EQUIVARIANT GROMOV-WITTEN INVARIANTS OF ALGEBRAIC

arXiv:1407.1370v1 [math.AG] 5 Jul 2014

EQUIVARIANT GROMOV-WITTEN INVARIANTS OF ALGEBRAIC GKM MANIFOLDS CHIU-CHU MELISSA LIU AND ARTAN SHESHMANI A BSTRACT. An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many one-dimensional orbits. Using virtual localization, we express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

C ONTENTS 1. Introduction 1.1. Gromov-Witten invariants of a smooth projective variety 1.2. Equivariant Gromov-Witten invariants and virtual localization 1.3. Algebraic GKM manifolds and their Gromov-Witten invariants 1.4. Outline Acknowledgments 2. Algebraic GKM manfolds 2.1. Basic notation 2.2. GKM graph 3. Gromov-Witten Theory 3.1. Moduli of stable curves and Hodge integrals 3.2. Moduli of stable maps 3.3. Obstruction theory and virtual fundamental classes 3.4. Gromov-Witten invariants 3.5. Equivariant Gromov-Witten invariants 4. Virtual Localization 4.1. Torus fixed points and graph notation 4.2. Virtual tangent and normal bundles 4.3. Contribution from each graph 4.4. Sum over graphs References

1 1 2 3 4 4 4 4 5 6 6 8 8 9 10 11 11 13 18 19 21

1. I NTRODUCTION In this paper, we work over C. 1.1. Gromov-Witten invariants of a smooth projective variety. Let X be a smooth projective variety. Gromov-Witten invariants of X are virtual counts of parametrized algebraic curves of X. More precisely, let M g,n ( X, β) be the Kontsevich’s moduli 1

2

CHIU-CHU MELISSA LIU AND ARTAN SHESHMANI

space of n-pointed, genus g, degree β stable maps to X, where β ∈ H2 ( X, β) is an effective curve class. It is a proper Deligne-Mumford stack with a perfect obstruction theory of virtual dimension dvir =

(1)

Z

β

c1 ( TX ) + (dim X − 3)(1 − g) + n,

R where stands for the pairing between the (rational) homology and cohomology. There is a virtual fundamental class [30, 4, 2] [M g,n ( X, β)]vir ∈ Advir (Mg,n ( X, β); Q )

which can also be viewed as an element in H2dvir (M g,n ( X, β); Q ). The virtual fundamental class defines a Q-linear map Z

[M g,n ( X,β)]vir

: H ∗ (Mg,n ( X, β); Q ) −→ Q.

For i = 1, . . . , n, let evi : M g,n ( X, β) → X be the evaluation map at the i-th marked point. Genus g, degree β descendant Gromov-Witten invariants of X are defined by (2)

hτa1 (γ1 ), . . . , τan (γn )i X g,β : = H ∗ ( X; Q ),

Z

n

a

∏(ev∗i γi ∪ ψi i ) ∈ Q [M g,n ( X,β)]vir i =1

H 2 (M

where γi ∈ ai ∈ Z ≥0 , and ψi ∈ g,n ( X, β ); Q ) are ψ-classes (to be d i defined in Section 3.4). If γi ∈ H ( X; Q ) then the (2) vanishes unless n

∑ (di + 2ai − 2) = 2( i =1

Z

β

c1 ( TX ) + (dim X − 3)(1 − g)).

If X is a smooth algebraic variety which is not projective, then M g,n ( X, β) is usually not proper, so [M g,n ( X, β)]vir is not defined. If X is not projective but Mg,n ( X, β) is proper for some particular g, n, β, then [M g,n ( X, β)]vir exists and the invariants in (2) are defined for such g, n, β. 1.2. Equivariant Gromov-Witten invariants and virtual localization. Suppose that T = (C ∗ )m acts algebraically on X. Then T acts on Mg,n ( X, β). There exists a T-equivariant perfect obstruction theory, and a T-equivariant virtual fundamental class T [Mg,n ( X, β)]vir T ∈ A dvir (M g,n ( X, β ); Q ). Let R T := HT∗ ({point}; Q ) = H ∗ ( BT; Q ) = Q [u1 , . . . , um ] be the T-equivariant cohomology of a point, where ui ∈ HT2 ( BT; Q ). There is an R T -linear map Z

[M g,n ( X,β)]vir T

: HT∗ (Mg,n ( X, β); Q ) −→ HT∗ ({point}; Q ) = R T .

Genus g, degree β T-equivariant descendant Gromov-Witten invariants of X are defined by (3)

T hτa1 (γ1T ), . . . , τan (γnT )i X g,β : =

Z

n

∏(ev∗i γiT ∪ (ψiT )ai ) ∈ RT ,

[M g,n ( X,β)]vir T i =1

EQUIVARIANT GW INVARIANTS OF ALGEBRAIC GKM MANIFOLDS

3

where γiT ∈ HT∗ ( X; Q ), and ψiT ∈ HT2 (M g,n ( X, β); Q ) is a T-equivariant lift of d

ψi ∈ H 2 (M g,n ( X, β); Q ) (see Section 3.5). If γiT ∈ HTi ( X; Q ) then ∑n ( d i +2a i −2)−2dvir

i =1 T hτa1 (γ1T ), . . . , τan (γnT )i X g,β ∈ HT

({point}; Q ).

In particular, (3) vanishes if ∑ni=1 (di + 2ai − 2) < 2dvir . The T-equivariant Gromov-Witten invariants (3) are related to the GromovWitten invariants (2) as follows. Let γi be the image of γiT under the ring homomorphism HT∗ ( X; Q ) −→ H ∗ ( X; Q ). Then T T XT hτa1 (γ1 ), . . . , τan (γn )i X = h τ ( γ ) , . . . , τ ( γ )i . a a n g,β n 1 g,β 1 u1 =···= u m =0

The torus fixed part of the restriction of the T-equivariant perfect obstruction theory to the T fixed substack M g,n ( X, β) T ⊂ M g,n ( X, β) defines a perfect obT T struction theory on M g,n ( X, β) T and a virtual class [M g,n ( X, β) T ]vir T ∈ A ∗ (M g,n ( X, β ) ; Q ); the torus moving part defines the virtual normal bundle N vir of the inclusion Mg,n ( X, β) T ⊂ M g,n ( X, β). By localization of virtual fundamental class [17, 3], the RHS of (3) is equal to (4)

Z

i ∗ (ev∗i γiT ∪ (ψiT ) ai ) e T ( N vir ) i =1 n

∏ [M g,n ( X,β) T ]vir T

where i : M g,n ( X, β) T ֒→ Mg,n ( X, β) is the inclusion, and e T ( N vir ) is the Tequivariant Euler class of N vir . It is know that e T ( N vir ) is invertible in HT∗ (Mg,n ( X, β); Q ) ⊗ R T Q T , where Q T = Q (u1 , . . . , um ) is the fracitonal field of R T = Q [u1 , . . . , um ]. Suppose that X is non-compact, and for some g, n, β, M g,n ( X, β) is not proper but M g,n ( X, β) T is. Then the RHS of (3) is not defined, but (4) is. In this case, T we define T-equivariant Gromov-Witten invariants hτa1 (γ1T ), . . . , τan (γnT )i X g,β by (4), which is an element in Q (u1 , . . . , um ) instead of Q [u1 , . . . , um ]. 1.3. Algebraic GKM manifolds and their Gromov-Witten invariants. In this paper, an algebraic GKM manifold, named after Goresky-Kottwitz-MacPherson, is a non-singular algebraic variety equipped with an algebraic action of T = (C ∗ )m , such that there are finitely many torus fixed points and finitely many one-dimensional orbits. Examples of algebraic GKM manifolds include toric manifolds, Grassmanians, flag manifolds, etc. If X is an algebraic GKM manifold then each connected component of M g,n ( X, β) is, up to some quasi-finite map, a product of moduli stacks of pointed stable curves, and the RHS of (4) can be expressed in terms Hodge integrals on moduli stacks of pointed stable curves. This algorithm was first described by Kontsevich for genus zero Gromov-Witten invariants of Pr in 1994 [28], before the construction of virtual fundamental class and the proof of virtual localization. The moduli spaces M0,n (Pr , d) of genus zero stable maps to Pr are proper smooth DM stacks, so there exists a fundamental class [M0,n (Pr , d)] ∈ H∗ (M0,n (Pr , d); Q ), and one may apply the classical Atiyah-Bott localization formula [1] in this case. H. Spielberg derived a formula of genus zero Gromov-Witten invariants of toric manifolds in his thesis [43]. Localization computations of all genus equivariant GromovWitten invariants of toric manifolds can be found in [32]. The main purpose of

4


this paper is to provide details of the virtual localization calculations of all genus equivariant Gromov-Witten invariants for general algebraic GKM manifolds. 1.4. Outline. In Section 2, we define algebraic GKM manifolds and their GKM graphs, following [18, 19]. In Section 3, we give a brief review of Gromov-Witten theory. In Section 4, we compute all genus equivariant descendant Gromov-Witten invariants of an arbitrary algebraic GKM manifold by virtual localization. Acknowledgments. The first author wish to thank Tom Graber for his suggestion of generalizing the computations for toric manifolds in [32] to GKM manifolds. The second author would like to thank the Columbia University for hospitality during his visits. This work is partially supported by NSF DMS-1159416 and NSF DMS-1206667. 2. A LGEBRAIC GKM

MANFOLDS

In this section, we review the geometry of algebraic GKM manifolds, following [18], and introduce the GKM graph associated to an algebraic GKM manifold, following [19]. The GKM graph in this paper can be non-compact since we consider algebraic GKM manifolds which are not necessarily compact. In Section 4, we will see that the GKM graph contains all the information needed for computing Gromov-Witten invariants and equivariant Gromov-Witten invariants of the GKM manifold. 2.1. Basic notation. Let X be a non-singular algebraic variety of dimension r. We say X is an algebraic GKM manifold if it is equipped with an algebraic action of a complex algebraic torus T = (C ∗ )m with only finitely many torus fixed points and finitely many one-dimensional orbits. Let N = Hom(C ∗ , T ) ∼ = Z m be the lattice of 1-parameter subgroups of T, and ∗ let M = Hom( T, C ) be the lattice of irreducible characters of T. Then M = Hom( N, Z ) is the dual lattice of N. Let NR = N ⊗Z R and MR = M ⊗Z R, so that they are dual real vector spaces of dimension k. Let K = U (1)m be the maximal compact subgroup of T. Then NR can be canonically identified with the Lie algebra of K. Let NQ = N ⊗Z Q and let MQ = M ⊗Z Q. Then MQ can be canonically identified with HT2 (point; Q ). We make the following assumption on X. Assumption 5. (1) The set X T of T fixed points in X is non-empty. (2) The closure of a one-dimensional orbit is either a complex projective line P1 or a complex affine line C. Note that (1) and (2) hold when X is proper. Indeed, if X is proper then the closure of any one-dimensional orbit is P1 . Example 6. If X is a non-singular toric variety defined by a finite fan, then X is an alegebraic GKM manifold. Example 7 (The Grassmannian Gr (k, m)). Let Gr (k, m) be the set of k-dimensional linear subspace of C m . It is a nonsingular projective variety of dimension k(m − k). Let T = (C ∗ )m act on C m by

( t1 , . . . , t m ) · ( z1 , . . . , z m ) = ( t1 z1 , . . . , t m z m ).


5

Given t ∈ T, let φt : C m → C m be defined by φt (z) = t · z. Let T act on Gr (k, m) by t · V = φt (V ), where V is a k-dimensional linear subspace of C m . Given J ⊂ {1, . . . , m}, let J c := {1, . . . , m} \ J, and define C J := {(z1 , . . . , zm ) ∈ C m : zi = 0 if i ∈ J c } ∼ = C| J| Note that φt (C J ) = C J for any t ∈ T, J ⊂ {1, . . . , n}. The torus-fixed points in Gr (k, m) are Gr (k, m) T = {C J : J ⊂ {1, . . . , n}, | J | = k}. So there are (mk) torus-fixed points in G (k, m). ′ ′ ′ Let C J and C J be distinct T-fixed points in Gr (k, m). Then C J ∩ C J = C J ∩ J . There ′ is a torus-fixed line connecting C J and C J if and only of | J ∩ J ′ | = k − 1. In this case, ′ | J ∪ J | = k + 1. The T-fixed lines in G (k, m) are

{ℓ I,K : I ⊂ K ⊂ {1, . . . , m}, | I | = k − 1, |K | = k + 1} where

∼ P1 . ℓ I,K = {V ∈ G (k, m) : C I ⊂ V ⊂ C K } = Suppose that I ⊂ K ⊂ {1, . . . , m}, and | I | = k − 1, |K | = k + 1. Then K = I ∪ { j1 , j2 }, m − k +1 m where j1 , j2 ∈ I c . So there are (k− 1)( 2 ) torus-fixed lines in G ( k, m ). 2.2. GKM graph. Let X be an algebraic GKM manifold of dimenison r, so that T = (C ∗ )m acts algebraically on X. Following [19], we define a graph Υ as follows. Let V (Υ) (resp. E(Υ)) denote the set of vertices (resp. edges) in Υ. (1) (Vertices) We assign a vertex σ to each torus fixed point p σ in X. (2) (Edges) We assign an edge ǫ to each one-dimensional Oǫ in X. Let ℓe be the closure of Oǫ . (3) (Flags) The set of flags in the graph Υ is given by F (Υ) = {(ǫ, σ ) ∈ E(Υ) × F (Υ) : σ ∈ ǫ} = {(ǫ, σ ) ∈ E(Υ) × F (Υ) : p σ ∈ ℓǫ }. The Assumption 5 can be rephrased in terms of the graph Υ. Assumption 8. (1) V (Υ) is non-empty. (2) Each edge in E(Υ) contains at least one vertex. Let E(Υ)c = {ǫ ∈ E(Υ) : ℓǫ ∼ = P1 } be the set of compact edges in Υ. Note that E(Υ)c = E(Υ) if X is proper. Given a vertex σ ∈ V (Υ), we denote by Eσ the set of edges containing σ, i.e. Eσ := {e ∈ E : (ǫ, σ ) ∈ F (Υ)}. Then | Eσ | = r for all σ ∈ V (Υ), so Υ is an r-valent graph. Given a flag (ǫ, σ ) ∈ F (Υ), let w(ǫ, σ ) ∈ M = Hom( T, C ∗ ) be the weight of T-action on Tpσ ℓǫ , the tangent line to ℓǫ at the fixed point pσ , namely, w(ǫ, σ ) := c1T ( Tpσ ℓǫ ) ∈ HT2 ( pσ ; Z ) ∼ = M. This gives rise to a map w : F (Υ) → M satisfying the following properties. (1) (GKM hypothesis) Given any σ ∈ V (Υ), and any two distinct edges ǫ, ǫ′ ∈ Eσ , w(ǫ, σ ) and w(ǫ′ , σ ) are linearly independent in MR . (2) Any edge ǫ ∈ Eσ connecting the vertices σ, σ ′ ∈ V (Υ) satisfies the property that: (a) w(ǫ, σ ) + w(ǫ, σ ′ ) = 0.

6


(b) Let Eσ = {ǫ1 , . . . , ǫr }, where ǫr = ǫ. For any ǫi ∈ Eσ there exists ǫi′ ∈ Eσ′ and ai ∈ Z such that w(ǫi′ , σ ′ ) = w(ǫi , σ ) − ai w(ǫ, σ ). In particular, ǫr′ = ǫr = ǫ and ar = 2. Let ǫ be as in (2). The normal bundle of ℓǫ ∼ = P1 in X is given by Nℓǫ /X ∼ = L1 ⊕ · · · ⊕ Lr −1 where Li is a degree ai T-equivariant line bundle over ℓǫ such that the weights of the T-action on the fibers ( Li ) pσ and ( Li ) pσ′ are w(ǫi , σ ) and w(ǫi′ , σ ′ ), respectively. The map w : F (Υ) → M is called the axial function. Example 9 (Gr (r, m)). The GKM graph of Gr (r, m) is a k(m − k)-valent graph Υ such that V ( Υ) E ( Υ) F ( Υ)

= {σJ : J ⊂ {1, . . . , n}, | J | = k} = E(Υ)c = {ǫ I,K : I ⊂ K ⊂ {1, . . . , m}| I | = k − 1, |K | = k + 1}, = {(ǫ I,K , σJ ) ∈ E(Υ) × V (Υ) : I ⊂ J ⊂ K

w(ǫ IK , σI ∪{ j1 } ) = −w(ǫ IK , σI ∪{ j2 } ) = u j2 − u j1 ,

j1 , j2 ∈ I c , K = I ∪ { j1 , j2 }.

We define the 1-skeleton of X to be the union of 1-dimensional orbit closures: X 1 :=

[

ℓǫ .

ǫ∈ E ( Υ)

The formal completion Xˆ of X along the 1-skeleton X 1 , together with the T-action, can be reconstructed from the graph Υ and w : F (Υ) → M. We call (Υ, w) the GKM graph of X with the T-action. If ρ : T ′ → T is a homomorphism between complex algebraic tori, then T ′ acts on X by t′ · x = ρ(t′ ) · x, where t′ ∈ T ′ , ρ(t′ ) ∈ T, x ∈ X. The GKM graph of X with this T ′ -action is given by (Υ, ρ ∗ ◦ w), where ρ∗ : M = Hom( T, C ∗ ) → Hom( T ′ , C ∗ ). 3. G ROMOV-W ITTEN T HEORY In this section, we give a brief review of Gromov-Witten theory and equivariant Gromov-Witten theory. 3.1. Moduli of stable curves and Hodge integrals. An n-pointed, genus g prestable curve is a connected algebraic curve C of arithmetic genus g together with n ordered marked points x1 , . . . , x n ∈ C, where C has at most nodal singularities, and x1 , . . . , x n are distinct smooth points. An n-pointed, genus g prestable curve (C, x1, . . . , xn ) is stable if its automorphism group is finite, or equivalently, HomOC (ΩC ( x1 + · · · + x n ), OC ) = 0. Let M g,n be the moduli space of n-pointed, genus g stable curves, where n, g are nonnegative integers. We assume that 2g − 2 + n > 0, so that M g,n is nonempty. Then M g,n is a proper smooth Deligne-Mumford stack of dimension 3g − 3 + n [8, 26, 24, 25]. The tangent space of M g,n at a moduli point [(C, x1 , . . . , x n )] ∈ M g,n is given by Ext1OC (ΩC ( x1 + · · · + x n ), OC ).


7

Since M g,n is a proper Deligne-Mumford stack, we may define Z

M g,n

: A∗ (Mg,n ; Q ) → Q.

We now introduce some classes in A∗ (Mg,n ). There is a forgetful morphism π : Mg,n+1 → M g,n given by forgetting the (n + 1)-th marked point (and contracting the unstable irreducible component if there is one):

[(C, x1 , . . . , xn , xn+1 )] 7→ [(C st , x1 , . . . , xn )] where (C st , x1 , . . . , x n ) is the stabilization of the prestable curve (C, x1 , . . . , x n ). π : Mg,n+1 → M g,n can be identified with the universal curve over Mg,n .

• (λ classes) Let ωπ be the relative dualizing sheaf of π : M g,n+1 → M g,n . The Hodge bundle E = π∗ ωπ is a rank g vector bundle over Mg,n whose fiber over the moduli point [(C, x1 , . . . , x n )] ∈ M g,n is H 0 (C, ωC ), the space of sections of the dualizing sheaf ωC of the curve C. The λ classes are defined by λ j = c j (E ) ∈ A j (M g,n ; Q ). • (ψ classes) The i-th marked point xi gives rise a section si : Mg,n → M g,n+1 of the universal curve. Let L i = s∗i ωπ be the line bundle over M g,n whose fiber over the moduli point [(C, x1 , . . . , x n )] ∈ M g,n is the cotangent line Tx∗i C of C at xi . The ψ classes are defined by ψi = c1 (L i ) ∈ A1 (M g,n ; Q ). Hodge integrals are top intersection numbers of λ classes and ψ classes: (10)

Z

a

M g,n

kg

k

ψ11 · · · ψnan λ11 · · · λ g ∈ Q.

By definition, (10) is zero unless a1 + · · · + an + k1 + 2k2 + · · · + gk g = 3g − 3 + n. Using Mumford’s Grothendieck-Riemann-Roch calculations in [36], Faber proved, in [11], that general Hodge integrals can be uniquely reconstructed from the ψ integrals (also known as descendant integrals): Z

(11)

M g,n

ψ1a1 · · · ψnan .

The descendant integrals can be computed recursively by Witten’s conjecture which asserts that the ψ integrals (11) satisfy a system of differential equations known as the KdV equations [45]. The KdV equations R and the string equation determine all the ψ integrals (11) from the initial value M 1 = 1. For example, from the initial 0,3 R value M 1 = 1 and the string equation, one can derive the following formula of 0,3 genus 0 descendant integrals: (12)

Z

a

M0,n

ψ11 · · · ψnan =

( n − 3) ! a1 ! · · · a n !

where a1 + · · · + an = n − 3 [28, Section 3.3.2]. The Witten’s conjecture was first proved by Kontsevich in [27]. By now, Witten’s conjecture has been reproved many times (Okounkov-Pandharipande [37],

8


Mirzakhani [34], Kim-Liu [23], Kazarian-Lando [22], Chen-Li-Liu [6], Kazarian [21], Mulase-Zhang [35], etc.). 3.2. Moduli of stable maps. Let X be a nonsingular projective or quasi-projective variety, and let β ∈ H2 ( X; Z ). An n-pointed, genus g, degree β prestable map to X is a morphism f : (C, x1 , . . . , x n ) → X, where (C, x1 , . . . , x n ) is an n-pointed, genus g prestable curve, and f ∗ [C ] = β. Two prestable maps f : (C, x1 , . . . , x n ) → X,

f ′ : (C ′ , x1′ , . . . , x n′ ) → X

are isomorphic if there exists an isomorphism φ : (C, x1 , . . . , x n ) → (C ′ , x1′ , . . . , x n′ ) of n-pointed prestable curves such that f = f ′ ◦ φ. A prestable map f : (C, x1 , . . . , x n ) → X is stable if its automorphism group is finite. The notion of stable maps was introduced by Kontsevich [28]. The moduli space M g,n ( X, β) of n-pointed, genus g, degree β stable maps to X is a Deligne-Mumford stack which is proper when X is projective [5]. 3.3. Obstruction theory and virtual fundamental classes. The tangent space T 1 and the obstruction space T 2 at a moduli point [ f : (C, x1 , . . . , x n ) → X ] ∈ M g,n ( X, β) fit in the tangent-obstruction exact sequence: 0 →Ext0OC (ΩC ( x1 + · · · + x n ), OC ) → H 0 (C, f ∗ TX ) → T 1

(13)

→Ext1OC (ΩC ( x1 + · · · + xn ), OC ) → H 1 (C, f ∗ TX ) → T 2 → 0

where

• Ext0OC (ΩC ( x1 + · · · + xn ), OC ) is the space of infinitesimal automorphisms of the domain (C, x1 , . . . , x n ), • Ext1OC (ΩC ( x1 + · · · + xn ), OC ) is the space of infinitesimal deformations of the domain (C, x1 , . . . , x n ), • H 0 (C, f ∗ TX ) is the space of infinitesimal deformations of the map f , and • H 1 (C, f ∗ TX ) is the space of obstructions to deforming the map f . T 1 and T 2 form sheaves T 1 and T 2 on the moduli space M g,n ( X, β). Let X be a nonsingular projective variety. We say X is convex if H 1 (C, f ∗ TX ) = 0 for all genus 0 stable maps f . Projective spaces P n , or more generally, generalized flag varieties G/P, are examples of convex varieties. When X is convex and g = 0, the obstruction sheaf T 2 = 0, and the moduli space M0,n ( X, β) is a smooth Deligne-Mumford stack. In general, M g,n ( X, β) is a singular Deligne-Mumford stack equipped with a perfect obstruction theory: there is a two term complex of locally free sheaves E → F on M g,n ( X, β) such that 0 → T 1 → F∨ → E∨ → T 2 → 0 is an exact sequence of sheaves. (See [4] for the complete definition of a perfect obstruction theory.) The virtual dimension dvir of M g,n ( X, β) is the rank of the virtual tangent bundle T vir = F ∨ − E∨ . (14)

dvir =

Z

β

c1 ( TX ) + (dim X − 3)(1 − g) + n


9

Suppose that M g,n ( X, β) is proper. (Recall that if X is projective then M g,n ( X, β) is proper for any g, n, β.) Then there is a virtual fundamental class

[Mg,n ( X, β)]vir ∈ Advir (Mg,n ( X, β); Q ). The virtual fundamental class has been constructed by Li-Tian [30], Behrend-Fantechi [4] in algebraic Gromov-Witten theory. The virtual fundamental class allows us to define Z

[M g,n ( X,β)]vir

: A∗ (M g,n ( X, β); Q ) −→ Q,

α 7→ deg(α ∩ [M g,n ( X, β)]vir ).

3.4. Gromov-Witten invariants. Let X be a nonsingular projective variety. GromovWitten invariants are rational numbers defined by applying Z

[M g,n ( X,β)]vir

: A∗ (Mg,n ( X, β)) → Q

to certain classes in A∗ (Mg,n ( X, β)). Let evi : M g,n ( X, β) → X be the evaluation at the i-th marked point: evi sends [ f : (C, x1 , . . . , xn ) → X ] ∈ Mg,n ( X, β) to f ( xi ) ∈ X. Given γ1 , . . . , γn ∈ A∗ ( X ), define (15)

h γ1 , . . . , γ n i X g,β =

Z

[M g,n ( X,β)]vir

ev1∗ γ1 ∪ · · · ∪ ev∗n γn ∈ Q.

These are known as the primary Gromov-Witten invariants of X. More generally, we may also view [M g,n ( X, β)]vir as a class in H2d (Mg,n ( X, β)). Then (15) is defined for ordinary cohomology classes γ1 , . . . , γn ∈ H ∗ ( X ), including odd cohomology classes which do not come from A∗ (Mg,n ( X, β)). Let π : M g,n+1 ( X, β) → M g,n ( X, β) be the universal curve. For i = 1, . . . , n, let si : M g,n ( X, β) → M g,n+1 ( X, β), be the section which corresponds to the i-th marked point. Let ωπ → M g,n+1 ( X, β) be the relative dualizing sheaf of π, and let L i = s∗i ωπ be the line bundle over M g,n ( X, β) whose fiber at the moduli point [ f : (C, x1 , . . . , xn ) → X ] ∈ M g,n ( X, β) is the cotangent line Tx∗i C at the i-th marked point xi . The ψ-classes are defined to be ψi := c1 (L i ) ∈ A1 (Mg,n ( X, β)),

i = 1, . . . , n.

We use the same notation ψi to denote the corresponding classes in the ordinary cohomology group H 2 (Mg,n ( X, β)). Genus g, degree β descendant Gromov-Witten invariants of X are defined by (16) hτa1 (γ1 ) · · · τan (γn )i X g,β : =

Z

a

[M g,n

( X,β)]vir

ev1∗ γ1 ∪ ψ11 ∪ · · · ∪ ev∗n γn ∪ ψnan ∈ Q.

Suppose that γi ∈ H di ( X ). Then (16) is zero unless n

(17)

∑ (di + 2ai − 2) = 2 i =1

Z

β

c1 ( TX ) + (dim X − 3)(1 − g) .

10


3.5. Equivariant Gromov-Witten invariants. Let X be a non-singular projective or quasi-projective algebraic variety, equipped with an algebraic action of T = (C ∗ )m . Then T acts on M g,n ( X, β) by t · [ f : (C, x1 , . . . , x n ) → X ] 7→ [t · f : (C, x1, . . . , x n ) → X ] where (t · f )(z) = t · f (z), z ∈ C. The evaluation maps evi : Mg,n ( X, β) → X are T-equivariant and induce ev∗i : A∗T ( X; Q ) → A∗T (M g,n ( X, β); Q ). Suppose that M g,n ( X, β) is proper, so that there are virtual fundamental classes

[M g,n ( X, β)]vir ∈ Advir (Mg,n ( X, β); Q ), where dvir =

Z

β

T [M g,n ( X, β)]vir T ∈ A dvir (M g,n ( X, β ); Q ),

c1 ( TX ) + (r − 3)(1 − g) + n.

Given γi ∈ Adi ( X; Q ) = H 2di ( X; Q ) and ai ∈ Z ≥0 , define hτai (γ1 ) · · · τan (γn )i X g,β as in Section 3.4: Z n a (18) hτa1 (γ1 ) · · · τan (γn )i X = ∏ ev∗i γi ∪ ψi i ∈ Q. g,β [M g,n ( X,β)]vir i =1

By definition, (18) is zero unless ∑ni=1 di = dvir . In this case, Z n (19) hτa1 (γ1 ) · · · τan (γn )i X = ∏ ev∗i γiT ∪ (ψiT )ai g,β [M g,n ( X,β)]vir T i =1

d

where γiT ∈ A Ti ( X ) is any T-equivariant lift of γi ∈ Adi ( X ), and ψiT ∈ A1T (Mg,n ( X, β)) is any T-equivariant lift of ψi ∈ A1 (Mg,n ( X, β)). In this paper, we fix a choice of ψiT as follows. A stable map f : (C, x1 , . . . , x n ) → X induces C-linear maps Txi C → T f ( xi) X for i = 1, . . . , n. This gives rise to ∗ L∨ i → evi TX. The T-action on X induces a T-action on TX, so that TX is a Tequivariant vector bundle over X, and ev∗i TX is a T-equivariant vector bundle ∗ over M g,n ( X, β). Let T act on L i such that L ∨ i → evi TX is T-equivariant, and define ψiT = c1T (L i ) ∈ A1T (Mg,n ( X, β)), i = 1, . . . , n. Then ψiT is a T-equivariant lift of ψi = c1 (L i ) ∈ A1 (Mg,n ( X, β)). d

Given γiT ∈ A Ti ( X; Q ), we define genus g, degree β T-equivariant descendant Gromov-Witten invariants Z n ∗ T T ai T ev γ ( ψ ) hτa1 (γ1T ), · · · , τan (γnT )i X : = ∏ i i i g,β (20)

[M g,n ( X,β)]vir,T i =1 n

∈ Q [u1 , . . . , um ]( ∑ di − dvir ). i =1

where Q [u1 , . . . , um ](k) is the space of degree k homogeneous polynomials in u1 , . . . , um with rational coefficients. In particular, ( 0, ∑ni=1 di < dvir , X hτa1 (γ1T ), · · · , τan (γnT )i g,βT = X hτa1 (γ1 ), · · · , τan (γn )i g,β ∈ Q, ∑ni=1 di = dvir . d

where γi ∈ Adi ( X; Q ) is the image of γiT under A Ti ( X; Q ) → Adi ( X; Q ).


11

Let M g,n ( X, β) T ⊂ M g,n ( X, β) be the substack of T-fixed points, and let i : Mg,n ( X, β) T → M g,n ( X, β) be the inclusion. Let N vir be the virtual normal bundle of substack M g,n ( X, β) T in M g,n ( X, β); in general, N vir has different ranks on different connected components of M g,n ( X, β) T . By virtual localization, (21) Z Z n i ∗ ∏ni=1 ev∗i γiT ∪ (ψiT ) ai ∗ T T ai . ∏ evi γi ∪ (ψi ) = [Mg,n ( X,β) T]vir e T ( N vir ) [M g,n ( X,β)]vir T i =1 T If Mg,n ( X, β) T is proper but Mg,n ( X, β) is not, we define

(22)

T hτa1 (γ1T ), . . . , τan (γnT )i X g,β

=

Z

[M g,n ( X,β) T ]vir T

i ∗ ∏ ni=1 ev∗i γiT ∪ (ψiT ) ai ∈ QT . e T ( N vir )

When M g,n ( X, β) is not proper, the right hand side of (22) is a rational function (instead of a polynomial) in u1 , . . . , um . It can be nonzero when ∑ di < dvir , and does not have a nonequivariant limit (obtained by setting ui = 0) in general. 4. V IRTUAL L OCALIZATION In this section, we compute all genus equivariant descendant Gromov-Witten invariants of any algebraic GKM manifold by virtual localization. This generalizes the toric case in [32, Section 5]. Let X be an algebraic GKM manifold of dimension r, with an algebraic action of T = (C ∗ )m . 4.1. Torus fixed points and graph notation. In this subsection, we describe the T-fixed points in M g,n ( X, β). Following Kontsevich [28], given a stable map f : (C, x1, . . . , xn ) → X such that

[ f : (C, x1 , . . . , xn ) → X ] ∈ M g,n ( X, β) T , we will associate a decorated graph ~Γ. We first give a formal definition.

~ ~g,~s) for n-pointed, genus g, degree β Definition 23. A decorated graph ~Γ = (Γ, ~f , d, stable maps to X consists of the following data. (1) Γ is a compact, connected 1 dimensional CW complex. We denote the set of vertices (resp. edges) in Γ by V (Γ) (resp. E(Γ)). Let F (Γ) = {(e, v) ∈ E(Γ) × V (Γ) | v ∈ e} be the set of flags in Γ. (2) The label map ~f : V (Γ) ∪ E(Γ) → V (Υ) ∪ E(Υ)c sends a vertex v ∈ V (Γ) to a vertex σv ∈ V (Υ), and sends an edge e ∈ E(Γ) to an edge ǫe ∈ E(Υ)c . Moreover, ~f defines a map from the graph Γ to the graph Υ: if (e, v) is a flag in Γ then (ǫe , σv ) is a flag in Υ. (3) The degree map d~ : E(Γ) → Z >0 sends an edge e ∈ E(Γ) to a positive integer de . (4) The genus map ~g : V (Γ) → Z ≥0 sends a vertex v ∈ V (Γ) to a nonnegative integer gv . (5) The marking map ~s : {1, 2, . . . , n} → V (Γ) is defined if n > 0. The above maps satisfy the following two constraints:

12


∑

(i) (topology of the domain)

gv + | E(Γ)| − |V (Γ)| + 1 = g.

v ∈V ( Γ )

(ii) (topology of the map)

∑

de [ℓǫe ] = β.

e∈ E(Γ)

~ ~g,~s) satisfying the above Let Gg,n ( X, β) be the set of all decorated graphs ~Γ = (Γ, ~f , d, constraints. We now describe the geometry and combinatorics of a stable map f : (C, x1 , . . . , x n ) → X which represents a T-fixed point in M g,n ( X, β). For any t ∈ T, there exists an automorphism φt : (C, x1 , . . . , x n ) such that t · f (z) = f ◦ φt (z) for any z ∈ C. Let C ′ be an irreducible component of C, and let f ′ = f |C′ : C ′ → X. There are two possibilities: Case 1: f ′ is a constant map, and f (C ′ ) = { pσ }, where pσ is a fixed point in X associated to some σ ∈ V (Υ) Case 2: C ′ ∼ = P1 and f (C ′ ) = ℓǫ , where ℓǫ is a T-invariant P1 in X associated to some ǫ ∈ E(Υ)c . We define a decorated graph ~Γ associated to f : (C, x1 , . . . , x n ) → X as follows. (1) (Vertices) We assign a vertex v to each connected component Cv of f −1 ( X T ). (a) (label) f (Cv ) = { pσ } for some σ ∈ V (Υ); we define ~f (v) = σv = σ. (b) (genus) Cv is a curve or a point. If Cv is a curve then we define ~g(v) = gv to be the arithmetic genus of Cv ; if Cv is a point then we define ~g(v) = gv = 0. (c) (marking) For i = 1, . . . , n, define ~s (i ) = v if x i ∈ Cv . (2) (Edges) For any ǫ ∈ E(Υ), let Oǫ ∼ = C ∗ be the 1-dimensional orbit whose closure is ℓǫ . Then G X1 \ X T = Oǫ ǫ∈ E ( Υ)

where the right hand side is a disjoint union of connected components. We assign an edge e to each connected component Oe ∼ = C ∗ of f −1 ( X 1 \ X T ). 1 (a) (label) Let Ce ∼ = P be the closure of Oe . Then f (Ce ) = ℓǫ for some ǫ ∈ E(Υ)c ; we define ~f (e) = ǫe = ǫ. (b) (degree) We define d~(e) = de to be the degree of the map f |Ce : Ce ∼ = 1. P P1 → ℓǫ ∼ = (3) (Flags) The set of flags in the graph Γ is defined by F (Γ) = {(e, v) ∈ E(Γ) × V (Γ) | Ce ∩ Cv 6= ∅}.

~ ~g,~s) satisfying the The above (1), (2), (3) define a decorated graph ~Γ = (Γ, ~f , d, constraints (i) and (ii) in Definition 23. Therefore ~Γ ∈ Gg,n ( X, β). This gives a map from M g,n ( X, β) T to the discrete set Gg,n ( X, β). Let F~Γ ⊂ M g,n ( X, β) T denote the preimage of ~Γ. Then M g,n ( X, β) T =

G

F~Γ

~Γ∈ Gg,n ( X,β)

where the right hand side is a disjoint union of connected components. We next describe the fixed locus F~Γ associated to each decorated graph ~Γ ∈ Gg,n ( X, β). For later convenience, we introduce some definitions.


13

Definition 24. Given a vertex v ∈ V (Γ), we define Ev = {e ∈ E(Γ) | (e, v) ∈ F (Γ)}, the set of edges emanating from v, and define Sv = ~s−1 (v) ⊂ {1, . . . , n}. The valency of v is given by val(v) = | Ev |. Let nv = |Sv | be the number of marked points contained in Cv . We say a vertex is stable if 2gv − 2 + val(v) + nv > 0. Let V S (Γ) be the set of stable vertices in V (Γ). There are three types of unstable vertices: V 1 ( Γ) V

1,1 2

= {v ∈ V (Γ) | gv = 0, val(v) = 1, nv = 0},

(Γ) = {v ∈ V (Γ) | gv = 0, val(v) = nv = 1},

V ( Γ)

= {v ∈ V (Γ) | gv = 0, val(v) = 2, nv = 0}.

Then V (Γ) is the disjoint union of V 1 (Γ), V 1,1 (Γ), V 2 (Γ), and V S (Γ). The set of stable flags is defined to be F S (Γ) = {(e, v) ∈ F (Γ) | v ∈ V S (Γ)}.

~ ~g,~s), the curves Ce and the maps f |C : Given a decorated graph ~Γ = (Γ, ~f , d, e ~ Ce → ℓǫe ⊂ X are determined by Γ. If v ∈ / V S (Γ) then Cv is a point. If v ∈ V S (Γ) then Cv is a curve, and y(e, v) := Ce ∩ Cv is a node of C for e ∈ Ev . Cv , {y(e, v) : e ∈ Ev } ∪ { xi | i ∈ Sv }

is a (val(v) + nv )-pointed, genus gv curve, which represents a point in M gv ,val( v)+nv . We call this moduli space M gv ,Ev ∪Sv instead of M gv ,val( v)+nv because we would like to label the marked points on Cv by Ev ∪ Sv instead of {1, 2, . . . , val(v) + nv }. Then M~Γ = ∏ Mgv ,Ev ∪Sv . v ∈V S ( Γ )

The automorphism group A~Γ for any point [ f : (C, x1 , . . . , x n ) → X ] ∈ F~Γ fits in the following short exact sequence of groups: 1→

∏

Z de → A~Γ → Aut(~Γ) → 1

e∈ E(Γ)

where Z de is the automorphism group of the degree de morphism f |Ce : Ce ∼ = P 1 → ℓ ǫe ∼ = P1 ,

~ ~g,~s). and Aut(~Γ) is the automorphism group of the decorated graph ~Γ = (Γ, ~f , d, There is a morphism i~Γ : M~Γ → M g,n ( X, β) whose image is the fixed locus F~Γ associated to ~Γ ∈ Gg,n ( X, β). The morphism i~Γ induces an isomorphism [M~Γ /A~Γ ] ∼ = F~Γ . 4.2. Virtual tangent and normal bundles. Given a decorated graph ~Γ ∈ Gg,n ( X, β) and a stable map f : (C, x1, . . . , x n ) → X which represents a point in the fixed locus F~Γ associated to ~Γ, let B1 = Hom(ΩC ( x1 + · · · + x n ), OC ), B4 = Ext1 (ΩC ( x1 + · · · + x n ), OC ),

B2 = H 0 (C, f ∗ TX ) B5 = H 1 (C, f ∗ TX )

14


f

T acts on B1 , B2, B4 , B5 . Let Bim and Bi be the moving and fixed parts of Bi . We have the following exact sequences: f

f

f

f

(25)

0 → B1 → B2 → T 1, f → B4 → B5 → T 2, f → 0

(26)

0 → B1m → B2m → T 1,m → B4m → B5m → T 2,m → 0

The irreducible components of C are

{Cv | v ∈ V S (Γ)} ∪ {Ce | e ∈ E(Γ)}. The nodes of C are

{yv = Cv | v ∈ V 2 (Γ)} ∪ {y(e, v) | (e, v) ∈ F S (Γ)} 4.2.1. Automorphisms of the domain. Given any (e, v) ∈ F (Γ), let y(e, v) = Ce ∩ Cv , and define w( e,v) := e T ( Ty( e,v)Ce ) =

w(ǫe , σv ) ∈ HT2 (y(e, v); Q ) = M ⊗Z Q. de

We have f

=

B1

M

Hom(ΩCe (y(e, v) + y(e, v′ )), OCe )

M

H 0 (Ce , TCe (−y(e, v) − y(e, v′ ))

e ∈ E(Γ ) ( e, v ), ( e, v ′ ) ∈ F ( Γ )

=

e ∈ E(Γ ) ( e, v ), ( e, v ′ ) ∈ F ( Γ )

B1m

M

=

Ty( e,v)Ce

v ∈V 1 ( Γ ),( e,v )∈ F (Γ )

4.2.2. Deformations of the domain. Given any v ∈ V S (Γ), define a divisor xv of Cv by xv =

∑ i ∈ Sv

xi +

∑

y(e, v).

e ∈ Ev

Then f

B4

=

M

Ext1 (ΩCv (xv ), OC ) =

v ∈V S ( Γ )

B4m

=

M

M

T M gv ,Ev ∪Sv

v ∈V S ( Γ )

Tyv Ce ⊗ Tyv Ce′ ⊕

v ∈V 2 ( Γ ),Ev ={ e,e ′ }

M

Ty( e,v)Cv ⊗ Ty( e,v)Ce

( e,v )∈ F S( Γ)

where e T ( Tyv Ce ⊗ Tyv Ce′ ) e T ( Ty( e,v)Cv ⊗ Ty( e,v)Ce )

= w(e,v) + w(e′ ,v) ,

v ∈ V 2 ( Γ)

= w(e,v) − ψ(e,v),

v ∈ V S ( Γ)


15

4.2.3. Unifying stable and unstable vertices. From the discussion in Section 4.2.1 and Section 4.2.2, e T ( B1m ) 1 w( e,v) = ∏ w + w( e′,v) e T ( B4m ) v∈V 1 ( Γ)∏ v ∈V 2 ( Γ ),E ={ e,e ′ } ( e,v ) ,( e,v )∈ F (Γ) v

(27)

1

∏

·

v ∈V S ( Γ )

∏e∈ Ev (w( e,v) − ψ( e,v) )

.

Recall that

∏

M~Γ =

Mgv ,Ev ∪Sv .

v ∈V S ( Γ )

To unify the stable and unstable vertices, we use the following convention for the empty sets M0,1 and M0,2 . Let w1 , w2 be formal variables. (i) M0,1 is a −2 dimensional space, and Z

(28)

M0,1

1 = w1 . w1 − ψ1

(ii) M0,2 is a −1 dimensional space, and Z

(29)

M0,2

1 1 = (w1 − ψ1 )(w2 − ψ2 ) w1 + w2 Z

(30) (iii) M~Γ =

∏

M0,2

1 = 1. w1 − ψ1

M gv ,Ev ∪Sv .

v ∈V ( Γ )

With the above conventions (i), (ii), (iii), we may rewrite (27) as e T ( B1m ) 1 . = ∏ ( w e T ( B4m ) ∏ e∈ Ev ( e,v) − ψ( e,v)) v ∈V ( Γ )

(31)

The following lemma shows that the conventions (i) and (ii) are consistent with the stable case M0,n , n ≥ 3. Lemma 32. For any positive integer n and formal variables w1 , . . . , wn , we have Z 1 1 1 1 n −3 = (a) ( +··· ) . n ( w − ψ ) w · · · w w w ∏ M n n i 1 1 Z 0,n i =1 i 1 = w21−n . (b) M0,n w1 − ψ1 Proof. (a) The cases n = 1 and n = 2 follow from the definitions (28) and (29), respectively. For n ≥ 3, we have Z

M0,n

=

1 1 = n w1 · · · w n ∏ i = 1 ( w i − ψi )

1 w1 · · · w n

∑ a1 +···+ a n = n −3

− a1

w1

where Z

a

M0,n

ψ11 · · · ψnan =

Z

1 M0,n

· · · wn− an

ψ ∏ni=1 (1 − wii )

Z

( n − 3) ! . a1 ! · · · a n !

a

M0,n

ψ11 · · · ψnan

16


So

1

1 1 n −3 1 ( ) . +··· = w · · · w w w − ψ ) M0,n n n 1 1 i (b) The cases n = 1 and n = 2 follow from the definitions (28) and (30), respectively. For n ≥ 3, we have Z Z 1 1 3− n 1 1 = w = w21−n = w1 M0,n 1 − ψ1 w1 1 M0,n w1 − ψ1 Z

∏ni=1 (wi

w1

4.2.4. Deformation of the map. Consider the normalization sequence M

0 → OC →

OCv ⊕

v ∈V S ( Γ )

(33)

M

→

OCe

e∈ E(Γ)

M

Oyv ⊕

v ∈V 2 ( Γ )

M

Oy(e,v) → 0.

( e,v )∈ F S( Γ)

We twist the above short exact sequence of sheaves by f ∗ TX. The resulting short exact sequence gives rise a long exact sequence of cohomology groups H 0 (Cv ) ⊕

M

→ B2 →

0

v ∈V S ( Γ )

M

→

H 0 (Ce )

e∈ E(Γ)

M

T f (yv) X ⊕

v ∈V 2 ( Γ )

T f ( y( e,v))X

( e,v )∈ F S( Γ)

H 1 (Cv ) ⊕

M

→ B5 →

M

v ∈V S ( Γ )

M

H 1 (Ce ) → 0.

e∈ E(Γ)

where H i (Cv )

=

H i (Cv , ( f |Cv )∗ TX ) ∼ = H i (Cv , OCv ) ⊗ Tpσv X,

H i (Ce )

=

H i (Ce , ( f |Ce )∗ TX )

for i = 0, 1. We have H 0 (Cv ) 1

H (Cv )

=

Tpσv X

=

H 0 (Cv , ωCv )∨ ⊗ Tpσv X.

Lemma 34. Let σ ∈ V (Υ), so that p σ is a T-fixed point in X. Define w( σ ) h(σ, g)

= e T ( Tpσ X ) ∈ HT2r (point; Q ) =

e T (E ∨ ⊗ Tpσ X ) 2r ( g −1) ∈ HT (Mg,n ; Q ). T e ( Tpσ X )

Then w( σ ) =

(35)

∏

w(ǫ, σ ).

ǫ ∈ Eσ

h(σ, g) =

(36)

∏ ǫ ∈ Eσ

where Λ∨ g ( u) =

g

∑ (−1)i λi ug−i. i =0

Λ∨ g ( w( ǫ, σ )) w(ǫ, σ )


Proof. Tpσ X =

M

17

Tpσ ℓǫ , where e T ( Tpσ ℓǫ ) = w(ǫ, σ ). So

ǫ ∈ Eσ

e T ( Tpσ )

=

∏

w(ǫ, σ ),

ǫ ∈ Eσ

e T (E ∨ ⊗ Tpσ ℓǫ ) e T ( Tpσ ℓǫ )

=

∏ ǫ ∈ Eσ

e T (E ∨ ⊗ Tpσ ℓǫ ) , w(ǫ, σ )

where e T (E ∨ ⊗ Tpσ ℓǫ ) =

g

g

i =0

i =0

∑ (−1)i ci (E )c1T (Tpσ ℓǫ )g−i = ∑ (−1)i λi w(ǫ, σ)g−i.

The map B1 → B2 sends H 0 (Ce , TCe (−y(e, v) − y(e′ , v))) 0 H (Ce , ( f |Ce )∗ T ℓǫe ) f , the fixed part of H 0 (Ce , ( f |Ce )∗ T ℓǫe ).

isomorphically to

Lemma 37. Given d ∈ Z >0 and ǫ ∈ E(Υ)c , define σ, σ ′ , ǫi , ǫi′ , ai as in Section ??, and let f d : P1 → ℓǫ ∼ = P1 be the unique degree d map totally ramified over the two T-fixed points pσ and pσ′ in ℓǫ . Define h(ǫ, d) =

e T ( H 1 (P1 , f d∗ TX )m ) . e T ( H 0 (P1 , f d∗ TX )m )

Then (38)

h(ǫ, d) =

(−1)d d2d (d!)2 w(ǫ, σ )2d

r −1

∏ b( i =1

w(ǫ, σ ) , w(ǫi , σ ), dai ) d

where (39)

b(u, w, a) =

(

∏ aj=0 (w − ju)−1 , a −1 ∏− j =1 ( w + ju),

a ∈ Z, a ≥ 0, a ∈ Z, a < 0.

Proof. We use the notation in Section ??. We have Nℓǫ /X = L1 ⊕ · · · ⊕ Lr −1 . The weights of T-actions on ( Li ) pσ and ( Li ) pσ are w(ǫi , σ ) and w(ǫi , σ ) − ai w(ǫ, σ ), respectively. The weights of T-actions on T0 P1 , T∞ P1 , ( f d∗ Li )0 , ( f d∗ Li )∞ are u := w ( ǫ,σ ) d ,

−u, wi := w(ǫi , σ ), wi − dai u, respectively. By Example [32, Example 19], ( da ai ≥ 0, ∑ j=i0 ewi − ju , 0 1 ∗ 1 1 ∗ chT ( H (P , f d Li ) − H (P , f d Li )) = −da i −1 w i + ju e , ai < 0. ∑ j =1

Note that wi + ju is nonzero for any j ∈ Z since wi and u are linearly independent for i = 1, . . . , n − 1. So e T H 1 (P1 , f d∗ Li )m e T H 1 (P1 , f d∗ Li ) = = b(u, wi , dai ) e T H 0 (P1 , f d∗ Li ) e T H 0 (P1 , f d∗ Li )m where b(u, w, a) is defined by (39). By Example [32, Example 19], chT ( H 0 (P1 , f d∗ T ℓǫ ) − H 1 (P1 , f d∗ T ℓǫ )) =

2d

d

j =0

j =1

∑ edu− ju = 1 + ∑ (e jw(ǫ,σ)/d + e− jw(ǫ,σ)/d).

18


So

e T ( H 1 (P1 , f d∗ T ℓǫ )m ) = e T ( H 0 (P1 , f d∗ T ℓǫ )m )

d

−d2

∏ j2 w(ǫ, σ)2

=

j =1

(−1)d d2d . (d!)2 w(ǫ, σ )2d

Therefore, e T ( H 1 (P1 , f d∗ TX )m ) e T ( H 0 (P1 , f d∗ TX )m )

=

e T ( H 1 (P1 , f d∗ T ℓǫ )m ) r −1 e T ( H 1 (P1 , f d∗ Li )m ) ·∏ e T ( H 0 (P1 , f d∗ T ℓǫ )m ) i=1 e T ( H 0 (P1 , f d∗ Li )m )

=

(−1)d d2d (d!)2 w(ǫ, σ )2d

r −1

∏ b( i =1

w(ǫ, σ ) , w(ǫi , σ ), dai ). d

Finally, f (yv ) = pσv = f (y(e, v)), and e T ( Tpσv X ) = w(σv ). From the above discussion, we conclude that e T ( B5m ) e T ( B2m )

∏

=

∏

=

∏

w(σv ) ·

v ∈V 2 ( Γ )

v ∈V ( Γ )

∏

w(σv ) ·

h(σv , gv ) ·

v ∈V S ( Γ )

( e,v )∈ F S( Γ)

h(σv , gv ) · w(σv )val( v) ·

∏

∏

h ( ǫe , d e )

e∈ E(Γ)

h ( ǫe , d e )

e∈ E(Γ)

where w(σ ), h(σ, g), and h(ǫ, d) are defined by (35), (36), (38), respectively. 4.3. Contribution from each graph. f

f

f

4.3.1. Virtual tangent bundle. We have B1 = B2 , B5 = 0. So f

T 1, f = B4 =

M

T M gv ,Ev ∪Sv ,

T 2, f = 0.

v ∈V S ( Γ )

We conclude that

M gv ,Ev ∪Sv ]vir =

∏

[

v ∈V S ( Γ )

∏

[M gv ,Ev ∪Sv ].

v ∈V S ( Γ )

4.3.2. Virtual normal bundle. Let N~vir be the pull back of the virtual normal bundle Γ

of F~Γ in M g,n ( X, β) under i~Γ : M~Γ → F~Γ . Then e T ( B1m )e T ( B5m ) 1 h(σv , gv ) · w(σv )val( v) = = · ∏ h ( ǫe , d e ) ∏ e T ( B2m )e T ( B4m ) e T ( N~vir ) ∏e∈ Ev (w( e,v) − ψ( e,v)) e∈ E( Γ) v ∈V ( Γ ) Γ

4.3.3. Integrand. Given σ ∈ V (Υ), let iσ∗ : A∗T ( X ) → A∗T ( pσ ) = Q [u1 , . . . , ur ] be induced by the inclusion iσ : pσ → X. Then n i~Γ∗ ∏ ev∗i γiT ∪ (ψiT ) ai

(40)

i =1

=

∏ v ∈ V 1,1 ( E ) Sv = { i } , E v = { e }

iσ∗v γiT (−w( e,v) ) ai ·

∏

a

i ∏ iσ∗v γiT ∏ ψ(e,v )

v ∈ V S ( Γ ) i ∈ Sv

e ∈ Ev


19

To unify the stable vertices in V S (Γ) and the unstable vertices in V 1,1 (Γ) , we use the following convention: for a ∈ Z ≥0 , Z

(41)

M0,2

ψ2a = (−w1 ) a . w1 − ψ1

In particular, (30) is obtained by setting a = 0. With the convention (41), we may rewrite (40) as (42)

n i~Γ∗ ∏ ev∗i γiT ∪ (ψiT ) ai = i =1

∏

∏

v ∈ V ( Γ ) i ∈ Sv

iσ∗v γiT

∏ e ∈ Ev

ai . ψ( e,v )

The following lemma shows that the convention (41) is consistent with the stable case M0,n , n ≥ 3. Lemma 43. Let n, a be integers, n ≥ 2, a ≥ 0. Then  a −1 Z  ∏ i =0 ( n − 3 − i ) a +2 − n ψ2a w1 , = a!  M0,n w1 − ψ1 0,

n = 2 or 0 ≤ a ≤ n − 3, otherwise.

Proof. The case n = 2 follows from (41). For n ≥ 3, Z

M0,n

ψ2a 1 = w1 − ψ1 w1

Z

ψ2a M0,n

( n − 3) ! = (n − 3 − a)!a!

= w1a+2−n

= w1a+2−n

ψ1 w1 ∏ia=−01 (n − 3 − i )

1−

a!

Z

M0,n

ψ1n−3− a ψ2a

w1a+2−n .

4.3.4. Integral. The contribution of Z

[M g,n ( X,β) T ]vir,T

i ∗ ∏ni=1 (ev∗i γiT ∪ (ψiT ) ai ) e T ( N vir )

from the fixed locus F~Γ is given by 1 | A~Γ |

·

∏ e∈ E(Γ)

∏

∏

h ( ǫe , d e )

w(σv )val( v)

∏

iσ∗v γiT

i ∈ Sv ai h(σv , gv ) · ∏e∈ Ev ψ( e,v )

v ∈V ( Γ )

Z

v ∈V ( Γ ) M g v ,Ev ∪ Sv

∏e∈ Ev (w( e,v) − ψ( e,v) )

where | A~Γ | = |Aut(~Γ)| · ∏e∈ E( Γ) de . 4.4. Sum over graphs. Summing over the contribution from each graph ~Γ given in Section 4.3.4 above, we obtain the following formula.

20


Theorem 44. T hτa1 (γ1T ) · · · τan (γnT )i X g,β

(45)

h ( ǫe , d e ) ~ de |Aut(Γ)| e∈ E(Γ) ( X,β) 1

∑

=

~Γ∈ Gg,n

·

∏

∏

∏

w(σv )val( v)

∏ iσ∗v γiT

i ∈ Sv

v ∈V ( Γ )

a

Z

v ∈V ( Γ ) M g,Ev ∪ Sv

h(σv , gv ) ∏i∈Sv ψi i . ∏e∈ Ev (w( e,v) − ψ( e,v))

where h(ǫ, d), w(σ ), h(σ, g) are given by (38), (35), (36), respectively, and we have the following convention for the v ∈ / V S ( Γ): Z

M0,1

Z

M0,2

1 1 1 = w1 , = , w1 − ψ2 ( w − ψ )( w − ψ ) w + w2 M0,2 1 1 2 2 1 ψ2a = (−w1 ) a , a ∈ Z ≥0 . w1 − ψ1 Z

Given g ∈ Z ≥0 , r weights w ~ = {w1 , . . . , wr }, r partitions ~µ = {µ1 , . . . , µr }, and i a1 , . . . , ak ∈ Z, let ℓ(µ ) be the length of µi , and let ℓ(~µ ) = ∑ri=1 ℓ(µi ). We define

hτa1 , . . . , τak i g,~µ,w ~ =

Z

r

∏

M g,ℓ(~µ)+k i =1

Λ∨ (wi )wℓ(~µ)−1 g i ℓ( µ i ) w i ∏ j=1 µi − ψij ) j

k

a

∏ ψb i .

b =1

Given v ∈ V (Γ), define w ~ (v) = {w(ǫ, σv ) | (ǫ, σv ) ∈ F (Υ)}. Given v ∈ V (Γ), and ǫ ∈ Eσv , let µv,ǫ be a (possibly empty) partition defined by {d e | e ∈ Ev , ~f (e) = ǫ}, and define ~µ (v) = {µv,ǫ | (ǫ, σv ) ∈ F (Υ)}. Then (45) can be rewritten as T hτa1 (γ1T ) · · · τan (γnT )i X g,β

(46)

=

h ( ǫe , d e ) ~Γ)| e∈ E(Γ) de | Aut ( ( X,β)

∑ ~Γ∈ Gg,n

1

∏

∏

∏ iσ∗v γi h ∏ τai igv,~µ(v),w~ (v)

v ∈ V ( Γ ) i ∈ Sv

i ∈ Sv

.

Recall that g=

∑

gv + | E(Γ)| − |V (Γ)| + 1

v ∈V ( Γ )

so 2g − 2 =

∑

(2gv − 2 + val(v)).

v ∈V ( Γ )

~ ~g,~s), let ~Γ′ = (Γ, ~f , d, ~ ~s) be the decorated graph obtained by Given ~Γ = (Γ, ~f , d, forgetting the genus map. Let Gn ( X, β) = {~Γ′ | ~Γ ∈ ∪ g≥0 Gg,n ( X, β)}. Define (47)

(48)

T hτa1 (γ1T ), · · · , τan (γnT ) | ui X β =

hτa1 , . . . , τak | ui~µ,w ~ =

X

∑ u2g−2hτa1 (γ1T ), · · · , τan (γnT )ig,βT

g ≥0

∑ u2g−2+ℓ(~µ) hτa1 , . . . , τak ig,~µ,w~ .

g ≥0

Then we have the following formula for the generating function (47).


21

Theorem 49. T hτa1 (γ1T ) · · · τan (γnT ) | ui X β =

(50)

h ( ǫe , d e ) ~ de ~Γ′ ∈ Gn ( X,β) |Aut( Γ)| e ∈ E ( Γ) · ∏ ∏ iσ∗v γiT h ∏ τai | ui~µ(v),w ~ (v) .

∑

v ∈ V ( Γ ) i ∈ Sv

1

∏

i ∈ Sv

R EFERENCES [1] M.F. Atiyah and R. Bott, “The moment map and equivariant cohomology,” Topology 23 (1984), no. 1, 1–28. [2] K. Behrend, “Gromov-Witten invariants in algebraic geometry,” Invent. Math. 127 (1997), no. 3, 601–617. [3] K. Behrend, “Localization and Gromov-Witten invariants,” Quantum cohomology, 3–38, Lecture Notes in Math., 1776, Springer, Berlin, 2002. [4] K. Behrend and B. Fantechi, “Intrinsic normal cone,” Invent. Math. 128 (1997), no.1, 45–88. [5] K. Behrend and Y. Manin, “Stacks of stable maps and Gromov-Witten invariants,” Duke Math. J. 85 (1996), no. 1, 1–60. [6] L. Chen, Y. Li, and K. Liu, “Localization, Hurwitz numbers and the Witten conjecture,” Asian J. Math. 12 (2008), no. 4, 511–518. [7] D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68, American Mathematical Society, Providence, RI, 1999. [8] P. Deligne and D. Mumford, “The irreducibility of the space of curves of given genus,” Inst. Hautes ´ Etudes Sci. Publ. Math. No. 36 (1969), 75–109. [9] D. Edidin, W. Graham, “Equivariant intersection theory,” Invent. Math. 131 (1998), no. 3, 595–634. [10] D. Edidin, W. Graham, “Localization in equivariant intersection theory and the Bott residue formula,” Amer. J. Math. 120 (1998), no. 3, 619–636. [11] C. Faber, “Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians,” New trends in algebraic geometry (Warwick, 1996), 93–109, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999. [12] C. Faber and R. Pandharipande, “Hodge integrals and Gromov-Witten theory,” Invent. Math 139 (2000), no.1, 173–199. [13] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1984. [14] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, the William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993. [15] W. Fulton, “Equivariant Cohomology in Algebraic Geometry,” Eilenberg lectures at Columbia University, Spring 2007, notes by Dave Anderson are available at http://www.math.washington.edu/∼dandersn/eilenberg/ [16] W. Fulton and R. Pandharipande, “Notes on stable maps and quantum cohomology,” Algebraic geometry—Santa Cruz 1995, 45–96, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997. [17] T. Graber and R. Pandharipande, “Localization of virtual classes,” Invent. Math. 135 (1999), no. 2, 487–518. [18] M. Goresky, R. Kottwitz, R. MacPherson, “Equivariant cohomology, Koszul duality, and the localization theorem,” Invent. Math. 131 (1998), no. 1, 25–83. [19] V. Guillemin and C. Zara, “Equivariant de Rham theory and graphs,” Sir Michael Atiyah: a great mathematician of the twentieth century. Asian J. Math. 3 (1999), no. 1, 49–76. [20] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, Clay Mathematics Monographs 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003. [21] M.E. Kazarian, “KP hierarchy for Hodge integrals,” Adv. Math. 221 (2009), no. 1, 1–21. [22] M.E. Kazarian, S. K. Lando, “An algebro-geometric proof of Witten’s conjecture,” J. Amer. Math. Soc. 20 (2007), no. 4, 1079–1089. [23] Y-S. Kim, K. Liu, “Virasoro constraints and Hurwitz numbers through asymptotic analysis,” Pacific J. Math. 241 (2009), no.2, 275–284.

22


[24] F. Knudsen, “The projectivity of the moduli space of stable curves. II. The stacks M g,n ,” Math. Scand. 52 (1983), no. 2, 161–199. [25] F. Knudsen, “The projectivity of the moduli space of stable curves. III. The line bundles on M g,n , and a proof of the projectivity of M g,n in characteristic 0,” Math. Scand. 52 (1983), no. 2, 200–212. [26] F. Knudsen and D. Mumford, “The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”,” Math. Scand. 39 (1976), no. 1, 19–55. [27] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function,” Comm. Math. Phys. 147 (1992), no. 1, 1–23. [28] M. Kontsevich, “Enumeration of rational curves via torus actions,” The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math., 129, Birkhäuser Boston, Boston, MA, 1995. [29] A. Kresch, “Cycle groups for Artin stacks,” Invent. Math. 138 (1999), no. 3, 495–536. [30] J. Li and G. Tian, “Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties,” J. Amer. Math. Soc. 11 (1998), no. 1, 119–174. [31] J. Li, G. Tian, “Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds,” Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47–83, First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998. [32] C.-C. M. Liu, “Localization in Gromov-Witten theory and orbifold Gromov-Witten theory,” Handbook of Moduli, Volume II, 353425, Adv. Lect. Math., (ALM) 25, International Press and Higher Education Press, 2013. [33] Y.I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, 47, American Mathematical Society, Providence, RI, 1999. [34] M. Mirzakhani, “Weil-Petersson volumes and intersection theory on the moduli space of curves,” J. Amer. Math. Soc. 20 (2007), no. 1, 1–23. [35] M. Mulase, N. Zhang, “Polynomial recursion formula for linear Hodge integrals,” Commun. Number Theory Phys. 4 (2010), no. 2, 267–293. [36] D. Mumford, “Towards an enumerative geometry of the moduli space of curves,” Arithmetic and geometry, Vol. II, 271–328, Progr. Math., 36, Birkhäuser Boston, Boston, MA, 1983. [37] A. Okounkov, R. Pandharipande, “Gromov-Witten theory, Hurwitz numbers, and matrix models,” Algebraic geometry—Seattle 2005. Part 1, 325–414, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009. [38] Y. Ruan, “Topological sigma model and Donaldson-type invariants in Gromov theory,” Duke Math. J. 83 (1996), no. 2, 461–500. [39] Y. Ruan, “Virtual neighborhoods and pseudo-holomorphic curves,” Proceedings of 6th Gokova ¨ Geometry-Topology Conference, Turkish J. Math. 23 (1999), no. 1, 161–231. [40] Y. Ruan, G. Tian, “A mathematical theory of quantum cohomology,” J. Differential Geom. 42 (1995), no. 2, 259–367. [41] Y. Ruan, G. Tian, “Higher genus symplectic invariants and sigma models coupled with gravity,” Invent. Math. 130 (1997), no. 3, 455–516. [42] B. Siebert, “Gromov-Witten invariants of general symplectic manifolds,” arXiv:dg-ga/9608005. [43] H. Spielberg, “A formula for the Gromov-Witten invariants of toric varieties,” Thése, Université Louis Pasteur (Strasbourg I), Strasbourg, 1999, 103 pp. “The Gromov-Witten invariants of symplectic manifold,” arXiv:math/0006156. [44] A. Vistoli, “Intersection theory on algebraic stacks and on their moduli spaces,” Invent. Math. 97 (1989), no. 3, 613–670. [45] E. Witten, “Two-dimensional gravity and intersection theory on moduli space,” Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991. C HIU -C HU M ELISSA L IU , D EPARTMENT OF M ATHEMATICS , C OLUMBIA U NIVERSITY, 2990 B ROAD N EW Y ORK , NY 10027 E-mail address: [email protected]

WAY,

A RTAN S HESHMANI , D EPARTMENT OF MATHEMATICS AT THE O HIO S TATE U NIVERSITY, 600 M ATH T OWER , 231 W EST 18 TH AVENUE , C OLUMBUS , OH 43210 E-mail address: [email protected]