a crystalline criterion for good reduction on semi-stable $k3$-surfaces over a $p$-adic field

A Crystalline Criterion for Good Reduction on

Semi-stable K3-Surfaces over a p-Adic Field

Thesis Advisor: Prof. Adrian Iovita

J. Rogelio Perez Buendıa

Concordia University

January 10 2014

J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field

Objective:

To give a criterion for the reduction of an algebraic K3-surface over

a p-adic field in terms of its p-adic etale cohomology.

The desired Criterion

A K3-surface over a Local field with semistable reduction has good

reduction if and only if its second etale cohomology group is crystalline.


Layout


Notations:

Fix a prime number p and let Qp the field of p-adic numbers. Consider:

1 k be a perfect field of characteristic p.

2 k be a fix algebraic closure.

3 W := W (k) the ring of Witt vectors with coefficients in k

4 K0 = Frac(W ) its field of fractions. It is an unramified extension of

Qp.

5 K = K0 if k is ifninite or K be a finite extension of K0 if

[k : Fp] <∞.

6 OK the ring of integers of K .

7 π be a (fixed) uniformizer. So mK = πOK and

k = OK/πOK = W /pW .


K3-surfaces

Definition

A K3-surface over K is a smooth proper surface XK −→ Spec(K) such

that

1 q := H1(XK ,OXK ) = 0 and

2 ωXK ' OXK . Equivalently KX = 0.

where ωXK stands for the canonical sheaf and KX its canonical divisor.

This definition is independent of the field K , so we can consider this

definition for K = C and we get complex algebraic K3-surfaces.


Examples:

K3-surfaces were named by Andre Weil in honour of three algebraic

geometers, Kummer, Kahler and Kodaira, and the mountain K2 in

Kashmir.

Figure: A quartic in P3K given by x2y 2 + y 2z2 + z2x2


Examples

Let S be a non-singular sixtic curve in P2k where k is a field and consider

a double cover i.e., a finite generically etale morphism, π : X → P2k which

is ramified along S . Then X is a K3 surface.


Example

Complete intersections: Let X be a smooth surface which is a complete

intersection of n hypersurfaces of degree d1, . . . , dn in Pn+2 over a field k.

The adjunction formula shows that Ω2X/k∼= OX (d1, . . . , dn − n− 3). So a

necessary condition for X to be a K3 surface is d1 + . . .+ dn = n + 3.

The first possibilities are:

n = 1 d1 = 4

n = 2 d1 = 2, d2 = 3

n = 3 d1 = d2 = d3 = 2.

For a complete intersection M of dimension n one has that

H i (M,OM(m)) = 0 for all m ∈ Z and 1 ≤ i ≤ n − 1. Hence in those

three cases we have H1(X ,OX ) = 0 and therefore X is a K3 surface.


Example

Let A be an abelian surface over a field k of characteristic different from

2. Let A[2] be the kernel of the multiplication by-2-map, let π : A→ A

be the blow up of A[2] and let E be the exceptional divisor. The

automorphism [−1]A lifts to an involution [−1]A on A. Let X be the

quotient variety of A by the group of automorphisms idA, [−1]A and

denote by ι : A→ X the quotient morphism. It is a finite map of degree

2. We have the following diagram of morphisms over k. The variety X is

a K3 surface and it is called the Kummer surface associated to A.


Semistable K3-surfaces

Definition

XK has semistable reduction if it has a semi-stable model:

XK//

X

Spec(K) // Spec(OK)

that is a proper (flat) model X → Spec(OK) whose special fibre X is

smooth over k or etale locally a normal crossing divisor.


Good reduction

If the special fibre X → Spec(k) of such a model X is smooth, then we

say that XK has good reduction.


Layout


p-adic representations

Let GK := Gal(K,K) be the absolute Galois group of K .

Definition

A p-adic representation V of GK is a finite dimensional Qp-vector space

with a continuous action of GK .


Examples of p-adic representations

The main example:

Main example

The etale cohomology of a K3-surface. Indeed, in general we have that if

X is a proper and smooth variety over K , then

H iet(XK ,Qp)

is a p-adic representation of GK .

Tate modules of abelian varieties.

The r -Tate twists of Qp, Qp(r).


Ring of periods

In order to study the p-adic representations, Fontaine defines what we

know as ring of periods, which are topological Qp-algebras B (or B•),

with a continuous linear action of GK and some additional structures

which are compatible with the action of GK (for example the monodromy

operator N, Frobenius, filtrations).


Poincare duality

For a smooth and projective variety X of dimension n over the complex

numbers C, we have the Betti cohomlogy H i (X (C),Z).

By Poincare duality we have that:

H2n−i (X ,C) ' H i (X (C),C).


Complex periods

Also we have a perfect pairing given by the periods:

H idR(X (C)/C)× H2n−i (X (C),C) −→ C

(ω, λ) 7→∫λ

ω.


Comparison isomorphism

We have a natural comparison isomorphism:

H idR(X (C)/C) ' H i (X (C),C).

We remark that in order to have this isomorphism it is very important to

have coefficients in C (for example, this is not an isomorphism over Q).

In this sense C is a ring of periods (it contains all the periods∫λω).


The p-adic case of Cp

We denote by

Cp = ˆK

the p-adic completion of K .

We want analogous comparison isomorphisms in the p-adic cases.

However the situation is not as easy as in the complex case, mainly

because Cp does not have enough periods.


Fontaine’s Idea

The original idea of Fontaine was to construct these ring of periods, in

order to be able to have analogous comparison isomorphism between the

different cohomologies in p-adic settings.


B•

Examples of this rings are

1 BHT :=⊕

q∈Z Cp(q) B Hodge-Tate. is a graded C p-algebra with

GK -action rescts gradings and BGK

HT = K .

2 BdR : B de Rham is a complete discrete valuation field over K with

residue field Cp. It contains K (but not Cp). It has an action of GK

and a filtration by its valuation, and its graded quotient

gr iBdR = Cp(i)


Bcris

Bcris : B crys is an algebra over K0 and a GK -stable subring of BdR . It

contains K0 but not K . We have a filtration coming from form BdR , a

σ-semilinear injective GK -equivariant endomorphism φ (Frobenius

endomorphism). BGK

cis = K0.


Bst

Bst : B semistable is an algebra over K0 and has a GK -action. It contains

Bcris and KO but not K The Frobenius of Bcris extends to Bst and has a

Bcris -derivation

N : Bst −→ Bst . Nφ = pφN

and

BGKst = K0, BN=0

st = Bcris .


Dieudonne Modules

These rings are such that the BGK -modules

DB(V ) := (B ⊗Qp V )GK

give us (or expected to give us) good invariants for V . For example

comparison isomorphisms for the p-adic etale cohomology and de Rham

cohomologie or crystalline cohomology or nice criterion for good

reduction of varieties.


B-admissible

Let L = BGK .

Definition

A p-adic representation V is B-admissible, if

dimL DB(V ) = dimQp V .

Definition

A p-adic representation V , is crystalline (semistable, Hodge-Tate,

semistable) if V is Bcris-admissible (B•-admissible).

B-admissibility translates to isomorphisms which are analogous to the

comparison isomorphisms in the complex case.


Fontaine has defined several subcategories of the category of all p-adic

representations, denoted by RepGK .

This categories are formed by the property of being B-admissible objects.

So for any of the period rings B we have a subcategory of the category of

p-adic representations denoted by RepB . These categories satisfy proper

contention relations as follows:

RepBcris⊂ RepBst ⊂ RepBdR

⊂ RepBHT ⊂ RepGK .


C•-conjectures

Let XK be a proper smooth variety over K .

CHT : The Hodge-Tate conjecture. There exists a canonical

iomorphism, which is compatible with the Galois action.

Cp ⊗Qp Hmet (XK ,Qp) '

⊕0≤i≤m

Cp(−i)⊗K Hm−i (XK ,ΩiXK/K

).

CdR : The de Rham conjecture. There exist a conaonical

isomorphism, which is compatible with Galois action and filtrations.

BdR ⊗Qp Hmet (XK ,Qp) ' BdR ⊗K Hm

dR(XK/K ).


C•-conjectures

Ccris : The Crystalline conjecture. Let X be a proper smooth model

of XK over OK . Let X be the special fibre of X . There exist a

canonical isomorphism which is compatible with the Galois action,

and Frobenius endomorphism.

Bcris ⊗Qp Hmet (XK ,Qp) ' Bcris ⊗W Hm

crys(X/W )

Barthelo-Ogus isomorphism:

K ⊗W Hmcrys(X/W ) ' Hm

dR(XK/K ).


Cst conjecture

1 Cst : The semistable conjecture: Let X be a proper semistable model

of XK over OK . Let Y be the special fiber of X , and MY be a

naural log-structure on Y . There is a canonical isomorphsim,

compatible with Galois action, Frobenius and operator N.

Bst ⊗Qp Hmet (XK ,Qp) ' Bst ⊗W Hm

log−crys((Y ,MY ), (W ,O∗))

2 Hydo-Kato isomorphism

K ⊗W Hmlog−crys((Y ,MY ), (W ,O∗)) ' HdR(XK/K )


History

For X/K a proper smooth variety over K .

Fontaine: Proved that Tate modules V = TpA⊗Zp Qp of abelian varieties A are de Rham

and that DdR(V ) ' (HdR(X/K))v .

Fontaine and Messing: Proved the comparison theorem for H iet(XK ,Qp) for i ≤ p − 1 and

K/Qp finite unramified.

Kato, Hydo Tsuji: Extended these results to finally prove that one can recover H idR(X/K)

from the data of V = H iet(XK ,Qp) as a p-adic representation.

Tsuji, Niziol, Faltings: Proved that if X is semistable, then V is Bst -admissible and that V

is crystalline if V has good reduction and BdR otherwise.


For abelian varieties

For Abelian varieties, Bcris and Bst are exactly what it takes to decide:

whether A has good reduction or semistable reduction.

Crystalline criterion for abelian varieties:

Coleman-Iovita Breuil: A has good reduction if and only if V is

crystalline. A has semistable reduction if and only if V is semistable.


My Thesis problem

Crystalline criterion for K3 surfacess:

Let X be a K3 surface over a p-adic field K with semistable reduction. X

has good reduction (X → Spec(k) is smooth) if and only if

V := H2et(XK ,Qp)

is Crystalline (Bcris-admissible).


One side is Falting’s result:

Remember that

RepBcris⊂ RepBst

Since X has semistable reduction, then V is Bst-admissible. If X has

good reduction, then by Falting’s result, V is Bcris-admissible.


Results of Y. Matsumoto

Theorem

Let K be a local field with residue characteristic p 6= 2 and X a Kummer

surface over K. Assume that X has at least one K-rational point. If

H2et(XK ,Qp) is crystalline, then XK ′ has good reduction for some finite

unramified extension K ′/K.

Theorem

Let K be a local field with residue characteristic p 6= 2, 3, and Y a K3

surface over K with Shioda-Inose structure of product type. If

H2et(YK ,Qp) is crystalline, then YK ′ has good reduction for some finite

extension K ′/K of ramification index 1, 2, 3, 4 or 6.


Main tool

The main tooll is what we call p-adic logarithmic degenerations of a

K3-surface. These will be p-adic analogous of degeneration of K3

surfaces over the complex numbers constructed via degenerations.


Complex degenerations of K3-surfaces

Definition

Over the complex numbers C, a semistable degeneration of a

K3-surface X is a proper flat and surjective morphism

π : X (C)→ ∆

over the open disc, whose general fibre Xt = π−1(t), for t 6= 0 is a

smooth K3-surface and X0 is reduced with normal crossings.


Modification of a degeneration

Definition

A modification of π : X (C)→ ∆ is a degeneration of surfaces

π′ : X (C)→ ∆ such that there exists a birational map

φ : X (C)→ X ′(C) given an isomorphism form

(X (C)− X0) −→ (X ′(C)− X0) and such that the diagram:

X (C)φ //

π""

X ′(C)

π′||∆

commutes.


Kulikov degenerations

We have the following theorem:

Theorem (Kulikov, Persson, Pinkham)

Let π : X (C)→ ∆ be a semistable degeneration of a K3-surface, then

there exists a modification π′ : X ′(C)→ ∆ such that the canonical

divisor of the total space X ′(C) is trivial.

A degeneration with trivial canonical divisor is called a good

degeneration or a Kulikov degeneration.


Kulikov criterion

Theorem

Let X (C)→ ∆ be a good degeneration of a K3-surface. The degenerate

fibre X0 is one of the following three types:

I. X0 is a nonsingular K3 surface.

II. X0 = ∪ni=1Vi where the Vi are rational surfaces and V2, . . . ,Vn−1 are

elliptic ruled surfaces.

III. X0 = ∪ni=1Vi where all the Vi are rational surfaces.


In therms of monodromy

Moreover, the three cases can be distinguished from each other by means

of the monodromy T acting on H2(Xt ,Z):

For Type I we have N := ln T = 0 that is T = id .

For Type II, N 6= 0 but N2 = 0.

For Type III, N2 6= 0 but N3 = 0


Layout


The plan:

We need:

A p-adic semistable degeneration of our K3-surface.

A monodromy operator N on the log-crystalline cohomology.

We relate it with the monodromy operator on

Dst(H2et(XK ,Qp)) ' H2

log−cris(XK/W ) appearing on Fontain’s

theory.

By p-adic Hodge theory if H2et(XK ,Qp) is crystalline then N = 0.

Indeed BN=0st = Bcris and Dst(V )N=0 = Dcris(V ) , so if a p-adic

representation is crystalline we most have N = 0.


Finally we base change to the complex numbers and use Kulikov’s

classification theorem to deduce that our crystalline K3-surface has good

reduction.

Here we use the Deligne’s work on the Monodromy expressed as the

residue at zero of the GM-conexion.


Layout


Logarithmic geometry is concerned with a method of finding and

using “hide smoothness” in singular varieties.

Let X be a nonsingular irreducible complex variety, S a smooth

curve with a point s and f : X → S a dominant morphism smooth

away from s, the fiber Xs := f −1(s) = Y1 ∪ · · · ∪ Yn reduced simple

normal crossing divisor.

ΩX/S = ΩX/f ∗ΩS fails to be locally free at the singular points of f .


Consider ΩX/S(log(Xs)) the sheaf of differentials with at most

logarithmic poles along the Yi , and similarly ΩS(log(s)), there is an

injective sheaf homomorphism

f ∗ΩX (log(Xs)) −→ ΩS(log(s))

and the quotient sheaf ΩX (log(XS))/f ∗ΩX (log(Xs)) is locally free.


pre-log.st

Definition

1 Let X be a scheme. A pre-log structure on X , is a sheaf of

monoids MX together with a morphism of sheaves of monoids:

α : MX −→ OX , called the structure morphism.

2 A pre-log structure is called a log structure (log.st for short) if

α−1(O∗X ) ' O∗X via α.

3 The pair (X ,MX ) is called a log scheme and it will be denoted by

X×.

4 Morphisms are morphisms of sheaves which are compatible with the

structure morphism.


Induced log.st

We have the forgetful functor i from the category of log.st of X to the

category of pre-log.st of X by sending a log.st M in X to itself considered

as a pre-log.st i(M).

Vice-versa given a pre-log.st we can construct a log.st M ls out of it in

such a way that ( )ls is left adjoint of i , hence M ls is universal.


Inverse image log.st

Definition

Let f : X → Y be a morphism of schemes. Given a log.st MY on Y we

can define a log.st on X , called the inverse image of MY , to be the log

structure associated to the pre-log.st

f −1(MY )→ f −1(OY )→ OX .

This is denoted by f ∗(MY ).


Morphisms of log-schemes

Definition

By a morphism of log-schemes X ∗ −→ Y ∗ we understand a morphism of

the underlying schemes f : X → Y and a morphism f # : f ∗MY → MX of

log.st on X .

We denote by LSch the category of log.schemes.


One of the main examples of interest for us is the following:

Example

Let X be a regular scheme (we can take for example a K3-surface over K

or a proper model of it). Let D be a divisor of X . We can define a log.st

M on X associated to the divisor D as

M(U) :=

g ∈ OX (U) : g |U\D ∈ O∗X (U \ D)


log.st to algebraizable formal schemes

Let X be a scheme and X is a formal completion of X along a closed

subscheme Y , then we have a morphism of ringed spaces:

Xφ−→ X

for which φ is the inclusion Y → X on topological spaces, and on

sheaves, it is the natural projection

OX −→ lim←−OX/I n = OX

where I is the sheaf of ideals defined by the closed immersion Y → X .

If we have a log.st on X , say M, we can give a log structure on X by

taking the inverse image of the log structure M so that X becomes a

log-formal scheme:

(X , φ∗M).


For K3-surfaces

1 When X is a proper model of a K3-surface XK , we have that the

special fibre X of X is a closed divisor with normal crossings.

This divisor induces a log.st on X . We denote by X× the log-formal

scheme obtained as in the previews paragraph; that is, by completing

X along X and giving to it the inverse image log.st of X induced by

X .

2 Notice that we have an inclusion of ringed spaces X → X×. We

denote by X× the log-scheme obtained by giving to X the inverse

image log.st of X×.


Layout


p-adic degeneration

Definition

A p-adic degeneration of a K3-surface with semistalbe reduction is a proper, flat morphism of

schemes X −→ Spec(W[[t]]) with geometrically connected fibres, such that:

1 We have an isomorphism of the semistable model X of XK with the fibre Xπ of

X→ Spec(W[[t]]) induced by the ring homomorphism: W [[t]]→ OK ; t −→ π.

2 We have an isomorphism (compatible with the previous one) of the special fibre X of the

semistable model of XK with the fibre X0 induced by the projection

W [[t]]→ W [[t]]/(p, t) ' k = OK/πOk .

3 X→ Spec(W[[t]]) is smooth in the complement of X that is

(X− X ) −→ (Spec(W[[t]])− Spec(k))

is smooth.


In a diagram

Then we have commutative Cartesian diagrams:

X //

X //

X

Spec(k) // Spec(OK) // Spec(W[[t]])

.

Note that Spec(W[[t]]) is the analogous, in p-adic settings, of the open

unit disc ∆ in the complex numbers, and so we call Spec(W[[t]]) the

p-adic unit disc denoted by D. Then X is a family of surfaces

parametrized by the p-adic unit disc D and removing the special fibre X

smooth over D∗ = D − Spec(k).


rig functor

1 Let Y be the fibre of f at t = 0, that is the fibre induced by the

morphism

W [[t]]→W ; t 7→ 0.

This is a scheme over Spec(W) whose special fibre is again X .

Moreover Y is a normal crossing divisor (but now in characteristic

zero).

2 Call X = (X×)rig, D = (D×)rig, and f = (f ×)rig the rigid analytic

spaces over K0.


rigid version of the degeneration

Lemma

Under the previous settings we have:

1 X −→ Spec(K0) is smooth

2 Y := f −1(0) = (Y×)rig is a semistable surface over K0.

3 f |X∗ : X ∗ := (X − Y) −→ D∗ := (D − 0) is smooth.


Layout


Complex of relative logarithmic differentials

Consider the complex of sheaves K ·X/D induced by the relative

logarithmic differential:

OXd1X/D−−−→ OX ⊗X/K0

Ω1X/D(log(Y))

d2X/D−−−→ OX ⊗X/K0

Ω2X/D(log(Y))


The connection

Denote by Hi the i-th logarithmic relative de Rham cohomology group of

X/D with coefficients in OX , i.e, it is the sheaf Rf∗(K ·X/D). For every i ,

Hi is a free OD-module with an integrable, regular-singular connection

∇i : Hi −→ Hi ⊗OD Ω1D/K0

(log(0)).


Monodromy

If s is a point in D, let His be the fibre of H1 at s. We now define the

monodromy Ni as the residue at 0 of this connection. That is

Ni = res0(∇i ).

In our case the only important value is for i = 2, so we define the

monodromy as N := N2.


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a crystalline criterion for good reduction on semi-stable $k3$-surfaces over a $p$-adic field

Science

eld of p

padic eld

good reduction

crystalline criterion

semistable k3surfaces

prime number p

eld of fractions

perfect eld of characteristic