Localization Computations of Gromov-Witten Invariants

Transcription

1 Localization Computations of Gromov-Witten Invariants Kaisa Taipale May 2, 2007 Introduction Gromov-Witten invariants are enumerative invariants of stable maps. Their definition in the context of mirror symmetry in physics allowed new approaches to old problems for instance, counting the number of plane rational curves of degree d through 3d 1 points and solved all at once enumerative problems that had thwarted mathematicians for years. Any such innovation gives rise to a host of new questions in the search for structure, rigor, and generalizations. For rational domain curves C and convex target spaces X, Gromov-Witten invariants give a fairly straightforward count of rational curves in X by looking instead at stable maps to X. As an example, we ll compute the number of lines through two points in P 2 in section 5. We can also use Gromov-Witten invariants to give a count of rational curves in a quintic threefold, a question related to the Clemens Conjecture (given X P 4 a generic quintic threefold, and d a positive integer, there are finitely many rational curves of degree d in X [Kat06]). This question is not qualitatively so different from the first, but already the enumerative significance of the invariant becomes less obvious. We must take into account multiple covers higher-degree reparametrizations of C which factor through lower-degree maps to X which contribute to the Gromov-Witten invariant but overcount things enumeratively. We will compute this multiple cover contribution in section 5.1, but we ll need the technique of localization to do so. As genus increases and non-homogeneous spaces are explored, it becomes impossible to claim that the Gromov-Witten invariant is a literal count of genus g curves in a space X. At the same time, more mathematical machinery is required to compute the invariants at all. For instance, Gromov-Witten invariants for g = 0 and X homogeneous are computed as an integral over the space of stable maps [M 0,n (X, β)], which is a smooth Deligne-Mumford stack of the expected dimension. (We will look at this space and stable maps in section 1.) For g > 0 and X non-homogeneous, [M g,n (X, β)] is a singular Deligne-Mumford stack with many components of a larger than expected dimension, and we can t just do usual intersection theory. We must define a virtual fundamental class defining virtual classes is our way of making spaces act smooth. This will occupy section 3. 1

2 There are two techniques for computing general Gromov-Witten invariants: degeneration and localization. Localization is the focus of this paper and my current research. It is a technique that allows explicit computation of Gromov- Witten invariants for spaces X with nice torus action. (By nice, we mean that the torus-fixed points and torus-invariant one-dimensional orbits on X must be isolated.) We will prove the virtual localization formula in section 4, so that we can apply the technique to situations like the higher genus multiple cover formula. In the proof of the higher genus multiple cover formula, we will see that localization is also used fruitfully as a technique for uncovering relations between integrals. These relations can be exploited to find helpful generating series, which can answer all at once a question that is hard to do piece by piece. The multiple cover formula is a nice example of this approach. 1 Moduli of Stable Maps In order to even begin, we must understand M g,n (X, β). M g,n (X, β) is the space of stable maps to X of irreducible, genus g, n-pointed curves, for which the image takes values in the homology class β on X. M g,n (X, β) is its compactification, described by Kontsevich in [Kon95]. 1.1 Stable maps What is a stable map? The points of M g,n (X, β) are triples (C, {p i }, µ), where C is a genus g complex curve with n distinct nonsingular marked points p 1,...,p n, and a map µ : C X, such that µ ([C]) = β. We require that C be projective, connected, reduced, and (at worst) nodal. A map (C, {p i }, µ) is stable if for every component E C, 1. If E = P 1 and µ maps E to a point in X, then E contains at least three special points. 2. If E is an elliptic component (arithmetic genus 1) and µ maps E to a point in X, then E contains at least one special point. Special point here means marked or nodal point. These requirements ensure that the data (C, {p i }, µ) has finite automorphism group. Contrast stability of maps with Deligne-Mumford stability of curves: for a stable map, only those components which contract to a point in X need be stable in the sense of stable curves. Families of stable maps A family of pointed maps (π : C S, {p i }, µ) is stable if each map on a geometric fiber (C s, {p i (s)}, µ) of π is stable. When X = P r, stability can be expressed in terms of ω C/S, the relative dualizing sheaf. A flat family of maps (π : C S, {p i }, µ) is stable if and only if ω C/S (p p n ) µ (O P r(3)) is π-relatively ample. 2

3 The moduli space Define a contravariant functor, M g,n (X, β), from the category of complex algebraic schemes to sets. Then M g,n (X, β)(s) is the set of isomorphism classes of stable families (over S) of maps to X of genus g, n- pointed curves, whose images lie in class β. This functor is represented by a proper Deligne-Mumford stack ([Kon95], [BM96]) and M g,n (X, β) is the projective, coarse moduli space of M g,n (X, β) (Theorem 1, [FP97]). In particular, M 0,n (X, β) is an orbifold if X is homogeneous. The points of the boundary of M 0,n (X, β) correspond to reducible domain curves. The boundary locus of M 0,n (X, β) is a divisor with normal crossing singularities when X is homogeneous (but only virtually so when X is not homogeneous). For M g,n (X, β) with g 1, there are two types of boundary divisors: virtual divisors ξ corresponding to stable splittings ξ = (g 1 + g 2 = g, A 1 A 2 = [n], β 1 + β 2 = β), each of whose points gives a map with reducible domain curve, and the additional divisor 0 whose points give maps with irreducible nodal domain curve. A recurring theme in considering M g,n (X, β) is that taking virtual classes (here, divisors) allows us to treat things as in some sense smooth: what this means will be discussed later. In genus zero, fundamental relations between Gromov-Witten invariants come from linear equivalences between boundary components in M 0,n (X, β). These relations come from looking at partitions of four marked points, and with some manipulation give us associativity of the quantum product. Associativity of the quantum cohomology ring of P 2 gives a recursive formula for the number of degree d rational plane curves passing through 3d 1 general points in P 2. Universal family As stacks, M g,n+1 (X, β) is the universal family over M g,n (X, β), by the forgetful functor, π n+1 : M g,n+1 (X, β) M g,n (X, β). 2 Gromov-Witten invariants In ideal cases, Gromov-Witten invariants count the number of curves of a given genus and degree with marked points p i mapping to fixed algebraic cycles γ i = [V i ] on a nonsingular projective algebraic variety W. However, only g = 0 and X = G/P a homogeneous space is ideal. This characterization is not very useful beyond initial intuition and enumerative geometry on projective space, but it shows the enumerative beginnings of the Gromov-Witten invariant. We will use it later in calculating the number of lines passing through two points in P 2. However, Gromov-Witten invariants don t always have obvious enumerative significance. For example, consider the degree d maps from rational curves to a P 1 embedded with normal bundle O P 1( 1) O P 1( 1) into a Calabi-Yau threefold X. Maps factoring through a d-fold cover of P 1 contribute 1/d 3 to the Gromov-Witten invariant of X but this is not an integer! Any time the 3

4 same class β can be multiply covered, the contribution to the Gromov-Witten invariant will be a rational number. This will be explored in the section on the multiple cover formula. Definition, with caveats It seems premature to attempt to define Gromov- Witten invariants; a definition that holds for M g,n (X, β) requires the virtual fundamental class, which will be defined over the next few pages. However, the reader would feel frustrated if I didn t define it until the end! Definition 1. Let e j : M g,n (X, β) X be the evaluation map taking [C, {p i }, µ] M g,n (X, β) to µ(p j ). Let γ 1,..., γ n be classes in A (X). Then the Gromov- Witten invariant γ 1,..., γ n X g,n,β is defined by the integral γ 1,...,γ n X 0,n,β = e 1(γ 1 ) e n(γ n ). [M g,n(x,β)] vir Notice that this will be a non-zero number only if the sum of the codimensions of the γ i is equal to the (virtual) dimension of M g,n (X, β). 3 Virtual Fundamental Class Example: Vector bundles Consider a vector bundle π : E X of rank r over a variety X of dimension n. A section s of E has an associated subscheme Z(s) on X, its zero locus. The dimension of Z(s) is n r when s is a regular section, but there are other possibilities: what if s is the zero section? Then Z(s) = X and has dimension n. What if s is some other section, which doesn t intersect transversely with the zero section? To do intersection theory on Z(s) in a variety of situations, we must find an object of the right dimension (here n r) which behaves like the fundamental class of Z in all situations. This is the virtual fundamental class. In the case of the vector bundle E of rank r, let s define the virtual fundamental class. Let I be the ideal sheaf of Z(s) in X. Then the normal cone of Z in X is defined as C Z X = Spec( k=0 Ik /I k+1 ). The normal cone will have pure dimension n, equal to the dimension of the variety. Notice that if Z is regularly embedded, the normal cone is equal to the total space of the normal bundle. C Z X is an affine cone over Z, and it embeds into the bundle E Z : the map O(E ) I is a surjection, inducing k Sym k (O(E )/IO(E )) k I k /I k+1, which in turn induces our desired embedding when Spec is applied. Thus [C Z X] A n (E Z ), the Chow group. Apply the Gysin morphism s induced by the zero section of E to get s [C Z X] A n r (Z), a class in the Chow group of our expected dimension. One last lemma finishes the picture: 4

5 Lemma 1. Lemma 7.1.5, [CK99] If i : Z X is the inclusion, then i (s [C Z X]) A n r (X) is the Euler class c r (E) [X] of E. As well as being a fine motivational example, this is actually directly applicable to calculation of Gromov-Witten invariants, as we ll see in the examples below. Moduli space of stable maps Only rarely does the dimension of the moduli space match the expected dimension. In the unobstructed situation, for instance, M g,n (X, β) has dimension (dim X 3)(1 g) + n + c 1 (TX). This expected dimension is correct for g = 0 and X convex, but in most situations M g,n (X, β) has components of higher dimension. In order to construct the virtual fundamental class of the moduli space of stable maps, a trip through obstruction theory is necessary. In Fulton s exposition of intersection theory for schemes [Ful], intersection products are constructed through deformation to the normal cone, and this theory carries over to stacks in many respects [Vis89]. We will need some additional tools, though. Perfect obstruction theory For the purposes of this paper, full generality is not necessary. We assume a scheme (or Deligne-Mumford stack) X admits a global closed embedding into a nonsingular scheme (or Deligne-Mumford stack) Y. Then a perfect obstruction theory [GP99] consists of a complex of vector bundles E = [E 1 E 0 ], and a morphism φ from E to the cotangent complex L X of X, satisfying the properties: φ induces an isomorphism in cohomology in degree 0 φ induces a surjection in cohomology in degree 1. We ll find that the expected dimension of the fundamental class is equal to rk(e 0 ) rk(e 1 ). In this situation we can use just the first two terms of the cotangent complex: β φ : E [I/I 2 Ω Y X ]. (1) Here, I is the ideal sheaf of X in Y. Choose φ to be an actual map of complexes; to do this in generality, φ would be a morphism in the derived category. 5

6 Cones Take the mapping cone of the sequence (1) to get the exact sequence E 1 E 0 I/I 2 γ Ω Y 0. (As Graber and Pandharipande point out, this sequence is exact if and only if φ is a perfect obstruction theory!) If we let Q be the kernel of the map γ above, there is an associated sequence of abelian cones, 0 TY C(I/I 2 ) X E 0 C(Q) 0. Notice the dualization happening here. We know that C X Y, the normal cone of X in Y, embeds naturally into C(I/I 2 ) as a closed subscheme. Thus we can define D = C X Y X E 0 C(I/I 2 ) X E 0. In the terminology used by Behrend and Fantechi [BF97], D is a TY -cone. We let D/TY = D vir. (By [CFK], it is in fact possible to choose an embedding nice enough that this quotient by TY is unnecessary in moduli problems.) Virtual fundamental class Define the virtual fundamental class from this. Let [X] vir = s E 1 [D vir ], where s E 1 is the Gysin map induced by the zero section of the bundle E 1. It is worth noting that the virtual fundamental class is independent of the quasi-isomorphism class in the derived category of the perfect obstruction theory, but not of the obstruction theory chosen [BF97]. Comparison to classical intersection When i : X Y is a codimension d regular embedding, Y is smooth, and V is another smooth k-dimensional variety, we get a cartesian diagram j W V g f. i X Y The normal cone C W V is a closed subcone of g N X Y, of pure dimension k, and Fulton [Ful] defines the intersection product X V to be s [C W V ], where s : W g N X Y is the zero section and s is the Gysin map. What would the perfect obstruction theory for W be? E = [g N X Y j Ω V ] works: E has a natural morphism to L W induced by g L X L W and j L V L W. Then the virtual fundamental class is [W]vir = i! [V ] = X V. Relative construction for M g,n (X, β) Let M g,n be the stack of n-pointed genus g prestable curves (that is, they are at worst nodal but do not need to satisfy the stabilization conditions). This is a smooth Artin stack. The functor F : M g,n (X, β) M g,n 6

7 is a well-defined map of stacks which forgets the map and does not stabilize the curves. We also have for M g,n+1 (X, β) the usual map π, which forgets the n + 1-th marked point, and the evaluation map e n+1 : M g,n+1 (X, β) X. For M g,n (X, β), the relative perfect obstruction theory is E = (R π e TX), which we can represent as a complex of vector bundles [Beh97]. Following the construction above with some modification for instance, the definition of a perfect relative obstruction theory is the same as the absolute definition, replacing L X with L X/Y [BF97] we obtain the relative virtual fundamental class [M g,n (X, β)/m g,n, Rπ e TX] in the bivariant Chow group A (M g,n (X, β) M g,n ), and we can define the virtual fundamental class of M g,n (X, β) to be [M g,n (X, β)] vir = M g,n [M g,n (X, β)/m g,n, Rπ e TX] in A rke +3g 3+n(M g,n (X, β)). [CK99] Cases of note, [Hor03] The case g = 0 and X convex gives [M 0,n (X, β)] vir = [M 0,n (X, β)]. This is true for all cases for which the moduli space is unobstructed; that is, all cases for which the obstruction space at a point of the moduli space Ob(C, p 1,...,p n, f) is zero. When X is convex, we have by definition that H 1 (C, f TX) = 0, and so the relative obstruction theory is trivial. In such a situation the moduli space will be of the expected dimension. If a moduli space is nonsingular but of unexpected dimension, the virtual fundamental class will be the Euler class (top Chern class) of the canonical obstruction bundle Ob = [H 1 (C, f TX)]. (Note that we denote bundles by their fibers.) Since the moduli space is nonsingular, the obstruction bundle is of constant rank, and is thus a vector bundle. As it is a vector bundle, [M g,n (X, β)] vir = c top (Ob) [M g,n (X, β)]. (Compare to Prop. 7.3 in [BF97].) This situation is of particular note because the space of maps to a line in a Calabi-Yau threefold is of this type, and this is exactly the situation examined in the section on the multiple cover formula. A last important case is that of X a hypersurface and domain curves of genus zero. If X is a hypersurface of degree l in P m, we get an inclusion Compare dimensions: while i : M 0,n (X, d) M 0,n (P m, d). dimm 0,n (P m, d) = d(m + 1) + m 3 + n vdim M 0,n (X, d) = d(m + 1 l) + (m 1) 3 + n. We can look at i [M 0,n (X, d)] vir as a cycle class on M 0,n (P m, d); it turns out that the virtual fundamental class pushed forward is the top Chern class of a rank dl + 1 vector bundle on M 0,n (P m, d). 7

8 Look at the fibers over each point of the moduli space: dimh 0 (C, f O P m(l)) = dl+1. We can glue these together to give us the sought-after bundle, π f O P m(l)): C π f P m M 0,n (P m, d) where π is the universal map. Since X is a degree l hypersurface, it is described by the vanishing of a section s of O P m(l). There is an induced section s of π f O(l) that vanishes precisely on the stable maps to X. This returns us to our motivating example, the zero locus of a section of a vector bundle on a space; by our first lemma, we have i [M 0,n (X, d)] vir = c top (π f O P m(l)) M 0,n (P m, d). These cases are not altogether representative, though. Consider the situation of our hypersurface with g > 0: if φ : M g,n (Y, d) M g,n (X, d) for some general X, Y, it may be that there is no α for which φ [M g,n (Y, d)] vir = α [M g,n (X, d)] vir. A better way to look at the construction of the virtual fundamental class is K-theoretically, or using dg-algebras [CFK]. 4 Localization: Concepts When we have a torus action, localization is a useful technique for simplifying computation of Gromov-Witten invariants. Consider the action of the torus T = (C ) n on a space Y. Atiyah and Bott s Localization Theorem gives us an isomorphism between equivariant cohomology on Y and equivariant cohomology on the components Z j of the fixed point locus of the torus action after inverting the equivariant parameters. [CK99] Moreover, if Y is a nonsingular variety, Bott s formula gives us an explicit isomorphism mapping classes α in equvariant cohomology of Y to i j (α) et (N j ) 1, which are sums of classes in the equivariant cohomology of each torus-fixed Z j. We can use these ideas to shift the integral over [M g,n (X, β)] in the formula for Gromov-Witten invariants to a sum of integrals over torus-fixed loci in the moduli space, indexed by graphs Γ. Equivariant cohomology: definitions Localization rests on the use of equivariant cohomology. Let G be a compact connected Lie group. Let EG BG be the principal G-bundle classifying G, and let X G = EG G X. (EG G X is the twisted product, just (EG X)/, where (eg 1, gx) (e, x).) The equivariant cohomology HG (X) of X is defined by H G(X) := H (X G ). The situation considered exclusively in this paper is that in which G = T = (C ) n. In this case, BG = (P ) n and EG = π 1 S π ns, with π i : BG P the projection to the ith factor and S the tautological bundle on CP, with sheaf of sections O CP ( 1) [CK99]. 8

9 In this case, if X is a point, then X T ET BT is an isomorphism, and so HT (pt) = H (BT) = C[α 1,..., α n ], where α i = c 1 (πi O P (1)). (We can also think of α i as the weight by which T acts on each one-dimensional subspace.) More generally, if X has trivial T-action, HT (X) = H (X) C C[α 1,..., α n ]. Pullback on equivariant cohomology is defined, so we get a functor from the category of topological spaces with torus actions and torus-equivariant morphisms to the category of C-algebras with algebra homomorphisms. This allows us to establish that HT (X) is a H T (pt)-module, as we alway have a T- equivariant map X pt, giving us C[α 1,, α n ] = HT (pt) H T (X). Atiyah-Bott Localization If X is a proper scheme or Deligne-Mumford stack with a T-action, then the fixed point set X T is the disjoint union of a finite set of closed connected subspaces Z j. If X is smooth, so are the Z j (theorem by Iversen). Let i j : Z j X be the inclusion. Then i j : A T (Z j) A T (X) is proper pushforward on cycles, preserving degree. Theorem 1 (Atiyah-Bott). For X a compact, projective variety with a T- action, there is an isomorphism H T (Z j) C[α] C(α) H T (X) C[α] C(α). For α j H T (Z j ), j i j α j = α for α H T (X). Bott s formula What we ll actually use in computation: Theorem 2. If X is a smooth, compact projective variety with a T-action, there i is an explicit inverse to the map j i j above. It is j j e T (N j). As a corollary, we obtain the following statement for orbifolds: Corollary 1 ([CK99], 9.1.4). Let X be an orbifold which is the variety underlying a smooth stack with a T-action. If α HT (X) R T, then α = i j ( (α) X j (Z j) a j e T (N j ) ), where a j is the order of the group H occurring in a local chart at the generic point of Z j. Kontsevich realized that this method could be applied to the smooth stack M 0,n (P r, d) [Kon95]. However, the moduli stacks of stable maps M g,n (X, d) are not smooth for g > 0 (or X not convex). An extension of Bott s formula to virtual fundamental classes is needed to allow computation in these situations. 4.1 Virtual localization Localization for virtual fundamental classes was proven in [GP99]. Following their format, we will consider the basic case of virtual localization for zero loci of sections of vector bundles, and then proceed to the general case. 9

10 The basic case starts with a nonsingular scheme Y endowed with a C action and a C -equivariant bundle V. An invariant section v of V has a zero locus, denoted by X. We would like to prove localization on X: [X] vir = ι [Xi ] vir e(n vir i ) (2) in H C (X) Q(α), where the X i are the connected components of the fixed point scheme and ι is the inclusion. Start with the perfect obstruction theory on X, as this will give the virtual structure on the X i and allow the computation of Ni vir. X has a natural perfect obstruction theory, as it is the zero locus of a C -invariant section of V. We have E = [V Ω Y ], and L X = [I/I 2 Ω Y ]. The morphism φ between them comes from the natural map V I/I 2. Following the construction laid out above, the virtual fundamental class of X is e ref (V ), the refined Euler class of V. Given the tools of equivariant cohomology, we can find the C -fixed obstruction theory on X i. Denote by Y i the components of the fixed locus of Y. Each Y i is smooth, with a vector bundle V Yi and a section v Yi vanishing strictly at X i = X Y i. The C -action induces a C -action on V. On each Y i, V decomposes into a direct sum of eigensheaves: V i = V f i V m i. V f is the eigensheaf invariant under C, called the fixed part, and V m is the direct sum of eigensheaves on which C acts nontrivially, the moving part. [GP99] Each X i carries a perfect obstruction theory [(V f i ) Ω Yi ]. (This follows from Proposition 1 of [GP99].) Thus the virtual fundamental class of X i is e ref (V f i ) on Y i, where V f i denotes the fixed part of V restricted to Y i. Last, we need the virtual normal bundle. It is the moving part of the complex E = [TY V ] for each connected component. The moving part of TY is the normal bundle to Y i, while the moving part of V is simply Vi m, so Ni vir = [N Yi/Y Vi m ]. Take the Euler class of a complex using multiplicativity, obtaining e(ni vir ) = e(n Y i/y ) e(vi m ). One must check that e(vi m ) is invertible. It is, essentially because it has no fixed component. 10

11 Let us return to equation (2). Use the expressions for e(ni vir ), [X i ] vir, and [X] vir to see that to establish (2), we must prove Usual localization for Y tells us i ) eref (V f i e ref (V ) = ι ) e(v m e(n Yi/Y ) [Y ] = ι [Yi ] e(n Yi/Y ), and intersecting both sides with e ref (V ) gives eref (V ) [Y i ] e ref (V ) = ι. e(n Yi/Y ). (3) Consider the numerators on the right-hand side: taking refined Euler class commutes with pullback, so e ref (V ) [Y i ] = e ref (V i ). V i splits on each component Y i into fixed and moving parts. The section v giving X is C -fixed, so it lives in Vi m, which implies e ref (V i ) = e ref (V f i ) e(v i m ). Thus we can rewrite the numerators on the right-hand side to obtain (3), proving (2) in the basic case. The general case is similar, but requires manipulation of cones as X may be singular. Let X be an arbitrary scheme admitting an equivariant embedding into a nonsingular scheme Y. Compare our desired formula, (2), with the formula obtained by intersecting the localization formula for Y with [X] vir on both sides. Then use Vistoli s rational equivalence and a lemma on Gysin morphisms to show the desired equality. Earlier, we constructed the virtual fundamental class using the exact sequences 0 TY D D vir 0 (4) and D = C X Y E 0. (5) D vir embeds as a closed subcone of E 1 and [X] vir = s E 1 [D vir ]. Another way of characterizing [X] vir is by the fiber square TY D, X 0 E1 E1 using 0 E1 to denote the zero section. Then [X] vir = s TY 0! E 1 [D]. We have the obvious C -fixed analogues of (4) and (5) for the embeddings X i Y i. Localization for Y tells us that [Y ] = ι [Yi ] e(ty m ) 11

12 in A T (Y ) α. Take refined intersection product with [X] vir gives [X] vir = ι [X] vir [Y i ] e(ty m ) in A T (X) α. Compare this with what we want, keeping in mind that the normal bundle to X i is defined to be the moving part of the complex E,i. Thus it suffices to show [X] vir [Y i ] e(ty m ) = [X i] vir e(e m 1 ) e(e m 0 ) (6) in A T (X i) α. From the discussion of [X] vir above, we can intersect [X] vir and [Y i ] to get [X] vir [Y i ] = ι! s TY 0! E 1 [D] = s TY 0! E 1 ι! [D], using commutativity of the intersection product for the second equality. Using Vistoli s rational equivalence, we can continue a step further: [X] vir [Y i ] = s TY 0! E 1 [D i E m 0 ]. (7) All this occurs in A T (X i). (Vistoli s rational equivalence [Vis89] in N Yi Y ι C X Y implies ι! [C X Y ] = [C Xi Y i ] in A (ι C X Y ). Translated to the C -equivariant case it gives the same equation in A T (ι C X Y ). Pull back this relation to ι D = ι C X Y E 0 to get ι! [D] = [D i E0 m ]. ) Consider the TY i -action on ι D. It leaves D i E0 m invariant, and since D i /TY i = Di vir, the class [D i E0 m] AT (ι D) is the pullback of [Di vir E0 m] A T (ι D/TY i ). Thus we can rewrite s TY 0! E 1 [D i E0 m ] as s TY m0! E 1 [Di vir E0 m ]. Now we need a lemma in order to rewrite [X] vir [Y i ] in terms of E0 m and E1 m rather than TY m. Consider the scheme-theoretic intersection 0 1 E 1 (Di vir E0 m). It lies in TY m. The map Di vir E0 m E 1 is the product of inclusion Di vir E f 1 and the natural map Em 0 Em 1. Thus 0 1 E 1 (Di vir E0 m) also lies in Em 0. Lemma 2. [GP99] Let B 0 and B 1 be C -equivariant bundles on X i. Let Z be a scheme equipped with two equivariant inclusions j 0, j 1 over X i : Let ζ A C (Z). Then Z B 1. B 0 X i s B 0 j 0 (ζ) e(b 1 ) = s B 1 j 1 (ζ) e(b 0 ) A C (X i). 12

13 We can apply this lemma to the diagram 0 1 E 1 (D vir i E0 m) Em 0 TY m using ζ = 0 1 E 1 [Di vir E0 m ] to get X i [X] vir [Y i ] = s E m 0 (0 1 E 1 [D vir i E m 0 ] e(ty m ) e(e m 0 ) (8) since we can invert e(e0 m). Consider the class 0 1 E 1 [Di vir E0 m ] to lie in A T (E0 m ), and notice that it does not depend on the bundle map E0 m E1 m. Thus we can assume the bundle map is trivial. Notice that if we divide by e(ty m ) on both sides of (8) we will have the left side of (6), the identity we wanted to show. To finish, we must show [X i ] vir e(e1 m ) = s E0 m(0! E 1 [Di vir E0 m ]). This follows from the definition of [X i ] vir and the excess intersection formula, proving the virtual localization formula. 4.2 Applying the virtual localization formula Explanation of graphs We would like to use localization to shift from integration over [M g,n (X, β)] to integration over torus-fixed loci in M g,n (X, β). The components of the torus-fixed loci can be indexed by graphs trees with vertices and edges when X has isolated torus-fixed points and isolated torusinvariant one-dimensional orbits. When this is true, we can transform a geometric problem into a more combinatorial problem, calculating the contribution to the Gromov-Witten invariant of each component and summing over the possible graphs. Projective space is an ideal example, and will be the case considered here. Each graph Γ corresponds to a substack M Γ. Let (C, f, p 1,...,p n ) be a stable map. Construct the graphs as follows: Vertices: The only torus-fixed points of P r are the points q 0 = [1 : 0 : : 0], q 1 = [0 : 1 : : 0], through q r = [0 : 0 : : 0 : 1]. For a map to be stable and T-equivariant, all nodes, marked points, ramification points, and contracted components of C must be mapped to T-fixed points. There is exactly one vertex of Γ for each connected component of f 1 ({q 0,...,q r }). Label vertices by the fixed point to which they correspond. Edges: The torus-invariant one-dimensional orbits in P r are the coordinate lines l j. Each edge in a graph corresponds to a rational (non-contracted) component C e of C mapped to a coordinate line l j. We label each edge e with the degree d(e) of the map taking C e to l j. 13

14 Flags: Flags are pairs (v, e) of a vertex and an adjacent edge. In addition to edges e labeled with degree d(e) and vertices v labeled with the corresponding fixed point in P r, v qi, we must label vertices with the genus of the corresponding contracted component, if any, and with the marked points. We will adopt the convention that the lack of a label for genus indicates a rational or trivial contracted component. Examples It is useful to actually try constructing such a graph. Consider the example of degree two maps from rational curves to P 1 with no marked points. Notice that we have two basic types of graphs, since P 1 has only two torus-fixed points and the degree is restricted: 2 q i q j q i q i q j Figure 1: Figure 1 The graph on the left corresponds to a map from a curve with two intersecting rational components, with the node mapping to q j. There can be no more than two components of the source curve, since we require all contracted components to be stable and here we have no marked points. The graph on the right corresponds to the double cover of P 1, with ramification point mapped to q j. Deformation/tangent/obstruction sequence As can be seen from Bott s Formula above, localization requires the calculation of the value of e(nj vir ), the Euler class of the (possibly virtual) normal bundle for each component Z j of the fixed-point locus. (The virtual bundle is discussed below.) Define M Γ = Π v Γ M g(v),val(v)+n(v). We must take into account A, the group of automorphisms acting on M Γ : 1 Π edges Z/d(e) A Aut(Γ) 1 (9) is an exact sequence of groups. We can get a closed immersion of Deligne- Mumford stacks: γ/a : M Γ /A M g,n (P r, d). [GP99] tells us that a C -fixed component of M g,n (P r, d) is supported on M Γ /A, and that through this we can relate Gromov-Witten invariants of P r to integrals over moduli spaces of pointed curves. [GP99] give us the following tangent-obstruction sequence on the substack M Γ /A: 14

15 0 Ext 0 (Ω C (D), O C ) H 0 (C, f T P r) T 1 Ext 1 (Ω C (D), O C ) H 1 (C, f T P r) T 2 0 (10) (Note again we label vector bundles by their fibers, and we use D to denote the divisor i p i on C.) This uses the canonical obstruction theory on M g,n (P r, d), which will be denoted by E. T 1 and T 2 are defined on M Γ /A by looking at the restriction of the dual canonical perfect obstruction theory: 0 T 1 E 0,Γ E 1,Γ T 2 0. (Check back to the non-virtual case: the definition above does give the usual normal bundle. We just work with a complex of one term! Also note, in particular, that on projective space when g = 0, we have [N i ] vir = [N i ].) Thus we can rewrite. Borrowing notation from Graber and Pandharipande, let B i be the vector bundle making up the ith term of the deformation/obstruction sequence. We can see that e(n vir i ) = e(bm 2 )e(bm 4 ) e(b m 1 )e(bm 5 ). One has e(n) = e(t 1 ) e(t 2 ) from the definition, and from the multiplicativity of the Euler class over complexes we get the above. Now it s time to consider a general graph Γ, representing a class of torusinvariant stable maps to projective space, and calculate e(n Γ ). We can do it piece by piece. 5 Localization: Computations Infinitesimal automorphisms of (C, p 1,..., p n ) Ext 0 (Ω C (D), O C ) is the space of infinitesimal automorphisms of the pointed curve (C, p 1,..., p n ). C consists of contracted and non-contracted components; for non-contracted components C e, a weight zero piece comes from the infinitesimal automorphism of C e fixing the two special points. This will cancel with a term from H 0 (C, f T P r ). If a vertex v has genus zero and valence one, then the action of T will have weight α i(v) α i(v ) d e, which we can write ω F(v). Over all vertices, this term will give us ω i(v). val(v)=1 n(v)=0 If a vertex has genus zero and valence two, we can use flag notation to write the contribution over all such vertices: (ω F1(v) ω F2(v)). val(v)=2 n(v)=0 15

16 The space of deformations of (C, p 1,..., p n ) Look at Ext 1 (Ω C (D), O C ). One can deform contracted components, but this will give only weight zero pieces of the bundle. Non-contracted components can t be deformed on their own; the other deformations come from smoothings of nodes joining contracted to noncontracted components. Look at one vertex v corresponding to such a node: the space of deformations of a node p at which components C 1 and C 2 intersect is isomorphic to T p (C 1 ) T p (C 2 ) [HM98]. Consider the tangent spaces separately: for C 1 the non-contracted component, then just as in the previous paragraph, the weight of the induced action of T will be ω F(v). The contracted component, C 2, could be any kind of stable curve, so the tangent space may vary and the weight of the torus action is trivial. We must take into account the variations possible for C 2. Define the ψ-classes by ψ F := c 1 (L F ) H 2 (M g(v),val(v)+n(v) ), where L F is the bundle whose fibers are the cotangent space to the curve at the marked point associated to the flag. The final contribution to e(n Γ ) of the bundle whose fibers are Ext 1 (Ω C (D), O C ) is (ω F(v) ψ F(v) ) val(f)+n(f)+2g(v)>2 where the conditions on the flag come from the fact that we re looking at a node of valence at least two, with non-trivial contracted component (i.e., having either marked points or genus greater than zero). Deformations and obstructions of f H 0 (C, f T P r ) is the space of the first-order deformations of the map f, while H 1 (C, f T P r ) can be seen as the obstructions to deformations of f. It is useful to look at these together. Contributions come from resolving all the nodes of C implicit in being associated to a graph Γ. Consider the normalization sequence resolving these nodes: 0 O C O Cv O xf 0 vertices edges O Ce flags Twist by f (T P r ) and take cohomology, remembering that since non-contracted components are rational, they have no higher cohomology: 0 H 0 (f T P r ) flags vertices H 0 (C v, f T P r ) H 0 (C e, f T P r ) T pi(f)p r H 1 (f T P r ) edges vertices H 1 (C v, f T P r ) 0. (11) K-theoretically, we can write H 0 H 1 = H 0 (C v, f T P r ) H 0 (C e, f T P r ) vertices T pi(f)p r flags vertices edges H 1 (C v, f T P r ). 16

17 Note that on the right-hand side, H 0 (C v, f T P r ) = T qi(v) P r, as C v is a connected, contracted component (on C v, f is constant). The weights of the induced torus action are α i(v) α j for all j i(v). Likewise, the weights of the action on T pi(f)p r are α i(f) α j for all j i(f). These contribute vertices flags j i(v) (α i(v) α j ) j i(f) (α i(f) α j ). To find the contribution of H 0 (C e, f T P r ), we ll need the Euler sequence 0 O P r O(1) V T P r 0. Pull it back to C e and take cohomology: 0 C H 0 (C e, O(d e )) V H 0 (C e, f T P r ) 0. The weight of the action on C is trivial. Look at each piece in the middle term: the weights on V are just α 0,..., α r, while the weights on H 0 (O(d e )) are a d e α i(v) + b d e α i(v ) for a+b = d e and v, v the vertices of the edge e. The weights a of the middle term are the pairwise sums of these, d e α i(v) + b d e α i(v ) α j. We have two weight zero terms, when a = 0 and i(v ) = j and when b = 0, i(v) = j. One of these cancels the zero weight from C, while the other cancels with the trivial weight from the infinitesimal automorphisms of (C, p 1,..., p n ). In addition, when j = i(v), the product of weights 1 a d e ( a d e α j + de a d e α i α j ) gives us de! (α d de j α i ) de. Something similar happens when j = i(v ). Final e contribution: a product over edges of ( 1) (d e!) 2 de (α i α j ) d 2de e a+b=d e k i,j ( a d e α i + b d e α j α k ). Finally, we look at H 1 (C v, f T P r ). Notice that H 1 (C v, f T P r ) = H 1 (C v, O Cv ) T qi(v) P r, and that H 1 (C v, O Cv ) = E, where E = π ω is the Hodge bundle. (Use Serre duality to see this.) Tensoring with T qi(v) P r gives r copies of E twisted by the weights α i(v) α j, for j i(v). Take the top Chern class to get: c (αi(v) α j) 1(E ) (α i(v) α j ) g(v), j i where for a bundle Q of rank q, c t (Q) = 1 + tc 1 (Q) +... t q c q (Q). Putting it all together Combining all the above contributions, e(n Γ ) = e v Γ ef Γ ee Γ (12) 17

18 e v Γ = e F Γ = flags val(f)+n(f)+2g(v)>2 vertices j i val(v)+n(v)+2g(v)>2 e e Γ = edges (d e!) 2 (α i(v) α i(v )) 2de ( 1) de d 2de e (ω F ψ F ) j i(f) 1 α i(f) α j 1 c (αi(v) α j) 1(E ) (α i(v) α j ) g(v) a+b=d e k i,j ( a d e α i + b d e α j α k ) val(v)=1 n(v)=0 1 ω i(v) A few notes on integration Before we begin some computational examples, a few notes on what we ll encounter. To evaluate local contributions we will need to integrate over all the terms above. Terms of the form α i α j and variations thereof will be constant terms that we can take out of the integral. Terms of the form 1 1 ψ i will have to be expanded as formal power series. We will then be left to evaluate integrals of the form M g,n i ψ ai i λ i, where λ i = c i (E). These are called Hodge integrals. For g = 0, we can use a consequence of the string equation: ( ) ψ a1 1 ψa k n 3 k =, a 1 a 2...a k [M 0,n] where a a k = n 3, as λ i = 0. (This is immediate from the string equation. In higher genus, pure ψ-integrals can be evaluated recursively by using Witten s conjecture, or Kontsevich s theorem.) General Hodge integrals are not always tractable and their evaluation is an active area of research. Even when we can evaluate these integrals, virtual localization gives us only an algorithm for computing Gromov-Witten invariants. The number of graphs grows quickly with degree, and the combinatorial expressions become difficult to evaluate. A primary goal is the simplification of computation, usually by judicious choices of weights, so that graph sums can be evaluated in closed form. Number of lines through two points in P 2 We would like to compute the number of lines (genus 0, degree 1) through two points in P 2. This answer should be known to the reader already. In fancier notation, we re computing i h 2 h 2 P2 0,2,1P 1 where h is the Poincare dual to the hyperplane class. First, consider the (C ) 3 -fixed points of P 2 : let q 0 = [1 : 0 : 0], q 1 = [0 : 1 : 0], q 2 = [0 : 0 : 1]. Then notice that the torus-invariant one-dimensional orbits of val(v)=2 n(v)=0 (ω F1(v) ω F2(v)) 18

19 P 2 are the coordinate lines joining the fixed points. To draw the graphs, we must notice that we are dealing with two marked points and that our maps are of degree one. Thus in any graph, there is exactly one edge, and all graphs correspond to one of the types in Figure 2. p 1 p 1 p 2 p 2 q i q j q i q j Figure 2: Figure 2 There are twelve graphs, as one can see by considering the six combinations of i and j. Now we face a choice of linearizations. A linearization is a lifting of the torus action on a space to a vector bundle on that space. Here, we re looking at liftings of the torus action from P 2 to the line bundle O(1), and the lifting is uniquely determined by the weights [l 0, l 1, l 2 ] of the fiber representations at the fixed points q 0, q 1, q 2. The standard linearization lifts the three-torus action to the line bundle as (t 0, t 1, t 2 ) [x 0 : x 1 : x 2 ] (t 0 x 0, t 1 x 1, t 2 x 2 ). h is the first Chern class of O(1), and the cohomology class we get as a lift is the first equivariant Chern class of O(1) with this action. To finish the computation, the sum of twelve terms is required. From the four graph with vertices corresponding to q 0 and q 1, we get the contributions α 2 0 α2 0 + α2 1 α2 1 (α 0 α 1 ) 2 (α 0 α 2 )(α 1 α 2 ) α 2 0 α2 1 + α2 1 α2 0 (α 0 α 1 ) 2 (α 0 α 2 )(α 1 α 2 ). (13) Add to this the contributions from the graphs with the two other possible combinations of vertices to get a great big sum. With careful cancellation, we find the result to be 1. Do not despair. There are easier ways. Instead of the standard lift, we could also use (t 0, t 1, t 2 ) [x 0 : x 1 : x 2 ] [x 0 : t 1 0 t 1x 1 : t 1 0 t 2x 2 ], lifting the class h to h α 0. The contribution of any graph with a vertex corresponding to q 0 vanishes. This reduces our sum to four terms total: (α 1 α 0 ) 4 + (α 2 α 0 ) 4 (α 1 α 2 ) 2 (α 1 α 0 )(α 2 α 0 ) (α 1 α 0 ) 2 (α 2 α 0 ) 2 + (α 2 α 0 ) 2 (α 1 α 0 ) 2 (α 1 α 2 ) 2. (α 1 α 0 )(α 2 α 0 ) (14) An even better linearization was suggested by Pandharipande: use the torus action that lifts h 2 to (h α 1 )(h α 2 ) on one factor, while using the lift (h α 0 )(h α 2 ) on the other. Then the only graph that contributes is the one in Figure 3. 19

20 p 1 p 2 q 0 q 1 Figure 3: Notice that the calculation is made very easy. (α 0 α 1 )(α 0 α 2 ) (α 1 α 0 )(α 1 α 2 ) (α 0 α 1 ) 2 (α 0 α 2 )(α 1 α 2 ) = The multiple cover formula In the section on Gromov-Witten invariants, it was observed that one reason Gromov-Witten invariants fail to be strictly enumerative even in nice situation is because of multiple covers. Localization can be applied to computing the contribution of these multiple covers. The genus zero case is relatively straightforward, but in higher genus Hodge integrals appear and integration becomes more difficult. Luckily, [FP00] established some very nice results for higher genus, illustrating once again the benefit of looking at generating series: it is easier to compute answers all at once (all g 2, here) than to do it one case at a time. Genus zero Manin computed the contribution of degree d multiple covers of a rigid smooth rational curve C on a Calabi-Yau threefold V to be 1/d 3, using Kontsevich s formulas and clever summation. We will prove this here. Let C = P 1, with normal bundle N = O P 1( 1) O P 1( 1). Denote by M d,c (V ) the component of M 0,0 (V, d[c]) consisting of stable degree d maps. It is connected, it is isomorphic to M 0,0 (P 1, d), and it has dimension 2d 2. The contribution of this component is the integral over it of the Euler class of the obstruction sheaf F d, C(0, d) := e(f d ). M d,c (V ) The obstruction sheaf can be written more informatively as R 1 π µ N, where π : M 0,1 (P 1, d) M 0,0 (P 1, d) is the universal stable map (forgetful functor with stabilization) and µ : M 0,1 (P 1, d) P 1 is the evaluation map. Part of the computation is relatively mechanical; the contributions e(n Γ ) are easy to compute with the formulas above. A clever choice of linearization (as in our calculation of the number of lines through two points) greatly reduces the number of graphs to consider. We must also calculate the contribution of the Euler class of the obstruction bundle to each localization term. Our clever linearization involves looking at the action of the torus T = C 1 T = (C ) 2, rather than the action of the full two-torus. The fixed-point 20

21 loci of T and T correspond, and we can lift the action to O P 1( 1) O P 1( 1) by letting t T act by t (l(x 0, x 1 ), m(x 0, x 1 )) = (l(t x 0, x 1 ), m(x 0, t 1 x 1 )), where l and m are homogeneous of degree 1 in (x 0, x 1 ). What kinds of graphs contribute to the sum? Examine first graphs with vertices of valence greater than one. If a graph has more than one edge, the domain curve C of the stable map must have a node (or more). Let C split into components C i, with nodes r i. The normalization exact sequence, then, tells us 0 f (N) i (f Ci ) (N) i f (N) ri 0. The long exact sequence in cohomology, then, tells us that i H 0 (C, f (N) ri ) injects into H 1 (C, f N). The weights of the T -action depend on where the nodes get mapped: if f(r i ) = q 0, then a basis for O P 1( 1) at q 0 is 1/x 0, in which case T acts with weight zero on the first factor f O P 1( 1). If f(r i ) = q 1, T acts trivially on the second factor. Since weights are multiplicative, the contribution of N vanishes over any graph with vertex of valence greater than one [CK99]. This leaves us to consider only the case where Γ has one edge of degree d. Use Cech cohomology to calculate the contribution of H 1 (P 1, N), which will give us the contribution of e(r 1 π µ N). Since the map is to P 1, use the usual open cover U i = {z i 0}, where (z 0, z 1 ) are our coordinates. A basis for H 1 (P 1, f N) is given by Cech cocycles ( 1 zoz k 1 d k, 0 ), ( 0, 1 z k 0 zd k 1 ), 1 k d 1, where cocycles are of degree d because f is the pullback of a degree d map. Since T acts with weights h and 0 on x 0 and x 1 in H 0 (P 1, O(1)), it will act on z 0 and z 1 with weights h/d and 0. Thus the weights on the cocycles would be kh/d and (d k)h/d. Over all k, this gives us ( 1) d 1 ((d 1)!) 2 h 2d 2 d 2d 2. Now it remains to calculate the contribution from the normal bundle. Using equation (12) from the previous section, we can compute contributions using α 0 = h and α 1 = 0. We end up with e(n Γ ) = ( 1)d (d!) 2 h 2d 2 d 2d 2. It is easy to forget our last factor: the contribution of automorphisms of M Γ. The exact sequence (9) reminds us that we get a factor of d in the denominator. Including this, we get Done! ( 1) d 1 ((d 1)!) 2 h 2d 2 /d 2d 2 d ( 1) d (d!) 2 h 2d 2 d 2d 2 = 1 d 3. 21

22 Higher genus In the genus one case, physics predicts the contribution to be 1/12d. Graber and Pandharipande computed this mathematically, using localization. For g 2, Faber and Pandharipande did something even nicer! In [FP00], they proved that Theorem 3 (Theorem 3, FaP ). For g 2, C(g, d) = B 2g d 2g 3 2g (2g 2)! = χ(m d 2g 3 g) (2g 3)!, where B 2g is the 2g th Bernoulli number and χ(m g ) = B 2g /2g(2g 2) is the Harer-Zagier formula for the orbifold Euler characteristic of M g. where This closed-form equation follows from an intermediate result, C(g, d) = d 2g 3 b g1 b g2, g 1+g 2=g g 1,g 2 0 { 1 g = 0 b g = M g,1 ψ 2g 2 1 λ g g > 0, which is obtained from a straightforward localization computation. Calculating b g, though, uses generating series, and along the way establishes a number of beautiful identities. We will sketch the proof here, giving a tour of the paper. Recall that we are focused on the integral C(g, d) = e(r 1 π µ N), [M g,0(p 1,d)] vir just as in the genus zero case. As in the genus zero case, we can look at different linearizations in order to simplify computation of the integral. Manin s trick involves choosing a localization so that only graphs with the vertex q 0 contribute, for instance, and the sum over graphs reduces to a sum over partitions of d. This is how Graber and Pandharipande computed the g = 1 contribution, C(1, d) = 1/12d, and how Manin originally computed C(0, d) = 1/d 3. But as we saw in the genus zero section, the linearization [0, 1], [ 1, 0] results in the vanishing of the contribution of any graph Γ containing a vertex with valence greater than one. Thus all contributing graphs have exactly one edge. In this situation, the localization sum reduces to a sum over partitions g 1 + g 2 = g of the genus: one rational component of the domain curve will be mapped (with degree d) to the torus-equivariant line represented by the edge in the graph, and all other components will be contracted to one of the fixed points. The components contracted to q 1 will have genus g 1, while the components mapped to q 2 will have genus g 2. We need not worry about marked points. For Γ such a graph, the contribution of e(n Γ ) can be computed, noting that [FP00] 22

23 use the linearization with weight 1 on q 1 and weight 0 on q 2 : e F Γ = (1 ψ1)( 1 ψ2) ( 1)(1) e v Γ = d 2 c 1(E )c 1(E ( 1) g 2 e e Γ = (d!)2 ( 1) d d 2d The contributions of the normal sheaf, on the other hand, are calculated in a way similar to the genus zero case. The difference is that now we must take into account the fact that H 1 (C, O( 1)) = H 1 (C g1, L 1 Cg1 ) H 1 (C g2, L 1 Cg2 ) H 1 (P 1, µ O( 1)), where L n denotes the line bundle with (z 0, z 1, χ) (λz 0, λz 1, λ n χ). (O(n) is its associated sheaf of sections, which is why we re looking at L 1.) The contribution of H 1 (P 1, µ O( 1)) is exactly the same as in the genus zero case: ( 1) d 1 ((d 1)!) 2 h 2d 2 d 2d 2. Notice how many of these terms will cancel with the contribution of e(n Γ ). The contribution of H 1 (C gi, L 1 Cgi ), for i = 1, 2, is a bit different. Using duality and definitions, H 1 (C gi, L 1 Cgi ) = H 0 (C gi, ω Cgi ) L 1 qi = E L 1 qi. Taking the Euler class of H 1 (C gi, L 1 Cgi ), then, is equivalent to g i e(e L 1 qi ) = c j (E )α gi j, where α represents the linearization used. When the above contributions are combined, many terms cancel. To simplify our notation, introduce the notation j=0 Λ 1 (k) = g 1 i=0 ki λ g1 i A (M g1,1) Λ 2 (k) = g 2 i=0 ki λ g2 i A (M g2,1) for k Z. After cancellation and change of notation, using the linearization [1, 0], we are left with the integrand Λ 1 (1)Λ 1 (0)Λ 1 ( 1)Λ 2 ( 1)Λ 2 (0)Λ 2 (1) (1/d ψ 1 )(1/d + ψ 2 )d 3. 23

24 We can use Mumford s relationship Λ i (1)Λ i ( 1) = ( 1) gi to simplify. Then, integrating, we get d 2g 3 λ g1 ψ 2g1 2 λ g2 ψ 2g2 2. (15) [M g1,1] vir [M g2,1] vir The λ gi comes from Λ gi (0), in which all terms vanish except for c gi (E ; d 2g 3 1 comes about because in expanding 1/d±ψ i we get a factor of d 2 g i 3 as the coefficient of ψ 2g 2 i. To simplify this, we can use the notation b g = [M g,1] λ vir g ψ 2g 2, with the understanding that g 0, b 0 = 1. Then we have C(g, d) = d 2g 3 b g1 b g2. g 1+g 2=g How to compute the right-hand side? The paper [FP00] generalizes the question. They find a generating function for b g, b g t 2g = ( t/2 ) (16) sin(t/2) g 0 as a special case of the following more general theorem. Theorem 4 (Theorem 2, [FP00]). Define the series F(t, k) Q[k][[t]] by F(t, k) = 1 + g 1 g t 2g k i i=0 ψ 2g 2+i 1 λ g i. (17) M g,1 Then for all k Z. t/2 F(t, k) = f k (t) = ( sin(t/2) )k+1 Proof. A sketch of the proof: [FP00] define an intermediate function f ξ (t) = 1 + g 1 t 2g M g,1 Λξ 1 ψ 1 = 1 + g 1 g t 2g ξ i i=0 ψ 2g 2+i 1 λ g i (18) M g,1 for ξ Z. They prove that for ξ Z, f ξ (t) = f 0 (t) ξ+1, using comparison of localization computations to establishes initial cases and induction to proceed. Two integrals are examined. Let x denote the top Chern class of R 1 π µ O P(V ), and y the top Chern class of R 1 π µ O P(v) ( 1)). Use the linearizations [α, α] on O P(v) and [β, β + 1] on O P(V ) ( 1), α, β Z, and compute [M g,0(p(v),1)] vir x y = ( 1) g I g (α, β) 24