and f(pi) has length Li.

Consider the group Diff+(S, P ∪ |Γ|) of orientation-preserving auto-diffeomorphisms of Sthat fix each point in P ∪ |Γ|. It acts on SΓ(S, P ;L) by precomposing with the markings.

137

Let Diff0(S, P ∪ |Γ|) be the component of identity in Diff+(S, P ∪ |Γ|). Define

TΓ(S, P ;L) := SΓ(S, P ;L)/Diff0(S, P ∪ |Γ|).

We have an Rk-action on TΓ(S, P ;L), called the shearing action along Γ, defined as fol-lows. Let τ = (τ1, . . . , τk) ∈ RΓ, [X, f, ds] ∈ TΓ(S, P ;L). Define τ [X, f, ds] := [X, τ(f), ds],where τ(f) is defined as follows. For each 1 ≤ i ≤ k, choose a neighborhood Ui of f(γi)in X with a parametrization ϕi :] − r, r[×S1 → Ui such that ϕi(0 × S1) = f(γi) andϕ|(]−r,r[−0)×S1 is an isometry onto U−γi, where we choose 0 < r < e−π the same numberfor each i. Then by Lemma 8.2 the Ui are mutually disjoint and their closure does notintersect with the boundary of X. Let ψi = f−1 ϕi :]−r, r[×S1 → S, and let ϵ : R → R≥0

be a smooth map with the property that

ϵ|[0,+∞[ ≡ 1, ϵ|]−∞,−r/2] ≡ 0.

Defineατ :] − r, r[×S1 →] − r, r[×S1, (t, u) 7→ (t, ϵ(t)eτ2π

√−1u).

Define τ(f) : S − P → X by

τ(f)(x) :=

f(x) , x ∈ S − P − ∪if

−1(Ui)

f(ψi α ψ−1i (x)) , x ∈ f−1(Ui)

.

Now extend Γ to be a pair of pants decomposition P of S −P . The standard result isthat the map

lP−Γ : TΓ(S, P ;L) → RP−Γ>0

is a principal RP -bundle, where the action of RP−Γ is the usual Fenchel-Nielsen shearingand the RΓ-action is defined as above. In particular we have a noncanonical isomorphismof real analytic manifolds

(lP−Γ, τP) : TΓ(S, P ;L) → RP−Γ>0 × RP .

Moreover, we can define a cornered manifold structure on T +Γ (S, P ;L) := T (S, P ;L) ∪

TΓ(S, P ;L) independent of the choice of a pair of pants decomposition of S − P . Thefunction lΓ on T (S, P ;L) can be extended to T +

Γ (S, P ;L) by setting it to be zero. Weknow that lP : T +

Γ (S, P ;L) → RP≥0 is a principal RP bundle, where the RΓ action is

the normalized Fenchel-Nielsen shearing on T (S, P ;L) and is the action defined above onTΓ(S, P ;L) . We have the following isomorphism of cornered manifolds

(lP−Γ, lΓ, τP) : RP−Γ>0 × RΓ

≥0 × RP , (8.1)

and TΓ(S, P ;L) is recovered as the locus lΓ = 0.Warning: The coordinates (lP , τP) in (8.1) restricted to T (S, P ;L) is different than

the Fenchel-Nielsen coordinates since in this case the RΓ action on T (S, P ;L) is definedto be the normalized shearing.

We now replace Tg′ by T +Γ (S) where S is a closed surface of genus g′ in the construction

in Section 8.2. Understanding that the normalized shearing along Γ on TΓ(S) is just the

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shearing defined above, we find that everything still works when extending to the caseL = 0 and we recognize that the principal T k-bundle Tg,n(0) → Tg,n(0) gives rise to thecomplex line bundle over Mg,n which is the pull back of Li on Mg,n. Here Modg,n actson Tg,n via the homomorphism Modg,n → Mod(S,Γ) where the latter acts on T +

Γ (S)naturally.

In conclusion we have

Theorem 8.3. There exists a Hamiltonian Tn-space (M,ω, T k,Φ) satisfying the hypothe-

ses of Theorem 8.1 such that for L ∈ Rn≥0, Φ−1(L2

2 )/T k is isomorphic to Tg,n(L) as

symplectic manifolds. In addition, the first Chern class ψi ∈ H2(Tg,n,C) of the ith

principal circle bundle in the principal T k-bundle Φ−1(0) → Tg,n is the pull back ofψi ∈ H2(Mg,n,C).

Taking coordinates we see that actually there exists an isomorphism (lΓ, lP , τP) : M →RΓ

≥0 × RP>0 × RP where P is a pair of pants decomposition for Sg,n, such that under these

coordinates Φ = l2Γ/2 and (lP , τP) restricted to each Φ−1(L2/2) = Tg,n(L), L ∈ Rn≥0 are

the Fenchel-Nielsen coordinates with respect to P. In particular we see that identifyingTg,n(L) with Tg,n by identifying their Fenchel-Nielsen coordinates with respect to a samepair of pants decomposition of Sg,n is natural in the sense of Ehresmann. Under thisidentification a fundamental domain in Tg,n(L) for the Modg,n-action is mapped to afundamental domain in Tg,n for the Modg,n-action. Applying Theorem 8.1, we have for(g, n) = (1, 1)

Vg,n(L) =

∫

Mg,n(L)

ω3g−3+nWP

(3g − 3 + n)!

=

∫

Mg,n

(ωWP +n∑

j=1

L2j

2 ψj)3g−3+n

(3g − 3 + n)!

=∑

|α|≤3g−3+n

CgαL

2α, (8.2)

where

Cgα =

1

2|α|α!(3g − 3 + n− |α|)!

∫

Mg,n

ψαω3g−3+n−|α|

=1

2|α|α!(3g − 3 + n− |α|)!

∫

Mg,n

ψαω3g−3+n−|α|. (8.3)

For (g, n) = (1, 1) we already know V1,1(L), and (3.4) is just a convention on the intersec-tion numbers over M1,1. Since the rest part of Theorem 3.3 has already been proved inSection 3.3, we have finished proving Theorem 3.3.

139

References

[1] R. Berndt. An introduction to symplectic geometry. Graduate Studies in Mathematics.AMS, 2001.

[2] J.S. Birman and C. Series. Geodesics with bounded intersection number on surfacesare sparsely distributed. Topology, 24:217–225, 1985.

[3] A. C. da Silva. Lectures on Symplectic Geometry. Lecture Notes in Mathematics.Springer-Verlag, 2001.

[4] P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus.Pubilications Mathematiques de l’IHES, 36(1):75–109, 1969.

[5] W. Goldman. The symplectic nature of fundamental groups of surfaces. Adv. Math.,146:475–507, 1997.

[6] M. Kontsevich. Intersection theory on the moduli space of curves and the matrixAiry function. Comm. Math. Phys., 147, 1992.

[7] E. Looijenga. Intersection theory on Deligne-Mumford compactifications (after Wit-ten and Kontsevich). In Seminaire Bourbaki, 1992/1993, pages 187–212. Asterisque,1993.

[8] E. Looijenga. Moduli spaces of riemann surfaces at tsinghua, 2011. Lecture notes,downloadable at the author’s homepage.

[9] G. MacShane. Simple geodesics and a series constant over Teichmuller space. Invent.Math., 132:607–632, 1998.

[10] M. Mirzakhani. Weil-Petersson volumes and intersection theory on the moduli spaceof curves. J. Amer. Math. Soc.

[11] M. Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spaces ofbordered Riemann surfaces. Invent. Math., 2007.

[12] E. Witten. Two-dimensional gravity and intersection theory on moduli spaces. Sur-veys in Diff. Geom., 1:243–310, 1991.

[13] S. Wolpert. An elementary formula for the Fenchel-Nielsen twist. Comment. Math.Helv., 56:132–135, 1981.

[14] S. Wolpert. On the Weil-Petersson geometry of the moduli space of curves. Amer.J. Math., 107:969–997, 1985.

140

Rational Homotopy Theory of Loop Spaces

潘雯瑜∗

指导教师：Marcel Bokstedt 教授

Abstract

This article introduces some basic concepts of Sullivan models and how to useSullivan models to calculate the rational cohomology rings and rational homotopygroups of a topological space.

Keywords: Sullivan models, rational cohomology rings, rational homotopy groups

1 Introduction

Homotopy theory is the study of the invariants and properties of topological spaces Xand continuous maps f that depend only on the homotopy type of the space and thehomotopy class of the map. The classical examples of such invariants are the singularhomology groups Hi(X) and the homotopy groups πn(X).

The groups Hi(X) and πn(X), n ≥ 2, are abelian and hence can be rationalized to thevector spaces Hi(X; Q) and πn(X)⊗Q. Rational homotopy theory begins with the discov-ery by Sullivan in the 1960’s of an underlying geometric construction: simply connectedtopological spaces and continuous maps between them can themselves be rationalized totopological spaces XQ and to maps fQ : XQ → YQ, such that H∗(XQ) = H∗(X; Q) andπ∗(XQ) = π∗(X) ⊗ Q. The rational homotopy type of a CW complex X is the homo-topy type of XQ and the rational homotopy class of f : X → Y is the homotopy class offQ : XQ → YQ, and rational homotopy theory is then the study of properties that dependonly on the rational homotopy type of a space or the rational homotopy class of a map.

Although rational homotopy theory has the disadvantage of discarding a considerableamount of information, it has the advantage of being remarkably computational. Thecomputational power of rational homotopy theory is due to the discovery by Quillen andby Sullivan of an explicit 1-1 correspondence between geometry and algebra:

rational homotopy types

of spaces

∼=−→

isomorphism classes of

minimal Sullivan algebras

and

homotopy classes of maps

between rational spaces

∼=−→

homotopy classes of maps

between minimal Sullivan algebras

∗基数83

141

These models make the rational homology and homotopy of a space transparent.The main objective of the article is to provide some usable calculation tools and tech-

nique in rational homotopy theory. Meanwhile, we will illustrate part the above 1-1correspondence.

The materials are divided into five chapters, and the following is an overview of thefive chapters, indicating some of the highlights and the role of each part within the article.

• In § 3 we construct the Sullivan’s functor from topological spaces X to commutativecochain algebras APL(X), which satisfies C∗(X) w APL(X).

• In § 4 we set up the Sullivan model

(ΛV, d)w−→ (A, d)

for any commutative cochain algebra satisfying H0(A, d) = k.

• In § 5 Sullivan algebras are extended to relative Sullivan algebras.

• In § 6 we use relative Sullivan algebras to model fibrations and show that the Sullivanfibre of the model is a Sullivan model for the fibre.

• In § 7 we calculate the minimal Sullivan model of Ω(SU(n+ 1)/Tn).

2 Conventions and notation

2.1 Fibrations

Let X and Y be topological spaces. XY denotes the space of all continuous maps Y → X.A Moore path in X is a pair (γ, ℓ) in which γ : [0,∞)→ X is a continuous map, ℓ ∈ [0,∞),and γ(t) = γ(ℓ) for t ≥ ℓ. The path starts at γ(0), ends at γ(ℓ) and has length ℓ. Thespace MX ⊂ X [0,∞) × [0,∞) of all Moore paths is the free Moore path space on X. TheMoore path space P (X,x0) ⊂ MX is the subspace of all Moore paths ending at x0 andthe Moore loop space Ω(X,x0) is the subspace of all Moore paths starting and ending atx0. We usually write simply ΩX and PX. We may also denote (γ, ℓ) simply by ℓ.

Let f : X → Y be a continuous map between path connected spaces and let q0, q1 :MY → Y denote the maps (γ, ℓ) 7−→ ℓ(0) and (γ, ℓ) 7−→ γ(ℓ). Let cy be the path of length0 at y. Form the commutative diagram

Xj //

f @@@

@@@@

@ X ×Y MY

qyyssssssssss

Y

where

• X ×Y MY is the fibre product with respect to f and q0,

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• j(x) = (x, cfx),

• q(x, γ) = q1(γ).

It is easy to verify that j is a homotopy equivalence and q is a fibration: we say thediagram converts f into the fibration q.

2.2 Universal G-bundles

Central to the study of principal G-bundles is Milnor’s universal G-bundle

pG : EG → BG

constructed as follows.Let G be a topological group. Then CG = (G× I)/G× 0 is the cone on G, and the

nth join, G∗n, is the subspace of CG × . . .× CG of points ((g0, t0), . . . , (gn, tn)) such that∑ti = 1. Thus G∗n ⊂ G∗(n+1) (inclusion opposite base point of CG). Set EG =

∪nG

∗n,equipped with the weak topology determined by the G∗n.

A continuous action of G in EG is given by

((g0, t0), . . . , (gn, tn)) · g = ((g0g, t0), . . . , (gng, tn)).

Set BG = EG/G and let pG : EG → BG be the quotient map. It is easy to verify that

• pG : EG → BG is a principal G-bundle.

• every continuous map from a compact space to EG is homotopic to a constant map.In particular, π∗(EG) = 0.

Since EG is contractible, G is weakly homotopic to ΩBG by Proposition 4.46 in [3].

2.3 Basic definitions of singular chains

Recall that a singular n-simplex in a space X is a continuous map

σ : ∆n → X,

where n ≥ 0 and ∆n = n∑0tiei|0 ≤ ti ≤ 1,

∑ti = 1 is the convex hull of the standard

basis e0, . . . , en of Rn+1. If X ⊂ Rm is a convex subset then any sequence x0, . . . , xn ∈ Xdetermines the linear simplex

⟨x0, . . . , xn⟩ : ∆n → X,∑

tiei 7−→∑

tixi.

Two important examples of linear simplices are the ith face inclusion of ∆n,

λi = ⟨e0 . . . ei . . . en⟩ : ∆n−1 → ∆n, (ˆmeans omit)

143

defined for n ≥ 1 and 0 ≤ i ≤ n; and the jth degeneracy of ∆n,

ρj = ⟨e0 . . . ejej . . . en⟩ : ∆n+1 → ∆n,

defined for n ≥ 0 and 0 ≤ j ≤ n. The image of λi is called the ith face of ∆n.Denote by Sn(X) the set of singular n-simplices on a space X(n ≥ 0), and by

Sn(φ) : Sn(X)→ Sn(Y ), Sn(φ) : σ 7−→ φ σ,

the set map induced from a continuous map φ : X → Y . The set maps

∂i : Sn+1(X)→ Sn(X), σ 7−→ σ λi and sj : Sn(X)→ Sn+1(X), σ 7−→ σ ρj

are called the face and degeneracy maps. A straightforward calculation shows that

∂i∂j = ∂j−1∂i, i < j;

sisj = sj+1si, i ≤ j;∂isj = sj+1∂i, i < j;

∂isj = id, i = j, j + 1;

∂isj = sj∂j−1, i > j + 1.

(2.1)

2.4 Free commutative graded algebras

Let V be a graded vector space over a field k of characteristic zero. Then ΛV denotesto be the free commutative graded algebra on V . The following notation and basic factsassociated to ΛV will be used in the article:

• ΛV=symmetric algebra (V even)⊗ exterior algebra (V odd). The subalgebras Λ(V ≤p),Λ(V >q), . . . are denoted ΛV ≤p,ΛV >q, . . . .

• If vα or v1, v2, . . . is a basis for V we write Λ(vα) or Λ(v1, v2, . . . ) for ΛV .

• ΛqV is the linear span of elements of the form v1 ∧ . . .∧ vq, vi ∈ V . Elements in ΛqVhave wordlength q.

• ΛV =⊕q

ΛqV and we write Λ≥qV =⊕i≥q

ΛiV and Λ+V = Λ≥1V .

• If V =⊕λ

Vλ then ΛV =⊗λ

ΛVλ.

• Any linear map of degree zero from V to a commutative graded algebra A extendsto a unique graded algebra morphism ΛV → A.

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3 Commutative cochain algebras for spaces and simplicialsets

In this chapter the ground ring is a field k of characteristic zero.Let X be a topological space then C∗(X; k) denotes the cochain algebra of singular

cochains on it. In the chapter, we introduce a naturally defined commutative cochainalgebra APL(X; k), and natural cochain algebra quasi-isomorphisms

C∗(X; k)w−→ D(X)

w←− APL(X; k),

where D(X) is a third natural cochain algebra. The construction of APL(X; k) is due toSullivan and the functor

X APL(X; k)

serves as the fundamental bridge which we use to transfer problems from topology toalgebra.

The quasi-isomorphisms above will be constructed in Theorem 3.8. They define anatural isomorphism of graded algebras,

H∗(X; k) = H(APL(X; k)), (3.1)

and we shall always identify these two algebras via this particular isomorphism. Thus forany continuous map, f , we identify H∗(f ; k) = H(APL(f ; k)).

3.1 Simplicial sets and simplicial cochain algebras

Definition 3.1. A simplicial object K with values in a category C is a sequence Knn≥0

of objects in C, together with C-morphisms

∂i : Kn+1 → Kn, 0 ≤ i ≤ n+ 1 and sj : Kn → Kn+1, 0 ≤ j ≤ n.

satisfying the identities

∂i∂j = ∂j−1∂i, i < j;

sisj = sj+1si, i ≤ j;∂isj = sj+1∂i, i < j;

∂isj = id, i = j, j + 1;

∂isj = sj∂j−1, i > j + 1.

(3.2)

Then the singular complex of a topological space X, S∗(X) is a simplicial object withthe ordinary face and degeneracy maps.

A simplicial morphism f : L→ K between two such simplicial objects is a sequence ofC-morphisms φn : Ln → Kn commuting with the ∂i and sj .

A simplicial set K is a simplicial object in the category of sets. A simplicial cochainalgebra A is a simplicial object in the category of cochain algebras. Similarly, we definesimplicial cochain complexes, simplicial vector spaces, . . . .

145

Let K be any simplicial set. By analogy with the singular complex, the elementsσ ∈ Kn are called n-simplices and σ is non-degenerate if it is not of the form σ = sjτ forsome j and some τ ∈ Kn−1. The subset of non-degenerate n-simplices is denoted by NKn.It follows from the relations (3.2) that for each k ≥ 0 a simplicial set K(k) ⊂ K is given asfollows: for n ≤ k,K(k)n = Kn and for n > k,K(k) = sjτ |0 ≤ j ≤ n− 1, τ ∈ K(k)n−1.The simplicial set K(k) is called the k-skeleton of K.

Finally, we use the notation of § 2.3 to introduce a sequence of important subsimplicialsets ∆[n] ⊂ S∗(∆n), n ≥ 0. Let e0, . . . , en be the vertices of ∆n. Define a subsimplicialset ∆[n] ⊂ S∗(∆n) as follows: ∆[n]k ⊂ Sk(∆

n) consists of the linear k-simplices of theform σ = ⟨ei0 . . . eik⟩ with 0 ≤ i0 ≤ . . . ≤ ik ≤ n. Thus σ is non-degenerate if and only ifi0 < . . . < ik and the identity map

cn = ⟨e0 . . . en⟩ : ∆n → ∆n

is the unique non-degenerate n-simplex; cn is called the fundamental simplex of ∆[n]. The(n− 1)-skeleton of ∆[n]is denoted by ∂∆[n] and is called the boundary of ∆[n].

Lemma 3.2. If K is any simplicial set then any σ ∈ Kn determines a unique simplicialset map σ∗ : ∆[n]→ K such that σ∗(cn) = σ.

Proof. The verification is straightforward using (3.2).

A simplicial object A in a category C is called extendable if for any n ≥ 1 and anyI ⊂ 0, . . . , n, the following condition holds: given Φi ∈ An−1, i ∈ I, and satisfying

∂iΦj = ∂j−1Φi, i < j,

there exists an element Φ ∈ An such that Φi = ∂iΦ, i ∈ I.

3.2 The construction A(K)

Let K be a simplicial set, and let A = Ann≥0 be a simplicial cochain complex or asimplicial cochain algebra. The the “ordinary” cochain complex (or cochain algebra)

A(K) = Ap(K)p≥0,

(the upper script is denoted to be the graded of the cochain algebra) is defined as follows:

• Ap(K) is the set of simplicial set morphisms from K to Ap = Apnn≥0. Thus anelement Φ ∈ Ap(K) is a mapping that assigns to each n-simplex σ ∈ Kn(n ≥ 0) anelement Φσ ∈ Apn, such that Φ∂iσ = ∂iΦσ and Φsjσ = sjΦσ.

• Addition, scalar multiplication and the differential are given by

(Φ + Ψ)σ = Φσ + Ψσ, (λ ·Ψ)σ = λ ·Ψσ, and (dΨ)σ = d(Ψσ).

• If A is a simplicial cochain algebra multiplication in A(K) is given by

(Φ ·Ψ)σ = Φσ ·Ψσ.

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• If φ : K → L is a morphism of simplicial sets then A(φ) : A(K) ← A(L) is themorphism of cochain complexes (or cochain algebras) defined by

(A(φ)Φ)σ = Φφσ.

• If θ : A → B is a morphism of simplicial conchain complexes (or simplicial cochainalgebras) then θ(K) : A(K)→ B(K) is the morphism defined by

(θ(K)Φ)σ = θ(Φσ).

• When X is a topological space we write A(K) for A(S∗(X)).

Remark 3.3. Note that the construction A(K) is a covariant functor with respect to Aand contravariant functor with respect to K.

Proposition 3.4. Let A be a simplicial cochain algebra.

(i) For n ≥ 0 an isomorphism A(∆[n])∼=−→ An of cochain algebras is given by Φ 7−→ Φcn ,

where cn is the fundamental simplex of ∆[n].

(ii) If A is extendable and L ⊂ K is an inclusion of simplicial sets, then A(K)→ A(L)is surjective.

If A is extendable simplicial cochain algebra and L ⊂ K is an inclusion of simplicialsets we denote by A(K,L) the kernel of the surjective morphism A(K) → A(L). Thus,A(K,L) is a differential ideal in A(K) and

0→ A(K,L)→ A(K)→ A(L)→ 0

is a short exact sequence of cochain complexes, natural with respect to (K,L) and withrespect to the simplicial cochain algebra A.

A key step in the proof of Theorem 3.8 is the following

Proposition 3.5. Suppose θ : D → E is a morphism of simplicial cochain complexes.Assume that

(i) H(θn) : H(Dn)→ H(En) is an isomorphism, n ≥ 0.

(ii) D and E are extendable.

Then for all simplicial sets K,

H(θ(K)) : H(D(K))→ H(E(K))

is an isomorphism.

First we establish a preliminary lemma. Define

α : A(K(n),K(n− 1))→∏

σ∈NKn

A(∆[n], ∂∆[n]), n ≥ 0,

by setting α : Φ 7−→ A(σ∗)Φσ∈NKn , where σ∗ : ∆[n] → K is the unique simplicial map(Lemma 3.2) such that σ∗(cn) = σ.

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Lemma 3.6. If K is a simplicial set and A is extendable simplicial cochain complex thenα is an isomorphism, natural in A and in K.

proof of 3.5. The proof is divided into two steps:

• First we show that for any simplicial set K, θ(K(n)) is a quasi-isomorphims. Notethat for any inclusion L ⊂M of simplicial sets we have the row exact commutativediagram:

0 // D(M,L) //

θ(M,L)

D(M) //

θ(M)

D(L) //

θ(L)

0

0 // E(M,L) // E(M) // E(L) // 0

Hence, if any two of the vertical arrows is a quasi-isomorphism, so is the third.Moreover Proposition 3.4 identifies θ(∆[n]) : D(∆[n]) → E(∆[n]) with θn : Dn →En. Hence θ(∆[n]) is a quasi-isomorphism, n ≥ 0. From Lemma 3.6 and the remarksabove we show by induction that θ(−) is a quasi-isomorphism when (−) is in turngiven by:

∂∆[n], (∆[n], ∂∆[n]), (K(n),K(n− 1)) and K(n).

• By Lemma 3.2 in [1], to show θ(K) is a quasi-isomorphism, it is enough to provethat for any given Φ ∈ D(K) and Ψ ∈ E(K) such that dΦ = 0 and θ(K) = Φ = dΨ,there exist Ω ∈ D(K) and Γ ∈ E(K) such that Φ = dΩ and Ψ = θ(K)Ω + dΓ.

To find Ω and Γ we construct inductively a sequence Ωn ∈ D(K,K(n − 1)) andΓn ∈ E(K,K(n− 1)) such that

Φ−∑

i≤ndΩi ∈ D(K,K(n))

andΨ−

∑

i≤n(θ(K)Ωi + dΓi) ∈ E(K,K(n)).

Thus we may define Ω ∈ D(K) and Γ ∈ E(K) by Ωσ =∑n

(Ωn)σ and Γσ =∑n

(Γn)σ.

ClearlyΦ = dΩ and Ψ = θ(K)Ω + dΓ.

3.3 The simplicial commutative cochain algebra APL, and APL(X)

In this section, we construct the specific simplicial cochain algebra APL and show it isextendable. Applying the construction above, we obtain APL(X).

First consider the free graded commutative algebra Λ(t0, . . . , tn, y0, . . . , yn) in whichthe basis elements ti have degree zero and basis elements yj have degree 1.

Now define APL = (APL)nn≥0 by:

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• The cochain algebra (APL)n is given by

(APL)n =Λ(t0, . . . , tn, y0, . . . , yn)

(∑ti − 1,

∑yj)

;

dti = yi and dyj = 0.

• The face and degeneracy morphisms are the cochain algebra morphisms

∂i : (APL)n+1 → (APL)n and sj : (APL)n → (APL)n+1

satisfying

∂i : tk 7−→

tk, k < i0, k = itk−1, k > i

and

sj : tk 7−→

tk, k < jtk + tk+1, k = jtk+1, k > j.

Notice that the inclusions ti → (APL)n, yj → (APL)n extend to an isomorphism ofcochain algebras,

(Λ(t1, . . . , tn, y1, . . . , yn), d)∼=−→ (APL)n.

Lemma 3.7. (i) (APL)0 = k · 1.

(ii) H((APL)n) = k · 1, n ≥ 0.

(iii) Each ApPL is extendable.

The construction APL() defined in § 3.2 assigns to each simplicial set K a commutativecochain algebra, APL(K), and to each map f of simplicial sets a morphism, APL(f)of commutative cochain algebras. Since APL is extendable, with every pair L ⊂ K ofsimplicial sets is associated

0→ APL(K,L)→ APL(K)→ APL(L)→ 0.

For topological spaces X and continuous maps f we apply this construction to thesimplicial set S∗(X) and to S∗(f). This defines a contravariant functor from spaces tocommutative cochain algebras, which will be denoted

X APL(X) and f APL(f).

In particular, associated with a subspace Y is the short exact sequence

0→ APL(X,Y )→ APL(X)→ APL(Y )→ 0.

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3.4 The simplicial cochain algebra CPL, and the main theorem

In this section, we construct another simplicial cochain algebra CPL and show that for each

simplicial set K, there are natural isomorphisms CPL(K)∼=−→ C∗(K), where C∗(K) is the

“ordinary” cochain algebra of K. In particular, CPL(X)∼=−→ C∗(X) for each topological

space X. Applying Proposition 3.5, we prove the main theorem by showing the morphismsCPL(K)→ (CPL ⊗APL)(K)← APL(K) are quasi-isomorphisms.

First recall that every topological space X is associated with its singular cochain alge-bra, C∗(X, k) (k is an arbitrary field with character zero). We simplify notation and writeC∗(X) for C∗(X, k). The construction can be generalized to any simplicial set K to givethe cochain algebra C∗(K), also written C∗(K, k) if we need to emphasize coefficients.More precisely

• C∗(K) = Cp(K)p≥0;

• Cp(K) consists of the set maps Kp → k vanishing on degenerate simplices.

• For f ∈ Cp(K), g ∈ Cq(K) the product is given by

(f ∪ g)(σ) = (−1)pqf(∂p+1 . . . ∂p+qσ) · g(∂0∂0 . . . ∂0σ), σ ∈ Kp+q.

• The differential, d, is given by

(df)(σ) =

p+1∑

i=0

(−1)p+i+1f(∂iσ), σ ∈ Kp+1, f ∈ Cp(K).

Observe that, by definition, C∗(X) = C∗(S∗(X)) for topological spaces X.Next we define a simplicial cochain algebra CPL using the simplicial sets ∆[n] ⊂ S∗(∆n)

introduced in § 3.1. Indeed since the face inclusions and degeneracy maps for the ∆n havethe form

λi = ⟨e0, . . . ei . . . en+1⟩ : ∆n → ∆n+1

andρj = ⟨e0, . . . ej , ej , . . . en⟩ : ∆n+1 → ∆n,

it follows that S∗(λi) and S∗(ρj) restrict to simplicial maps

[λi] : ∆[n]→ ∆[n+ 1] and [ρj ] : ∆[n+ 1]→ ∆[n].

Thus we define CPL = (CPL)nn≥0 by

• (CPL)n is the cochain algebra C∗(∆[n]).

• The face and degeneracy morphisms are the C∗([λi]) and C∗([ρj ]).

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Finally, letCPL ⊗APL = (CPL)n ⊗ (APL)n; ∂i ⊗ ∂i; sj ⊗ sj

be the tensor product simplicial cochain algebra. Morphisms

CPL → (CPL ⊗APL)← APL

are defined by γ 7−→ γ⊗1 and Φ 7−→ 1⊗Φ. Thus, for any simplicial set K they determinethe natural cochain algebra morphisms

CPL(K)→ (CPL ⊗APL)(K)← APL(K).

Theorem 3.8. Let K be a simplical set. Then

(i) There is a natural isomorphism CPL(K)∼=−→ C∗(K) of cochain algebras.

(ii) The natural morphisms of cochain algebras,

CPL(K)→ (CPL ⊗APL)(K)← APL(K)

are quasi-isomorphisms.

Substituting S∗(X) = K in Theorem 3.8 we obtain

Corollary 3.9. For topological spaces X there are natural quasi-isomorphisms of cochainalgebras.

C∗(X)w−→ (CPL ⊗APL)(X)

w←− APL(X)

This gives the isomorphisms H∗(X) ∼= H(APL(X)).For the proof of Theorem 3.8 we require two lemmas.

Lemma 3.10. There are natural isomorphisms CPL(K)∼=−→ C∗(K).

Lemma 3.11. (i) H((CPL)n) = k = H((CPL)n ⊗ (APL)n), n ≥ 0.

(ii) CPL is extendable.

(iii) CPL ⊗APL is extendable.

Then the first assertion of Theorem 3.8 is Lemma 3.10. The second follows by applyingProposition 3.5 to the morphisms

θC : CPL → CPL ⊗APL and θA : APL → CPL ⊗APLgiven by γ 7−→ γ ⊗ 1 and Φ 7−→ 1⊗ Φ.

4 Sullivan model

In this chapter the ground ring is an arbitrary field k of characteristic zero.In § 4.1, the definitions of Sullivan algebra, Sullivan model and minimal Sullivan model

will be presented. And we will show the existence of these models under mild conditions.In § 4.2, the ideal “homotopy” is extended in Sullivan algebras and the homotopy relationbetween topological spaces will be “preserved” between their Sullivan models.

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4.1 Sullivan algebras and models: constructions and examples

Definition 4.1. A Sullivan algebra is a commutative cochain algebra of the form (ΛV, d),where

• V = V pp≥1 and, as usual, ΛV denotes the free graded commutative algebra on V ;

• V =∞∪k=0

V (k), where V (0) ⊂ V (1) ⊂ . . . is an increasing sequence of graded sub-

spaces such that

d = 0 in V (0) and d : V (k)→ ΛV (k − 1), k ≥ 1.

The second condition is called the nilpotence condition on d. It can be restated as:d preserves each ΛV (k), and there exist graded subspaces Vk ⊂ V (k) such that ΛV (k) =ΛV (k − 1)⊗ ΛVk, with d : Vk → ΛV (k − 1).

Definition 4.2. • A Sullivan model for a commutative cochain algebra (A, d) is aquasi-isomorphism

m : (ΛV, d)→ (A, d)

from a Sullivan algebra (ΛV, d).

• If X is a path connected topological space then a Sullivan model for APL(X),

m : (ΛV, d)∼=−→ APL(X),

is called a Sullivan model for X.

• A Sullivan algebra (or model), (ΛV, d) is called minimal if

Imd ⊂ Λ+V · Λ+V.

Our first step is to show the existence of Sullivan models:

Proposition 4.3. Any commutative cochain algebra (A, d) satisfying H0(A) = k has aSullivan model

m : (ΛV, d)∼=−→ (A, d).

Proof. We construct this so that V is the direct sum of graded subspaces Vk, k ≥ 0 with

d = 0 in V0 and d : Vk → Λ(k−1⊕i=0

Vi).

First, set V0 = H+(A) and extend the inclusion map V0 → H(A) to the gradedmorphism m0 : (ΛV0, 0)→ (A, d). Since H0(A) = k, H(m0) is surjective.

Suppose m0 has been extended to mk : (Λ(k⊕i=0

Vi), d) → (A, d). Let zα be cocycles in

Λ(k⊕i=0

Vi) such that [zα] is a basis for kerH(mk). Let Vk+1 be a graded space with basis

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vα in 1-1 correspondence with the zα, and with deg vα = deg zα − 1. Extend d to a

derivation in Λ(k+1⊕i=0

Vi) by setting dvα = zα. Since d has odd degree, d is a derivation of

Λ(i=k+1⊕i=0

Vi).

Since H(mk)[zα] = 0,mkzα = daα, aα ∈ A. Extend mk to a graded algebra mor-

phism mk+1 : Λ(k+1⊕i=0

Vi) → A by setting mk+1vα = aα.Then mk+1dvα = dmk+1vα, and so

mk+1d = dmk+1.

This completes the construction of m : (ΛV, d)→ (A, d) with V =∞⊕k=0

Vk and m|ΛVk=

mk. Since m|ΛV0 = m0, and H(m0) is surjective, H(m0) is surjective as well. If H(m)[z] =

0 then, since z is necessarily in some Λ(k⊕i=0

Vi), H(mk) = [z] = 0. By construction, z is a

coboundary in Λ(k+1⊕i=0

Vi). Thus H(m) is an isomorphism.

Next we show by induction on k that Vk is concentrated in degrees ≥ 1. This iscertainly true for k = 0, because V0

∼= H+(A). Assume it true for Vi, i ≤ k. Any element

in Λ(k⊕i=0

Vi) of degree 1 then has the form

v = v0 + . . .+ vk, vi ∈ V 1i .

Thus if dv = 0 then dvk ∈ d(Λ(k−1⊕i=0

Vi)). By construction, this implies vk = 0. Repeating

this argument we find v = v0 and H(mk)[v0] = H(m0)[v0] = 0, unless v0 = 0. ThuskerH(mk) vanishes in degree 1; i.e., it is concentrated in degrees ≥ 2. It follows that Vk+1

is concentrated in degrees ≥ 1.Finally, the nilpotence condition on d is built into the construction.

Example 4.4. The spheres, Sk.Let [Sk] ∈ Hk(S

k; Z). This determines a unique class ω ∈ Hk(APL(Sk)) such that⟨ω, [Sk]⟩ = 1, and 1, ω is a basis for H(APL(Sk)). Let Φ be a representative cocycle for ω.

Now if k is odd then a minimal Sullivan model for Sk is given by

m : (Λ(e), 0)w−→ APL(Sk), dege = k,me = Φ.

Suppose, on the other hand, that k is even. We may still define m : (Λ(e), 0) →APL(Sk) by dege = k,me = Φ. But now, because dege is even, Λ(e) has a basis 1, e, e2, e3, . . .and this morphism is not a quasi-isomorphism. However, Φ2 is certainly a coboundary.Write Φ2 = dΨ and extend m to

m : (Λ(e, e′), d)→ APL(Sk)

by setting dege′ = 2k − 1, de′ = e2 and me′ = Ψ. Thus this is a minimal model for Sk.

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Example 4.5. Products of topological spaces.Suppose mX : (ΛV, d) → APL(X) and mY : (ΛW,d) → APL(Y ) are Sullivan models

for path connected topological spaces X and Y . Assume further that the rational homologyof one of these spaces has finite type. Let pX : X × Y → X and pY : X × Y → Ybe the projections. Then APL(pX) · APL(pY ) : APL(X) ⊗ APL(Y ) → APL(X × Y ) is aquasi-isomorphism of cochain algebras.

Since APL(pX) ·APL(pY ) is a quasi-isomorphism so is

mX ·mY : (ΛV, d)⊗ (ΛW,d)w−→ APL(X × Y ),

where (mX ·mY )(a ⊗ b) = APL(pX)mXa · APL(pY )b. This exhibits (ΛV, d) ⊗ (ΛW,d) asa Sullivan model for X × Y . Furthermore, if (ΛV, d) and (ΛW,d) are minimal Sullivanmodels then so is their tensor product.

Example 4.6. H-spaces have minimal Sullivan models of the form (ΛV, 0).An H-spaces is a based topological space (X, ∗) together with a continuous map µ :

X ×X → X such that the self maps x 7−→ µ(x, ∗) and x 7−→ µ(∗, x) of X are homotopicto the identity. Also assume X is path connected and H∗(X; k) has finite type. We claimthat the minimal Sullivan model of X is of the form (ΛV, 0), where V is a graded spacewith a basis 1-1 correspondence to the generators of H+(X; k).

To see this, observe first that because H∗(X; k) has finite type, H∗(µ) can be identifiedas a morphism of graded algebras,

H∗(µ) : H∗(X; k)→ H∗(X; k)⊗H∗(X; k).

Moreover, the conditions µ(x, ∗) v id,µ(∗, x) v id imply that for h ∈ H+(X; k),

H∗(µ)h = h⊗ 1 + Φ + 1⊗ h, some Φ ∈ H+(X; k)⊗H+(X; k).

This implies that H∗(X; k) is a Hopf algebra. Let V be a graded with basis vi which is1-1 correspondence to the generators of H+(X; k). Then by Theorem 3C.4 in [3], there isan isomorphism between algebras

φ : ΛVw−→ H∗(X; k)

by mapping vi to the corresponding generator of H∗(X; k).Finally, let wi ∈ APL(X) be cocycles representing the cohomology classes which are the

images of vi in the above morphism. The correspondence vi 7−→ wi defines a linear mapV → APL(X) which extends to a unique morphism m : (ΛV, 0)→ APL(X). Since φ is anisomorphism it follows that m is a quasi-isomorphism:

m : (ΛV, 0)v−→ APL(X)

Thus m is a minimal Sullivan model for the H-space X.

Again, suppose (A, d) is a commutative cochain algebra. In fact if H0(A) = k then(A, d) has a (unique) minimal Sullivan model; this follows from a more general result aboutrelative Sullivan algebras. However, if also H1(A) = 0 then there is a simple inductiveconstruction for a minimal model. We carry this out here.

Thus suppose given (A, d) with H0(A) = k and H1(A) = 0.

154

• Choose m2 : (ΛV 2, 0)→ (A, d) so that H2(m2) : V 2∼=−→ H2(A). Note that H1(m2) is

an isomorphism because H1(A) = 0 and that H3(m2) is injective because (ΛV 2)3 =0.

• Supposing that mk : (ΛV ≤k, d) → (A, d) is constructed, we extend to mk+1 :(ΛV ≤k+1, d)→ (A, d) by the following procedure.

Choose cocycles aα ∈ Ak+1 and zβ ∈ (ΛV ≤k)k+2 so that

Hk+1(A) = ImHk+1(mk)⊕⊕

α

k · [aα] and kerHk+2(mk) =⊕

β

k · [zβ ].

In particular, mkzβ = dbβ , some bβ ∈ A.

Let V k+1 be a vector space (in degree k+1) with basis v′α, v

′′β in 1-1 correspondence

with the elements aα, zβ. Write ΛV ≤k+1 = ΛV ≤k ⊗ ΛV k+1. Extend d and mk,respectively, to a derivation in ΛV ≤k+1 and to a morphism mk+1 : ΛV k+1 → A ofgraded algebras, by setting

dv′α = 0, dv′′

β = zβ , mv′α = aα, mv′′

β = bβ.

Since d has degree 1, d2 is a derivation. By construction, d2 = 0 in V k+1 and inΛV ≤k. Thus d2 = 0. In the same way mk+1d = dmk+1 in V k+1 and ΛV ≤k, and somk+1d = dmk+1.

Proposition 4.7. Suppose (A, d) is a commutative cochain algebra such that H0(A) = kand H1(A) = 0. Then

(i) The morphism m : (ΛV, d)→ (A, d) constructed above is a minimal Sullivan model.

(ii) If r is the least integer greater than zero such that Hr(A) = 0, then V i = 0, 1 ≤ i < rand

Hr(m) : V r ∼=−→ Hr(A).

(iii) If dimHk(A) <∞, k ≥ 1, then dimV k <∞,k ≥ 1.

Corollary 4.8. Let X be a simply connected topological space such that each Hi(X; Q) isfinite dimensional. Then X has a minimal Sullivan model

m : (ΛV, d)w−→ APL(X)

such that V = V ii≥2 and each V i is finite dimensional.

4.2 Homotopy in Sullivan algebras

First of all, we introduce the definition of homotopy in Sullivan algebras. Define augmen-tations

ε0, ε1 : Λ(t, dt)→ k by ε0(t) = 0, ε1(t) = 1,

where Λ(t, dt) is the free commutative graded algebra on the basis t, dt with deg t =0,deg t = 1, and d is the differential sending t 7−→ dt.

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Definition 4.9. Two morphisms φ0, φ1 : (ΛV, d) → (A, d) from a Sullivan algebra to acommutative cochain algebra are homotopic if there is a morphism

Φ : (ΛV, d)→ (A, d)⊗ (Λ(t, dt), d)

such that (id · ϵi)Φ = φi, i = 0, 1. Here Φ is called a homotopy from φ0 to φ1 and we writeφ0 v φ1.

Observe that Λ(t, dt) is precisely the cochain algebra (APL)1 and by Proposition 3.4(i)is the commutative cochain algebra of the standard 1-simplex. Given that APL is a con-travariant functor, homotopy in Sullivan algebras is the obvious analogue of the topologicalhomotopy X ← X × I.

In fact, APL ‘preserves homotopy’ in the following sense. Let X and Y be twotopological spaces. Suppose given continuous maps f0, f1 : X → Y and a morphismψ : (ΛV, d)→ APL(Y ) from a Sullivan algebra (ΛV, d).

Proposition 4.10. If f0 v f1 : X → Y then APL(f0)ψ v APL(f1)ψ : (ΛV, d)→ APL(X).

Thus the proposition establish an injection:

rational homotopy type

between path connected spaces

→

homotopy classes of maps

between Sullivan algebras

The proof of the proposition is based on the following two lemmas, which are importanttricks in the latter proofs. The first one is the lifting property. Consider a diagram ofcommutative cochain algebra morphisms

(A, d)

w η

(ΛV, d)

ψ// (C, d)

in which η is a surjective quasi-isomorphism, and (ΛV, d) is a Sullivan algebra.

Lemma 4.11. (Lifting lemma) There is a morphism φ : (ΛV, d) → (A, d) such thatηφ = ψ (φ is a lift of ψ through η).

Second, with a graded vector space U = U ii≥0 associate the commutative cochainalgebra (E(U), δ),defined by

E(U) = Λ(U ⊕ δU) and δ : U∼=−→ δU.

It is augmented by ε : (E(U), δ)→ k, where ε(U) = 0. Note that if U = U ii≥1 then thisis a Sullivan algebra. Cochain algebras of this form are called contractible.

Lemma 4.12. ε : (E(U), δ)→ k is a quasi-isomorphism; i.e.

H(E(U), δ) = k.

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As a direct consequence of this lemma we obtain:The surjective trick: If (A, d) is any commutative cochain algebra, then the iden-

tity of A extends uniquely to a surjective morphism σ : (E(A), δ) → (A, d). Thus anymorphism φ : (B, d)→ (A, d) of commutative cochain algebras factors as

(B, d)λ−→w

(B, d)⊗ (E(A), δ)φ·σ−−→ (A, d)

in which the inclusion λ : b 7−→ b⊗ 1 is a quasi-isomorphism and φ · σ is surjective.

proof of Proposition 4.10. Identify Λ(t, dt) as a subcochain algebra of APL(I), bymapping t 7−→ u ∈ A0

PL(I), where u restricts to 0 at 0 and to 1 at 1 (the existence ofthe required u follows from the fact that ApPL is extendable). Denote by j0, j1 : X → X×Ithe inclusions at the endpoints and by pX : X×I → X and pI : X×I → I the projections.Then

APL(X)⊗ Λ(t, dt)

APL(pX)·APL(pI)

(id·ε0,id·ε1)

**TTTTTTTTTTTTTTT

APL(X)×APL(X)

APL(X × I)(APL(j0),APL(j1))

44jjjjjjjjjjjjjjj

is a commutative diagram of cochain algebra morphisms.Since H(APL(pX)) = H∗(pX ; k) is an isomorphism, APL(pX) · APL(pI) is a quasi-

isomorphism. We now make it surjective. Let U ⊂ APL(X × I) be the kernel of(APL(j0), APL(j1)). The inclusion of U extends to a unique cochain algebra morphism

ρ : (E(U), δ)→ APL(X × I).Extend the diagram above to the commutative diagram

APL(X)⊗ Λ(t, dt)⊗ (E(U), δ)

APL(pX)·APL(pI)·ρ

(id·ε0·ε,id·ε1·ε)

++VVVVVVVVVVVVVVVVVVV

APL(X)×APL(X)

APL(X × I)(APL(j0),APL(j1))

33hhhhhhhhhhhhhhhhhhh

Here APL(pX) · APL(pI) · ρ is obviously, surjective, and it follows from Lemma 4.12 thatit is a quasi-isomorphism too.

Let H : X × I → Y be a homotopy from f0 to f1. Use Lemma 4.11 to lift APL(H)ψ :(ΛV, d) → APL(X × I) through the surjective quasi-isomorphism APL(pX) · APL(pI) · ρ.This produces a morphism

Ψ : (ΛV, d)→ APL(X)⊗ Λ(t, dt)⊗ E(U).

Then set Φ = (id⊗ id⊗ ε)Ψ; it is the desired homotopy from APL(f0)ψ to APL(f1)ψ.

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As with homotopy of continuous maps, we have

Proposition 4.13. Homotopy is an equivalence relation in the set of morphisms φ :(ΛV, d)→ (A, d) from a Sullivan algebra.

Notation The set of homotopy classes of morphisms (ΛV, d)→ (A, d) will be denotedby [(ΛV, d), (A, d)]. The homotopy class of a morphism φ will be denoted by [φ].

Definition 4.14. The linear part of a morphism φ : (ΛV, d)→ (ΛW,d) between Sullivanalgebras is the linear map

Qφ : V →W

defined by:φv −Qφv ∈ Λ≥2W, v ∈ V .

Note that Q(φ) commutes with the linear parts of the differentials (denoted by d0):Q(φ)d0 = d0Q(φ). The next proposition illustrates a property of Qφ which will be usedin studying the relation between Sullivan models of the homotopy groups.

Proposition 4.15. If φ0 v φ1 : (ΛV, d) → (ΛW,d) are homotopic morphisms betweenminimal Sullivan algebras, and if H1(ΛV, d) = 0, then Qφ0 = Qφ1.

4.3 Quasi-isomorphisms, Sullivan representatives, uniqueness of mini-mal models

Suppose η : (A, d) → (C, d) is a morphism of commutative cochain algebras. If Φ ishomotopy between φ0, φ1 : (ΛV, d) → (A, d) then (η ⊗ id)Φ : ηφ0 v ηφ1. Thus we candefine

η# : [(ΛV, d), (A, d)]→ [(ΛV, d), (C, d)] by η#[φ] = [ηφ].

Now suppose given commutative cochain algebra morphisms

(A, d)

wη

(ΛV, d)

ψ// (C, d)

in which η is a (not necessarily surjective) quasi-isomorphism and (ΛV, d) is a Sullivanalgebra. The Lifting lemma extends to the fundamental

Proposition 4.16. There is a unique homotopy class of morphism φ : (ΛV, d) → (A, d)such that ηφ v ψ. Thus

η# : [(ΛV, d), (A, d)]∼=−→ [(ΛV, d), (C, d)]

is a bijection.

158

Proof. • First prove the assertion under the additional hypothesis that η is surjective.In this case the surjectivity of η# follows from the Lifting lemma 4.11.

To show η# is injective, suppress the differentials from the notation (for simplicity).

Use the morphisms C⊗Λ(t, dt)(idC ·ε0,idC ·ε1)−−−−−−−−−→ C×C η×η←−− A×A to construct a fibre

product, and observe that

(η ⊗ id, idA · ε0, idA · ε1) : A⊗ Λ(t, dt)→ [C ⊗ Λ(t, dt)]×C×C (A×A)

is a surjective quasi-isomorphism of cochain algebras.

Now suppose γ0, γ1 : ΛV → A are cochain algebra morphisms, and that Ψ is ahomotopy from ηγ0 to ηγ1. Lift the morphism

(Ψ, γ0, γ1) : ΛV → [C ⊗ Λ(t, dt)]×C×C (A×A)

through the surjective quasi-isomorphism (η ⊗ id, idA · ε0, idA · ε1). This defines amorphism Φ : ΛV → A ⊗ Λ(t, dt), and it is immediate from the construction thatΦ : γ0 v γ1.

• As for the general case, let E(C, δ) be the acyclic cochain algebra of Lemma 4.12and ρ : E(C)→ C is defined by c 7−→ c, δc 7−→ dc. Consider the morphisms

(A, d)λ−→ (A, d)⊗ (E(C), δ)

η·ρ−−→ (C, d).

Supposeα : (A, d)→ (A′, d)

is an arbitrary morphism of commutative cochain algebras that satisfy H0(−) = k. Let

m : (ΛV, d)w−→ (A, d) and m′ : (ΛV ′, d)→ (A′, d) be Sullivan models. By Proposition 4.16

there is a unique homotopy class of morphisms

φ : (ΛV, d)→ (ΛV ′, d)

such that m′φ v αm.

Definition 4.17. A morphism φ : (ΛV, d) → (ΛV ′, d) such that m′φ v αm is calleda Sullivan representative for α. If f : X → Y is a continuous map then a Sullivanrepresentative of APL(f) is called a Sullivan representative of f .

It follows at once from Proposition 4.10 that Sullivan representatives of homotopicmaps are homotopic morphisms.

As a second application of Proposition 4.16 we deduce the uniqueness of minimalmodels in the simply connected case.

Proposition 4.18. If (A, d) is a commutative cochain algebra and H0(A) = k, H1(A) = 0,then the minimal models of (A, d) are all isomorphic.

Thus we establish an injection

rational homotopy type

of simply connected spaces

→

isomorphic classes of

minimal Sullivan algebras

159

5 Relative Sullivan algebras

In this chapter we will generalize the Sullivan algebras to the relative Sullivan algebrasand extend the Lifting lemma to the relative case.

5.1 The existence of relative Sullivan models

Definition 5.1. A relative Sullivan algebra is a commutative cochain algebra of the form(B ⊗ ΛV, d) (identify B = B ⊗ 1 and ΛV = 1⊗ ΛV for simplicity) where

• (B, d) = (B ⊗ 1, d) is a sub cochain algebra, and H0(B) = k.

• 1⊗ V = V = V pp≥1.

• V =∞∪k=0

V (k), where V (0) ⊂ V (1) ⊂ . . . is an increasing sequence of graded sub-

spaces such that

d : V (0)→ B and d : V (k)→ B ⊗ ΛV (k − 1), k ≥ 1.

The third condition is called the nilpotence condition on d. It can be restated asfollows: if we put V (−1) = 0 and write V (k) = V (k − 1) ⊕ Vk for some graded subspaceVk then

B ⊗ ΛV (k) = [B ⊗ ΛV (k − 1)]⊗ ΛVk, and d : Vk → B ⊗ ΛV (k − 1), k ≥ 0.

Note that a Sullivan algebra is just a relative Sullivan algebra with B = k. Nowgeneralizing the Sullivan models, we consider morphisms of commutative cochain algebras

φ : (B, d)→ (C, d)

such that H0(B) = k and make the

Definition 5.2. A Sullivan model for φ is a quasi-isomorphism of cochain algebras

m : (B ⊗ ΛV, d)w−→ (C, d)

such that (B ⊗ ΛV, d) is a relative Sullivan algebra with base (B, d) and m|B = φ.

If f : X → Y is a continuous map then a Sullivan model for APL(f) is called a Sullivanmodel for f .

As in the case of Sullivan algebras, we define the minimal Sullivan algebras:

Definition 5.3. A relative Sullivan algebra (B ⊗ ΛV, d) is minimal if

Imd ⊂ B+ ⊗ ΛV +B ⊗ Λ≥2V.

160

A minimal Sullivan model for φ : (B, d) → (C, d) is a Sullivan model (B ⊗ ΛV, d)w−→

(C, d) such that (B ⊗ ΛV, d) is minimal.A useful fact for the relative Sullivan algebras is the ‘preservation under pushout’.

Suppose (B ⊗ ΛV, d) is a relative Sullivan algebra and ψ : (B, d)→ (B′, d) is a morphismof commutative cochain algebras with H0(B′) = k. Then, immediately from the definition,the cochain algebra

(B′, d)⊗(B,d) (B ⊗ ΛV, d) = (B′ ⊗ ΛV, d)

is a relative Sullivan algebra. It is called the pushout of (B⊗ΛV, d) along ψ. We have theresult:

Lemma 5.4. If ψ is a quasi-isomorphism so is ψ ⊗ id : (B ⊗ ΛV, d)→ (B′ ⊗ ΛV, d).

Now we can establish the existence of Sullivan models for morphisms:

Proposition 5.5. A morphism φ : (B, d) → (C, d) of commutative cochain algebras hasa Sullivan model if H0(B) = k = H0(C) and H1(φ) is injective.

Proof. Choose a grade subspace B1 ⊂ B so that

(B1)0 = k, (B1)

1 ⊕ d(B0) = B1 and (B1)n = Bn, n ≥ 2.

Clearly (B1, d) is a sub cochain algebra and the inclusion φ : (B1, d) → (B, d) is a quasi-

isomorphism. In particular the restriction φ1 : (B1⊗ΛV, d)w−→ (C, d) of φ satisfies: H1(φ1)

is injective.Now because (B1)

0 = k the argument of Proposition 4.3 applies verbatim to show the

existence of a Sullivan model m1 : (B ⊗ ΛV, d)w−→ (C, d) for φ1. Thus a commutative

diagram of cochain algebra morphisms is given by

(B, d)⊗(B1,d) (B1 ⊗ ΛV, d) m // (C, d)

(B1 ⊗ ΛV, d)

j

jjTTTTTTTTTTTTTTT m1

88qqqqqqqqqqq

in which j(z) = 1⊗B1 z.This may be rewritten as

(B ⊗ ΛV, d)m // (C, d)

(B1 ⊗ ΛV, d)

i⊗id

hhPPPPPPPPPPPP m1

w88qqqqqqqqqq

Lemma 5.4 and the preceding remarks show that i⊗ id is a quasi-isomorphism and (B ⊗ΛV, d) is a Sullivan algebra; hence m : (B⊗ΛV, d)

w−→ (C, d) is a Sullivan model for φ.

161

Finally, we extend the lifting lemma to the relative case and define relative homotopy.To begin, suppose given a commutative diagram of morphisms of commutative cochainalgebras

(B, d)α //

i

(A, d)

w η

(B ⊗ ΛV, d)

ψ// (C, d)

in which i is the base inclusion of a relative Sullivan algebra and η is a surjective quasi-isomorphism. An argument identical to that of Lemma 4.11 establishes

Lemma 5.6. There is a morphism φ : (B ⊗ΛV, d)→ (A, d) such that φi = α (φ extendsα) and ηφ = ψ (φ lifts ψ).

6 Fibrations and homotopy groups

In § 6.1 we will first construct models of fibrations, which provide a method to computethe Sullivan models of fibres using the Sullivan models of total spaces and base spaces.

SupposeX is a simply connected space with a minimal Sullivan modelmX : (ΛVX , d)→APL(X). Using models of fibrations, a bijection between VX and HomZ(π∗(X),k) will beestablished under mild conditions.§ 6.3 studies the relationship among the minimal models of fibres, total spaces, and

base spaces and a long exact sequence among them will be established.§ 6.4 constructs models for path fibrations.

6.1 Models of fibrations

Consider a Serre fibration of path connected spaces

p : X → Y,

whose fibres are also path connected. Let j : F → X be the inclusion of the fibre aty0 ∈ Y . Then APL converts the diagram

Fj //

X

p

y0 // Y

to

APL(F ) APL(X)APL(j)oo

k

OO

APL(Y ),εoo

APL(p)

OO(6.1)

where ε is the augmentation corresponding to y0.

162

Observe that H1(APL(p)) is injective. Then proposition 5.5 asserts the existence of aSullivan model for p,

m : (APL(Y )⊗ ΛV, d)w−→ APL(X).

The augmentation ε : APL(Y )→ k defines a quotient Sullivan algebra

(ΛV, d) = k⊗APL(Y ) (APL(Y )⊗ ΛV, d),

which is called the fibre of the model at y0.Since APL(j)APL(p) reduces to ε in APL(Y ), APL(j)m factors over ε · id to give the

commutative diagram of cochain algebra morphisms

APL(F ) APL(X)APL(j)oo

(ΛV, d)

m

OO

(APL(Y )⊗ ΛV, d)ε·id

oo

m

OO(6.2)

We show now that, under mild hypotheses, m is a quasi-isomorphism. Thus in this casem : (ΛV, d)

w−→ APL(F ) is a Sullivan model for F : the fibre of a model is a model of thefibre.

Theorem 6.1. Suppose Y is simply connected and one of the graded spaces H∗(Y ; k),H∗(F ; k)has finite type. Then

m : (ΛV, d)→ APL(F )

is a quasi-isomorphism.

Proof. • First suppose that p is a fibration. Apply Theorem 7.1 in [1] with diagram(6.2) corresponding to diagram (7.9) in [1].

• Suppose only that p is a Serre fibration. Using the diagram constructed in § 2.1

Xλ //

p

X ×Y MY

qvvmmmmmmmmmmmmmmm

Y

with λx = (x, const. path at px), we apply the Lifting lemma 4.11 to the diagram

APL(Y )APL(q) //

APL(X ×Y MY )

w APL(λ)

(APL(Y )⊗ ΛV, d)wm

// APL(X)

to construct the quasi-isomorphism n extending APL(q). Then n : (ΛV, d) →APL(X ×Y PY ) is a quasi-isomorphism by the argument for fibrations, and m isthe quasi-isomorphism APL(λ)n.

163

Theorem 6.1 establishes a fundamental property for relative Sullivan models (APL(Y )⊗ΛV, d)

w−→ APL(X). It is, however useful to replace APL(Y ) by a Sullivan model and thisis done as follows.

Choose a Sullivan model mY : (ΛVY , d)w−→ APL(Y ). Since VY = V i

Y i≥1 by definition,there is a unique augmentation ε : (ΛVY , d) → k. Construct a commutative diagram ofcochain algebra morphisms

APL(Y )APL(p) // APL(X)

APL(j) // APL(F )

(ΛVY , d)

mY wOO

i// (ΛVY ⊗ ΛV, d)

w m

OO

ε·id// (ΛV, d)

m

OO(6.3)

by requiring that

• i is the inclusion of a relative Sullivan algebra, and

• m is a Sullivan model for the composite APL(p)mY .

Then APL(j)m factors over ε · id to yield m.As in Theorem 6.1, we now restrict to the case that Y is simply connected and one of

H∗(F ; k),H∗(Y ; k) has finite type. With these hypotheses we have

Proposition 6.2. The three morphisms mY ,m and m in (6.3) are all Sullivan models.

Example 6.3. The model of an Eilenberg-MacLane space.Let X be an Eilenberg-MacLane space of type (π, n), n ≥ 1. Assume π is abelian

(this is automatic for n > 1) and set V n =Hom(π,k). The Hurewicz theorem asserts

that there is a natural isomorphism π∼=−→ Hn(X,Z). This extends to an isomorphism

π ⊗Z k∼=−→ Hn(X; k), which dualizes to an isomorphism V n

∼=←− Hn(X; k). Suppose π⊗Z kis finite dimensional. Then we claim that the minimal Sullivan model of X is given by

m : (ΛV n, 0)w−→ APL(X).

§15 in [1] provides a proof of the assertion.

6.2 Homotopy groups

Suppose f : (Y, ∗) → (X, ∗) is a continuous map between simply connected topologicalspaces. Recall that a choice of minimal Sullivan models,

mX : (ΛV, d)w−→ APL(X) and mY : (ΛW,d)

w−→ APL(Y )

determines a unique homotopy class of morphism φf : (ΛV, d) → (ΛW,d) such thatAPL(f)mX v mY φf . Moreover, Proposition 4.15 asserts that these Sullivan representa-tives of f all have the same linear part: Q(φf ) is independent of the choice of φf . Thiswill therefore be denoted by

Q(f) : V →W.

164

Note that (Proposition 4.10) Q(f) only depends on the homotopy class of f .We use this construction to define a natural pairing, ⟨; ⟩ between V and π∗(X), de-

pending only on the choice of mX : (ΛV, d)→ APL(X).First, recall the minimal Sullivan models mk : (Λ(e), 0) → APL(Sk), k odd, and mk :

(Λ(e, e′), de′ = e2) → APL(Sk), k even, as constructed in Example 4.4. Now supposeα ∈ πk(X) is represented by a : (Sk, ∗)→ (X, ∗). Then Q(a) : V k → k · e depends only onα and the choice of the morphism mX : (ΛV, d)→ APL(X). Define the pairing

⟨−;−⟩ : V × π∗(X)→ k (6.4)

by the equations

⟨v;α⟩e =

Q(a)v, if v ∈ V k;

0, if degv = degα.

It is immediate from the definition that for f : (Y, ∗)→ (X, ∗) as above,

⟨Q(f)v;β⟩ = ⟨v;π∗(f)β⟩, v ∈ v, β ∈ π∗(Y ). (6.5)

In [1], Lemma 13.11 shows that the map ⟨−;−⟩ is bilinear.Thus the bilinear map ⟨−;−⟩ determines the linear map

νX : VX → HomZ(π∗(X),k)

given by νX(v)(α) = ⟨v;α⟩. It follows from (6.5) that νX is a natural transformation:

νX Q(f) = HomZ(π∗(f),k) νY .Theorem 6.4. Suppose X is simply connected and H∗(X; k) has finite type. Then thebilinear map VX × π∗(X)→ k is non-degenerate. Equivalently,

νX : VX∼=−→ HomZ(π∗(X),k)

is an isomorphism.

Remark 6.5. It follows from Theorem 6.4 that ifH∗(X,Q) has finite type so does π∗(X)⊗Q,since in this case the Sullivan model has finite type. In fact, the converse is true, i.e., ifπ ⊗Q has finite type then H∗(X; Q) also finite type.

Example 6.6. Rational Homotopy groups of spheres.Let ι ∈ πn(Sn) be the class represented by the identity map of Sn. Then

πn(S2k+1)⊗Q =

Q · ι, n = 2k + 1

0, otherwise

and

πn(S2k)⊗Q =

Q · ι, n = 2k

Q · [ι, ι]W , n = 4k − 1

0, otherwise.

Indeed in the first case the minimal model is (Λ(e), 0) and ⟨e; ι⟩ = 1. In the second theminimal model is (Λ(e, e′), de′ = e2). Again ⟨e; ι⟩ = 1 while Proposition 13.6 in [1] gives⟨e′; [ι, ι]W ⟩ = −⟨e2; ι, ι⟩ = −2. Now apply Theorem 6.4.

165

6.3 The long exact homotopy sequence

Let p : X → Y be a Serre fibration of path connected spaces, and with path connectedfibre j : F → X. Assume that Y is simply connected and that all spaces have homologyH∗(−; k) of finite type. In Section 6.1 we constructed the diagram, labeled (6.3):

APL(Y )APL(p) // APL(X)

APL(j) // APL(F )

(ΛVY , d)

mY wOO

i// (ΛVY ⊗ ΛV, d)

w m

OO

ε·id// (ΛV, d)

m

OO

in which all three vertical morphisms are Sullivan models. Moreover we can (and dohere) take (ΛVY , d) and (ΛV, d) to be minimal. However, the central Sullivan algebra,(Λ(VY ⊕ V ), d) need not be minimal. Let d0 be the linear part of d: it is the differentialin V ⊕ VY defined to by dx− d0x ∈ Λ≥2(V ⊕ VY ), x ∈ V ⊕ VY . Then i and ε · id restrictto a short exact sequence of cochain complexes,

0→ (VY , 0)i−→ (VY ⊕ V, d0)

ε·id−−→ (V, 0)→ 0. (6.6)

Recall now from Theorem 14.9 in [1](applied with B = k) that (Λ(VY ⊕ V ), d) ∼=(ΛW,d) ⊗ (Λ(U ⊕ dU), d) with (ΛW,d) a minimal Sullivan algebra and Λ(U ⊕ dU) con-

tractible. This defines a quasi-isomorphism λ : (ΛW,d)w−→ (Λ(VY ⊕V ), d) and mX = mλ :

(ΛW,d)w−→ APL(X) is its minimal Sullivan model. Moreover (Proposition 14.13 in [1]),

the linear part of λ induces an isomorphism W∼=−→ H(VY ⊕ V, d0). Use this to replace

H(VY ⊕V, d0) by W is the long exact cohomology sequence of (6.6), which takes the form

. . .→ V kY →W k → V k d0−→ V k+1

Y → . . . . (6.7)

Apply the long exact homotopy sequence to a topological group. Suppose G is a pathconnected topological group with finite dimensional rational homology. (This holds for allLie groups.) As observed in Example 4.6, the minimal Sullivan model of G has the form

mG : (ΛPG, 0)w−→ APL(G),

where PG is a finite dimensional graded vector space concentrated in odd degrees.Now consider the principal bundle pG : EG → BG constructed in § 2.2. It determines

a long exact sequence

→ V kBG→W k → P kG

d0−→ V k+1BG→

connecting the generating spaces for minimal Sullivan models of BG, EG and G. But

H∗(EG) = k and so W = 0. Thus d0 : P ∗G

∼=−→ V ∗+1BG

. It follows that VBGis finite

dimensional and concentrated in even degrees. Thus ΛVBGis concentrated in even degrees

and so the differential must be zero. We thus established

166

Proposition 6.7. The minimal Sullivan model for BG has the form

mBG: (ΛVBG

, 0)w−→ APL(BG),

where V ∗BG

∼= P ∗+1G . In particular, H∗(BG) is the finitely generated polynomial algebra,

ΛVBG.

Since G is weakly equivalent to ΩBG, BG is simply connected. Since (ΛVBG, 0) is the

Sullivan model of BG, it follows from Theorem 6.4 that VBG∼= HomZ(π∗(BG),k). Thus

π∗(BG)⊗Q is finite dimensional and concentrated in even degrees.

6.4 The minimal Sullivan model of the path space fibration

Suppose (X,x0) is a based topological space, X is simply connected and H∗(X; k) hasfinite type. Let mX : (ΛVX , d)→ APL(X) be a minimal Sullivan model for X. Then thepath space, PX, has a Sullivan model of the form

m : (ΛVX ⊗ ΛV, d)w−→ APL(PX).

It factors to give a minimal Sullivan model m : (ΛV, d) → APL(ΩX) for the loop spaceΩX (Corollary to Proposition 15.5 in [1] applied to the path space fibration PX → X).Recall that the linear part of d is the differential d0 in VX ⊕ V defined by requiring(d − d0)x ∈ Λ≥2(VX ⊕ V ). Because of the minimality of (ΛVX , d) and (ΛV, d) we haved0 = 0 in VX and d0 : V → VX .

Now observe thatd0 : V

∼=−→ VX .

Indeed, (ΛVX⊕ΛV, d) is the tensor product of a contractible algebra and a minimal modelfor PX (Theorem 14.9 in [1]). Because PX is contractible the minimal model is trivialand (ΛVX ⊗ ΛV, d) itself is contractible. This implies H(VX ⊕ V, d0) = 0 and hence that

d0 : V∼=−→ VX .

Next, recall that VX is a graded vector space of finite type (Proposition 4.7). Henceso is H∗(ΩX; k) ∼= H(ΛV, d). Since ΩX is an H-space it follows (Example 4.6) that thedifferential d is zero,

m : (ΛV, 0)w−→ APL(ΩX).

7 The minimal Sullivn model of Ω(SU(n + 1)/T n)

First we use models of fibrations to calculate the models of some matrix Lie groups.

Example 7.1. Matrix Lie groups.The linear action of SO(n) in Rn restricts to a transitive action on Sn−1 which iden-

tifies SO(n)/SO(n− 1)w−→ Sn−1, i.e. it gives principal bundles:

SO(n− 1)→ SO(n)→ Sn−1.

167

In particular, SO(2) ∼= S1. Thus, the Sullivan model for S1 is also the Sullivan model forSO(2). We claim that the Sullivan models of SO(n) are:

SO(2n+ 1) : Λ(x1, . . . , xn), degxi = 4i− 1.

SO(2n) : Λ(x1, . . . , xn−1, x′n), degxi = 4i− 1, degx′

n = 2n− 1.

The principal bundle gives the following long exact sequence:

. . .→ πk(SO(n− 1))→ πk(SO(n))→ πk(Sn−1)→ πk−1(SO(n− 1))→ . . . .

Since Q is flat, the above exactness is preserved when the above sequence is tensor with Q.As remarked above, for any connected Lie group G, it has Sullivan model of the form

(ΛPG, 0), where PG is concentrated in odd degrees, and PG ∼= π∗(G)⊗Q as graded vectorspaces (since G is weakly equivalent to ΩBG). Also apply the calculation of π∗(Sk) ⊗ Qin Example 6.6 to the long exact homotopy sequence, the claim can be proved by an easyinductive calculation.

Similarly, using fibrations:

SU(n− 1)→ SU(n)→ S2n−1

andU(n− 1)→ U(n)→ S2n−1

we can prove that the minimal models for SU(n) and U(n) are respectively:

SU(n) : Λ(x2, . . . , xn), degxi = 2i− 1

andU(n) : Λ(x1, x2, . . . , xn), degxi = 2i− 1.

Let Tn be the maximal torus in SU(n+ 1). Because Tn ∼= U(1)n, it follows from themodels of product spaces (Example 4.5) and the models of U(1) (Example 7.1) that theminimal model of Tn is (Λ(x1, . . . , xn), 0), deg xi = 1. The minimal model of SU(n + 1)is (Λ(u3, . . . , u2n+1), 0), deg ui = i. Since SU(n + 1)/Tn is simply connected and allthe spaces have homology H∗(−; k) of finite type, the long exact sequence (6.7) can beapplied to the fibration Tn → SU(n + 1) → SU(n + 1)/Tn. Thus the minimal model ofSU(n+ 1)/Tn has the form (Λ(y1, . . . , yn)⊗Λ(v1, . . . , vn), d), deg yi = 2, deg vj = 2j+ 1.

Then apply the minimal Sullivan model of path fibration to Ω(SU(n + 1)/Tn) →P (SU(n+ 1)/Tn)→ SU(n+ 1)/Tn. The minimal model of Ω(SU(n+ 1)/Tn) is

m : (Λ(a1, . . . , an)⊗ Λ(b1, . . . , bn), 0)→ APL(Ω(SU(n+ 1)/Tn)),

where deg ai = 1, and deg bj = 2j.

168

References

[1] Yves Felix, Stephen Halperin, and Jean-Claude Thomas. Rational homotopy theory,volume 205 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001.

[2] Jelena Grbic and Svjetlana Terzic. The integral Pontrjagin homology of the based loopspace on a flag manifold. Osaka J. Math., 47(2):439–460, 2010.

[3] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.

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毕业感言

朱艺航∗

就要毕业了，大学四年如梦幻一般，眨眼而过。这里面有太多的欢笑，忧伤，希望，

和颓唐。我还记得每年拿到《荷思》的毕业特刊时读里面的毕业感言，然后在心里妄加臧

否；现在到了我自己写毕业感言的时候了。心头的思绪太多，反而不知道从何处说起。现

在大学四年的无数个场景，在我眼前浮现。我还记得自己在新馆的英文数学区流连忘返，

赞叹于这浩瀚的知识海洋；在系图书馆的午后的炽热的阳光下闻兰花和老旧的书散发的香

味；在五教的大阶梯教室里复习数分时演算场论的公式；在有一个学期拿到数分的第一名

后得到的姚家燕老师的赠书；大三上和几个志同道合的好友一起搞黎曼面讨论班时的热火

朝天，意兴昂扬；准备丘赛时每天意犹未尽的在中厅熬到凌晨两点半时那透进窗来的路灯

光和虫唱；在大三的夏天那容易下雨的傍晚去参加代数几何的习题课；申请美国的研究生

院时心中的不安和决心；以及大四下在荷兰度过的三个月的美好时光。当然，还有那无数

个车流中赶着上课的早晨，和三教、四教、五教、六教、老馆、新馆自习的夜晚。离开学

校的那天越来越近，学士服的袍带下，是我的青春记忆。

对我来说，大学四年是自己心智和知识的极大的成长。要研究数学，这个志向自从进

大学以来从来没有一丝的动摇过。当初的能让心跳加快的单纯的热爱，经过了四年，现在

变为了生活的一个部分。然而大学之于我的意义，不仅在于我在这里受到最初的数学训

练，更在于它让我对生活做出的思考。我有时会觉得自己仿佛一个旁观者，看着年轻的自

己在做着年轻的事情，充满热切的干劲，或是不羁的莽撞。不得不承认，现在的我在很多

方面都比四年前的上一次毕业时要成熟了；但是成长和成熟不是一件坏事，只要我们不忘

记最初的梦想和内心深处的柔软。

我想每个毕业的人都有一大把的踌躇满志和一大把的留恋不舍，也许也有或多或少的

遗憾。然而这些情绪还是留在自己的心里为好。每个人都有不同的资质和趣味，也将会拥

有不同的独一无二的大学经历，我还是不要做那个因为已先看过一遍电影就在新的观影人

身边喋喋不休的介绍自己也只一知半解的剧情的人了。不过也许我可以对尚未毕业的读者

旁敲侧击的写下一条“观影提示”，聊以凑齐字数，如下：

请你在听到任何学长或学姐介绍的所谓经验时慎之又慎，因为你有自己的梦想，有自

己的实现梦想的能力，也应该有自己的实现梦想的方式。当然前一句话是一个明显的罗素

悖论，这源于我观点的自相矛盾。

梦想是什么呢？这个带着歧义的问题，让我们一起用今后的人生来思考，回答。

∗基数82

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主办：清华大学数学科学系《荷思》编辑部

特刊主编：潘锦钊

编委：（按姓氏笔画为序）

车子良 乐鹏宇 冯 鑫 朱艺航 刘诗南 刘琳媛

苏 桃 李奇芮 李梦龙 李嘉伦 余成龙 郑志伟

张 铭 郭家胤 程经睿 谢松晏 赖 力 潘锦钊

封面：陈凌骅

排版：潘锦钊

联系本刊：THUmath@googlegroups.com

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