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.