Degeneration formulae and its applications to local GW and DT - PowerPoint PPT Presentation

Degeneration formulae and its applications to local GW and DT invariants Jianxun Hu Department of Mathematics, Sun Yat-sen University stsjxhu@mail.sysu.edu.cn 1

§ 1 Motivations Example: Consider the projective plane P 2 • Fix two general points P , Q , there is only one line C passing through P and Q , with the homology class β = [ ℓ ] . 2

• Fix two general points P , Q , and a line ℓ , there is only one line C passing through P , Q and intersecting the line ℓ , with the homology class β = [ ℓ ] . Question : Fix a homology class A ∈ H 2 ( X, Z ) and some cycles Z i in a projective (or symplectic) manifold X , assuming the Z i are in general position. The basic question is : How many curves on X satisfy: C ⊂ X of genus g , homology class A , and C ∩ Z i ̸ = ∅ for all i. (1) 3

Naively, Gromov-Witten invariant is defined as the number of curves (1). 4

Physical origin of Gromov-Witten invariant The origins in physics of Gromov-Witten invariants is the topological sigma model coupled to gravity. In particular, the genus zero (sometimes called tree level) Gromov- Witten invariants originate from the topological sigma model, which is a topological quantum field theory. In fact, in topological quantum field theory, the Gromov-Witten invariants appear as correlation functions. 5

§ 2 Gromov-Witten invariant Notations: • ( X, ω ) : a compact symplectic (or projective) manifold of dim 2 n , here ω is a nondegenerate closed 2 -form, i.e. ω n ̸ = 0 . • There exist almost complex structures J : TX → TX such that J 2 = − id . Fact:The space of all tamed almost complex structures is contractible( This implies that symplectic geometry is much more flexible than complex geometry). • Example: ( R 2 n , ∑ n i =1 dx i ∧ dy i ) . 6

Stable map An n -pointed stable map consists of a connected marked curve ( C, p 1 , · · · , p n ) and a morphism f : C − → X satisfying the following conditions: (i) The only singularities of C are ordinary double points(nodal Riemann Surface). (ii) p 1 , · · · , p n are distinct ordered smooth points in C . (iii) If C i is a component of C such that C i ∼ = P 1 and f | C i is constant, then C i contains at least 3 special points(nodal points and marked points). (iv) If C has arithmetic genus one and n = 0 , then f is not constant. (v) f | C i : C i − → X is holomorphic where C i is a smooth component of C . 7

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• Equivalence of stable map ( C, p 1 , · · · , p n ; f ) is isomorphic to ( C ′ , p ′ 1 , · · · , p ′ n ; f ′ ) if there is an isomorphism i for all i and f ′ ◦ τ = f . → C ′ such that τ ( p i ) = p ′ τ : C − ∃ τ ( ∼ =) ( C ′ , p ′ 1 , · · · , p ′ ( C, p 1 , · · · , p n ) − → n ) ↙ f ′ f ↘ X Denote by [( C, p 1 , · · · , P n ; f )] the equivalence class. 9

Moduli space of stable maps For A ∈ H 2 ( X, Z ) , define the moduli space of stable maps as follows M g,n ( X, A ) : = { [( C, p 1 , · · · , p n ; f )] | ( C, p 1 , · · · , p n ; f ) is a genus g stable map and f ∗ [ C ] = A } . In general, M g,n ( X, A ) is very singular. Remark: In many case, different For example, assume that X = P 1 , g > 0 , component has different dimension. A = dL with d > 2 where L is the homology class of a line in P 1 . Then M g,n ( X, A ) has more than one components. The most interesting one consists (generically) Call this one M g, 0 ( P 1 , A ) o . of irreducible genus g curves. The second consists (generically) of two intersecting components, one of genus g and mapping to a point, and the other rational and mapping to P 1 with degree d . The first one has dimension 2 d + 2 g − 2 , and the second has dimension 2 d + 3 g − 3 , so the second is not in the closure of the first. 10

Proposition:[Ruan1, LT, FO, S] M g,n ( X, A ) has a virtual fundamental class [ M g,n ( X, A )] vir with the expected dimension C 1 ( A ) + (dim X − 3)(1 − g ) + n. 11

Gromov-Witten invariant Once we have the moduli space of stable maps, then we may define the following evaluation maps: ev i : M g,n ( X, A ) − → X [ C, p 1 , · · · , p n , f ] �→ f ( p i ) , i = 1 , 2 , · · · , n. Given cohomology classes α i ∈ H ∗ ( X, R ) , Definition: roughly define the (primitive) Gromov-Witten invariant n ∫ ∏ ev ∗ Ψ ( A,g ) ( α 1 , · · · , α n ) = i α i , [ M g,n ( X,A )] vir i =1 if ∑ n i =1 deg α i = 2 C 1 ( A ) + 2(dim X − 3)(1 − g ) + 2 n . Otherwise, we simply define the invariants to be zero. 12

Remark: (Enumerative meaning) If Z i is a cycle in X dual to α i , then the primitive Gromov-Witten invariant ⟨ α 1 , · · · , α n ⟩ X g,A should count genus g curves ( C, p 1 , · · · , p n ) for which we can find f such that f : ( C, p 1 , · · · , p n ) − → X is stable and f ∗ [ C ] = A, f ( p i ) ∈ Z i . 13

Relative Gromov-Witten invariants Let Z ⊂ X be a real codimension 2 symplectic submanifold. Suppose that J is an ω − tamed almost complex structure on X preserving TZ , i.e. making Z an almost complex submanifold. The relative GW invariants are defined by counting stable J − holomorphic maps intersecting Z at finitely many points with prescribed tangency. More precisely, fix a k -tuple T k = ( t 1 , · · · , t k ) of positive integers, consider a marked pre-stable curve ( C, x 1 , · · · , x l , y 1 , · · · , y k ) → X such that the divisor f ∗ Z is and stable J − holomorphic maps f : C − ∑ f ∗ Z = t i y i . i We consider the moduli space of such curves, M g,T k ( X, Z, A ) . Unfortunately, this moduli space is not compact. Similar to the case of absolute Gromov-Witten invariant, we may compactify this moduli space by relative stable maps. Denote by [ M g,T k ( X, Z, A )] vir the virtual fundamental class. Then use the virtual technique to define the relative Gromov-Witten invariant. 14

Evaluation maps: M g,T k ( X, Z, A ) − → X ev i : ( C, x 1 , · · · , x l ; y 1 , · · · , y k ; f ) �→ f ( x i ) , 1 ≤ i ≤ l. ev Z M g,T k ( X, Z, A ) − → Z j : ( C, x 1 , · · · , x l ; y 1 , · · · , y k ; f ) �→ f ( y j ) , 1 ≤ j ≤ k. Definition: (relative Gromov-Witten invariant) Let α i ∈ H ∗ ( X, R ) , 1 ≤ i ≤ l , β j ∈ H ∗ ( Z, R ) , 1 ≤ j ≤ k . Define the relative Gromov-Witten invariant 1 ∫ [ M g,Tk ( X,Z,A )] vir Π i ψ d i ∧ ev ∗ ⟨ Π i τ d i α i | Π j β j ⟩ X,Z j ) ∗ β j . i α i ∧ ( ev Z g,A,T k = | Aut ( T k ) | 15

§ 3 Degeneration formula for symplectic cutting • Symplectic cutting Suppose that X 0 ⊂ X is an open subset with a hamiltonian S 1 -action such that → R is a Hamiltonian function with 0 as a regular value and H − 1 (0) is a H : X 0 − separating hypersurface in X . Cut X along H − 1 (0) , we obtain two connected manifolds X ± with boundary ∂X ± = H − 1 (0) . Denote by Z = H − 1 (0) /S 1 the symplectic reduction. Collapsing the S 1 -action on ∂X ± = H − 1 (0) , we obtain closed smooth manifolds ¯ X ± . Definition: Two symplectic manifolds ( ¯ X ± , ω ± ) are called the symplectic cuts of X along H − 1 (0) . 16

Here is the geometric description of symplectic cut: • Symplectic bow-up Let Y ⊂ X be a symplectic submanifold of X of codimension 2 k , N Y | X the normal bundle of Y in X . Perform the symplectic cut along the sphere bundle of N Y | X , we obtain two symplectic cuts ¯ X ± : ¯ X + := P Y ( N Y | X ⊕ C ) ¯ ˜ X − := X, symplectic blowup of X along Y . X + = P n , ¯ X − = ˜ • Example: Y = pt . Then ¯ X . 17

Denote by p : ˜ X − → X the natural projection of the blow-up. E = P Y ( N Y | X ) the exceptional divisor. • Symplectic blow-down: the opposite operation from ˜ X to X . 18

Degeneration formula Denote the reduction map by X + ∪ Z ¯ → ¯ X − . π : X − So we have a map X + ∪ Z ¯ → H 2 ( ¯ X − , Z ) . π ∗ : H 2 ( X, Z ) − For A ∈ H 2 ( X, Z ) , define [ A ] = A + ker π ∗ and define ∑ ⟨ Π i τ d i α i ⟩ X ⟨ Π i τ d i α i ⟩ X g, [ A ] := g,B . B ∈ [ A ] 19

Degeneration Formula:(gluing formula) X + ,Z X − ,Z ¯ ¯ i | ˇ ⟨ Π i τ d i α i ⟩ X ∑ ⟨ Π i ∈ I 1 τ d i α + g 1 ,A 1 ,T k ∆( T k ) ⟨ Π i ∈ I 2 τ d i α − i | β j ⟩ β j ⟩ g, [ A ] = g 2 ,A 2 ,T k , where the summation runs over all the splittings of g and A , all distribution of the insertion α ± i , all intermediate cohomology weighted partitions ( T j , β j ) and all configurations of connected components yielding a connected total domain, ∆( T k ) := Π j t j | Aut ( T k ) | , I 1 ∪ I 2 = { 1 , 2 , · · · , l } , and ˇ β j is dual to β j . 20

Local Gromov-Witten invariants Let S be a Fano surface and K S its canonical bundle. For β ∈ H 2 ( S, Z ) , denote by M g,k ( S, β ) the moduli space of k -pointed stable maps of degree β to S . Then the following diagram ev M g, 1 ( S, β ) − → S ρ ↓ M g, 0 ( S, β ) defines the obstruction bundle R 1 ρ ∗ ev ∗ K S whose fiber over a stable map f : C − → S is given by H 1 ( C, f ∗ K S ) . Chiang-Klemm-Yau-Zaslow defined the local Gromov-Witten invariants of K S as follows ∫ [ M g, 0 ( S,β )] vir e ( R 1 ρ ∗ ev ∗ K S ) . K S g,β = (2) 21