cosection localization and quantum singularity theory€¦ · cosection localization [k.-li...
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Cosection localization and quantum
singularity theory
Young-Hoon Kiem
Department of Mathematics
Seoul National University
2018.09.15 - Cetraro, Italy
Enumerative geometry
Schubert (1874): Enumerative geometry = How many geomet-
ric figures of fixed type satisfy certain given conditions?
Hilbert’s 15th problem: Provide rigorous foundation of Schu-
bert’s enumerative calculus.
Example. Lines in P3.• How many lines in P3 meet 4 general given lines?
lines in P3=Gr(2, 4) ⊂ P5 quadric hypersurface
lines in P3 meeting a given line=Gr(2, 4) ∩ H ⊂ P5
lines in P3 meeting 4 given lines=Gr(2, 4)∩H1∩ · · ·∩H4=2 lines.
Kleiman (1972): The beauty of enumerative geometry lies in
finding the number of geometric figures without finding the fig-
ures themselves.
Enumerative geometry = intersection theory on moduli space
(+ checking the intersection numbers actually enumerate the
desired objects).
Example. Rational curves in Fermat quintic CY3.
Q = zero(∑5i=1 z
5i ) ⊂ P4 Fermat quintic CY 3-fold.
The set of rational curves of degree d in Q is expected to be
finite but not much seems known.
A stable map to a projective variety W is a morphism f : C → W
such that C is a reduced curve with at worst nodal singularities
and Aut(f) = τ : C∼=−→C | f τ = f is finite.
Mg,n(W,d) = f : C→W | stable maps, f∗[C] = d ∈ H2(W,Z)/ ∼= .
GW invariant = virtual intersection numbers on Mg,n(W,d).
[Kontsevich-Manin] Q ⊂ P4 ⇒ X =M0,0(Q, d) ⊂ Y =M0,0(P4, d).
C f //
π
P4
Y univ. family
⇒ E = π∗f∗OP4(5)
X = s−1(0)
//Y
sII
E is a vector bundle of rank 5d+1 and Y is smooth, dim Y = 5d+1.
s = f∗∑5i=1 z
5i where H0(P4,OP4(5))
f∗−→H0(C, f∗OP4(5)) = H0(Y, E).
[X]vir := s![Y] = [Y] ∩ e(E, s) ∈ H0(X) refined Euler class of E
GW0(Q, d) = #[X]vir ∈ Q.
[Givental, Lian-Liu-Yau] Mirror theorem was proved for GW0(Q, d)
about 20 years ago.
Refined Euler class
E
X = s−1(0)
//Y
s
DD
• Cx VBs are oriented ⇒ orE ∈ H2r(E, E − 0E) where r = rank(E).
• s : (Y, Y − X) → (E, E− 0E) ⇒ e(E, s) = s∗orE ∈ H2r(Y, Y − X) = H2rX (Y).
• Cap product ∩ : Hi(Y)× H2rX (Y) → Hi−2r(X).
• Hi(·) is Borel-Moore homology of locally finite closed chains.
The Gysin map is defined by
s! = (·) ∩ e(E, s) : Hi(Y) −→ Hi−2r(X).
Topologically, s!(ξ) is the intersection of ξ with small perturbation
s ′ of s. [X]vir = s![Y] is the correct fundamental class of X.
Refined Euler class in algebraic geometry
• Chow groups
Ai(X) = Zi-dim’l irred subvarieties/(rational equiv.).
• Proper pushforward and flat pullback
f : X→ Y is proper ⇒ f∗ : A∗(X) → A∗(Y), f∗[V ] = deg(f|V) · [f(V)red].
f : X→ Y is flat ⇒ f∗ : A∗(Y) → A∗(X), f∗[W] = [W ×Y X].
• Cycle class map (functorial)
hX : Ai(X) → H2i(X).
E
X = s−1(0)
//Y
s
DD
• CX/Y = SpecX(⊕
n InX/I
n+1X
)normal cone of X in Y.
• s∨ : E∨ IX ⊂ OY ⇒ Sym(E∨|X)⊕n InX/I
n+1X ⇒ CX/Y → E|X.
• π∗ : Ai(X)∼=−→Ai+r(E|X) ⇒ 0!
E|X= (π∗)−1 : Ai(E|X) → Ai−r(X).
[X]vir := s![Y] = 0!E|X
[CX/Y] ∈ Adim Y−rankE(X).
s!(ξ) = 0!E|X
[Cξ∩X/ξ] ∈ Ai−r(X) ⇒ s! : Ai(Y) → Ai−r(X).
flat deformation Y CX/Y ⇒ Ai(Y)s! //
hY
Ai−r(X)hX
H2i(Y)s!//H2i−2r(X)
Virtual fundamental class
For g > 0, X = Mg,n(Q, d) ⊂ Y = Mg,n(P4, d) is not the zero locusof a section of any vector bundle globally as H1(C, f∗OP4(5)) 6= 0.
For any scheme/DM stack X locally of finite type, it is alwayspossible to find an open cover X = ∪αXα such that
Eα
Xα = s−1α (0)
//Yα
sαFF
for a vector bundle Eα, sα ∈ H0(Eα), and Yα smooth. So we have[Xα]vir = s!α[Yα] ∈ Adim Yα−rankEα(Xα).
When can we glue [Xα]vir to a class [X]vir ∈ A∗(X)?
Perfect obstruction theory
X has a perfect obstruction theory if we have an open cover
X = ∪αXα and diagrams
Eα
Xα = s−1α (0)
//Yα
sαFF
such that the tangent obstruction complex
Eα = [TYα|Xαdsα−→Eα|Xα]
glue to an E ∈ Db(Coh(X)), i.e. E |Xα ∼= Eα.
Examples. Moduli of stable maps, moduli of stable sheaves on
CY3 or surfaces, moduli of line bundles with sections on curves.
Tangent-Obstruction complex
Eα = [TYα|Xαdsα−→Eα|Xα], Xα = s−1α (0).
• TXα = ker(TYα|Xαdsα−→Eα|Xα) = h
0(Eα) tangent sheaf of Xα.
• Xα is smooth if dsα : TYα|Xα → Eα|Xα is surjective, i.e. h1(Eα) = 0.
• ObXα := coker(TYα|Xαdsα−→Eα|Xα) = h
1(Eα) obstruction sheaf of Xα.
• Virtual (expected) dimension vd = dim Yα− rankVα = rank(Eα).
• Infinitesimal lifting problem.0→ I→ A→ A→ 0, (A,m)=Artin local ring, I ·m = 0.
SpecA // _
Xα = Spec(Rα/Jα) _
SpecA //
∃? 44
Yα = SpecRα.
∃ob(A, A, g) ∈ ObXα|x ⊗ I whose vanishing guarantees the lifting.
[Li-Tian, Behrend-Fantechi]
If there is a perfect obstruction theory on X, ∃[X]vir ∈ Avd(X) such
that [X]vir|Xα = [Xα]vir ∈ Avd(Xα) for all α.
Assume E = [E0 → E1] for vector bundles E0, E1 over X such that
we have a surjective quasi-isomorphism
E0|Xα//
E1|Xα
TYα|Xαdsα//Eα|Xα.
Then CXα/Yα ×Eα|Xα E1|Xα glue to a cone CX,E in E1. Then
[X]vir = 0!E1[CX,E ].
Since 1995, enumerative invariants are defined as the integralsof cohomology classes over the virtual fundamental classes [X]vir
on suitable moduli spaces X, e.g. Gromov-Witten, Donaldson-Thomas, Pandharipande-Thomas, etc.
By definition, the actual computation takes place only at thesupport of [X]vir. Hence it helps enormously if we can confinethe support to a smaller subset of X.
[Graber-Pandharipande] When C∗ acts on X and the perf. obst.th. is equivariant, [X]vir is localized to the fixed point locus XC
∗
[X]vir = ı∗[XC
∗]vir
e(Nvir), ı : XC
∗→ X
If X is not proper,∫[X]vir is not defined. However if XC
∗is proper,
the torus localization enables us to define invariants.
Cosection localization [K.-Li 2006∼2013]
X = scheme/DM stack with perfect obstruction theory E.
σ : ObX = h1(E) → OX cosection of the obstruction sheaf.
D(σ) = subscheme defined by the image of σ; σ|X−D(σ) is surjec-
tive. Let ı : D(σ) → X denote the inclusion. Then
∃ [X]virloc ∈ A∗(D(σ)) with ı∗[X]vir
loc = [X]vir and other nice properties
like deformation invariance.
The proof is algebraic and consists of two parts:
(1) The glued normal cone CX,E ⊂ E1 sits in ker(σ : E1 → OX).
(2) 0!E1
: Ai(E1) → Ai−r(X) localizes to 0!
E1,σ: Ai(ker σ) → Ai−r(D(σ)).
Example. GW invariants of general type surfaces.
S = surface of general type, θ ∈ H0(KS) a holomorphic 2-form,
C = zero(θ) canonical curve.
X =Mg,n(S, β) has cosection
σ : ObX = H1(f∗TS)θ−→H1(f∗ΩS)
f∗−→H1(ωC) = C.
D(σ) =Mg,n(C, d) if β = d[C] and D(σ) = ∅ otherwise.
[Mg,n(S, β)]virloc ∈ A∗(D(σ)) is zero unless β is a multiple of [C].
The computation of GW(S) is effectively reduced to the curve
case Mg,n(C, d). [Lee-Parker, K.-Li]
Example. Hilbert scheme of divisors on surface with hol. 2-form.
Fix θ ∈ H0(KS) = H2,0(S) whose vanishing locus is C.σ : ObHilb(S)|D = H1(OD(D)) → H2(OS)
θ−→H2(KS) = C.D(σ) = C. May assume smooth point by Green-Lazarsfeld.
[Hilb(S)]virloc = (−1)dim[pt] gives (−1)dim = SWKS
(S)⇒ algebro-geometric theory of Seiberg-Witten invariant.
By wall crossing of stable pairs [Mochizuki], Donaldson invariant(enumerating rank 2 stable bundles) ⇔ Seiberg-Witten invariant(enumerating line bundles with sections).
Many more applications including the proof of Katz-Klemm-Vafaformula about curves on K3 surfaces [Maulik-Pandharipande-Thomas].
Cosection localized virtual cycle (single chart case)
E
σY //OY
X = s−1(0)
//Y
s
EE
σYs=0@@
Since σY (t−1s) = 0 for t ∈ C∗, by MacPherson’s graph construc-tion, we find that CX/Y ⊂ ker(E|X
σ−→OX) where σ = σY |X.
Let D(σ) = X ∩ D(σY). The Gysin map 0! : Ai(E|X) → Ai−r(X) and[X]vir ∈ Avd(X) localize to D(σ) as
0!σ : Ai(ker(σ)) −→ Ai−r(D(σ)) and [X]virloc = 0!σ[CX/Y] ∈ Avd(D(σ)).
Similarly, we have a homomorphism
s!σ : Ai(Y) → Ai−r(D(σ)), s!σ(ξ) = 0!σ[Cξ∩X/ξ].
Cosection localized Gysin map
For σ : E1 → OX, let us define 0!σ : Ai(ker(σ)) → Ai−r(D(σ)).
Pick a proper birational ρ : X→ X such that we have an exact
0 −→ E ′ −→ ρ∗E1 σ−→OX(−D) −→ 0
for Cartier divisor D ≥ 0 lying over D = D(σ). (E.g. X = blDX.)For any ξ ∈ Ai(ker(σ)), ∃ ζ ∈ Ai(E ′) and η ∈ Ai(E1|D) such that
ξ = ρ∗ζ + ∗η
where : E1|D → ker(σ) is the inclusion and ρ : E ′ → ker(σ) is therestriction of the natural ρ∗(E1) → E1. Define
0!σ(ξ) = −(ρD)∗(D · 0!E ′ζ) + 0!E1|D
(η)
where ρD = ρ|D
: D→ D. Then s!σ(ξ) is independent of all choices.
Topological construction [K.-Li, 1806.00116]
E
σY //OY
X = s−1(0)
//Y
s
EE
σYs=0@@
Let us define a cosection localized Gysin map
s!σ : IHi(Y) −→ Hi−2r(D(σ)).
such that if Y is smooth, the following is commutative
Ai(Y)hY
s!σ //Ai−r(D(σ))hD(σ)
H2i(Y)s!σ //H2i−2r(D(σ))
Pick a proper birational ρY : Y → Y (e.g. blowup along DY = D(σY))
such that we have an exact
0 −→ E ′Y −→ ρ∗YEσY−→O
Y(−DY) −→ 0
for a Cartier DY ≥ 0 over DY. As σY s = 0, the induced section
s = ρ∗Ys of ρ∗YE is actually a section of E ′Y. By [Beilinson-Bernstein-
Deligne], any ξ ∈ IHi(Y) lifts to a class in IHi(Y) so that we have
ξ = (ρY)∗ζ + (Y)∗η, ζ ∈ Hi(Y), η ∈ Hi(DY)
where Y : DY → Y. We then define
s!σ(ξ) = −(ρD)∗(DY · s!ζ) + s!E|DY(η) ∈ Hi−2r(D)
Again, this is independent of all choices. Moreover s!σ commutes
with natural functors.
Quantum singularity theory
Consider the stack
w : X = [C5 × C/C∗] −→ C, w(z1, · · · , z5, z0) = z05∑i=1
z5i
where the C∗-weights are (1, · · · , 1,−5). ∃ two GIT quotients
w : X+ = OP4(−5) → C, w : X− = C5/µ5 → C, w =5∑i=1
z5i .
The critical locus of w on X+ is the Fermat quintic Q ⊂ P4.
[Witten 1993] GW(Q) ⇔ curve counting on Crit(w) ⊂ X+⇔ curve counting on (X+, w) = curve counting on (X−, w).
What is the curve counting invariant on (w : X− = C5/µ5 → C)?
[Witten 1993] It should be an integral on the space of spin curves
with sections satisfying Witten’s equation.
A morphism f : C → C5/µ5 should mean a 5-spin curve (1-dim’l
DM stack C - orbifold structures only at nodes or marked points
whose stabilizer group is µ5 - together with a line bundle L on
C such that ϕ : L5 ∼= ωlogC ) and five sections x1, · · · , x5 of L. We
say f is stable if the coarse moduli space of C with markings is a
stable curve.
[Fan-Jarvis-Ruan 2013] The stack S = Sg,n of stable 5-spin curves
is a smooth proper DM stack with projective coarse moduli. The
morphism Sg,n →Mg,n sending a spin curve (C, pj, Li, ϕ) to (|C|, |pj|)
is flat proper and quasi-finite.
Let Xg,n be the moduli space of spin curves (C, pj, Li, ϕ) with
sections (x1, · · · , xN) ∈ ⊕iH0(Li). Then Xg,n = zero(sM) by con-
struction; separated but not proper.
Xg,n comes with a natural perfect obstruction theory (basically a
lift of Rπ∗L⊕5) and hence the virtual fundamental class
[Xg,n]vir ∈ A∗(Xg,n)
However we cannot integrate cohomology classes over the virtual
cycle because Xg,n is not proper!
[Fan-Javis-Ruan 2013] Mathematical theory of curve counting byanalysis (The solution space to Witten’s equation is compact.)⇒FJRW invariants satisfying nice properties (axioms of Manin’scohomological field theory).
[Polishchuk-Vaintrob 2016] Algebraic theory for FJRW by matrixfactorization and Hochschild homology. (There is a universalmatrix factorization, compactly supported in Sg,n.)
[Chang-Li-Li 2015] Algebro-geometric theory by (algebraic) co-section localization; works only for narrow sectors.(The virtualcycle localizes to [Xg,n]vir
loc ∈ A∗(Sg,n) when x5i ∈ H0(ωC) for xi ∈
H0(L).)
[K.-Li 2018] Purely topological construction by cosection local-ization; works for all sectors and more (GLSM).
[Polishchuk-Vaintrob 2016] ∃ cosection σM : EM → OM that fit
into the diagram (M,qM,pM are smooth)
EM
σM//OM M× C //C
X s−1M (0)
pX$$
ı //M
sMGG
wqM
33
qM""
pM
S Bw
//C
Let Z = w−1(0) where w = wγ1 · · ·wγn. Let Y = Z×BM so that
we have a fiber diagram
X
qX
ı //Y //
qY
MqM
Z
//B
Here qM and qY are smooth morphisms.
• By the Thom-Sebastiani property⊗j
Hγj =⊗j
IHNγj(w−1γj
(0)) ∼= IH∑Nγj
(w−1(0)) = IH∑Nγj
(Z),
we have a map Hγ1 ⊗ · · · ⊗ Hγn −→ IH∑Nγj
(Z).
• Composing it with cosection localized Gysin map s!σ and the
smooth pullback q∗Y, we have
Ψg,γ : Hγ1 ⊗ · · · ⊗ Hγn −→ IH∗(Z)q∗Y−→ IH∗(Y)
s!σ−→H∗(S) −→ H∗(Mg,n).
satisfying the axioms of a cohomological field theory.
The FJRW invariant with vj ∈ Hγj is topologically defined by
FJRWw,G(v1, · · · , vn) =∫Mg,n
Ψg,n(v1, · · · , vn).
Thank you for your attention.