Localization Computations of Gromov-Witten

Invariants

Kaisa Taipale

May 2, 2007

Introduction Gromov-Witten invariants are enumerative invariants of stablemaps. Their definition in the context of mirror symmetry in physics allowednew approaches to old problems — for instance, counting the number of planerational curves of degree d through 3d − 1 points — and solved all at onceenumerative problems that had thwarted mathematicians for years. Any suchinnovation 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-Witteninvariants give a fairly straightforward count of rational curves in X by lookinginstead at stable maps to X . As an example, we’ll compute the number oflines through two points in P2 in section 5. We can also use Gromov-Witteninvariants to give a “count” of rational curves in a quintic threefold, a questionrelated to the Clemens Conjecture (given X ⊂ P4 a generic quintic threefold,and d a positive integer, there are finitely many rational curves of degree din 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 reparametrizationsof C which factor through lower-degree maps to X — which contribute to theGromov-Witten invariant but overcount things enumeratively. We will computethis multiple cover contribution in section 5.1, but we’ll need the technique oflocalization to do so.

As genus increases and non-homogeneous spaces are explored, it becomesimpossible to claim that the Gromov-Witten invariant is a literal count of genusg curves in a space X . At the same time, more mathematical machinery isrequired to compute the invariants at all. For instance, Gromov-Witten invari-ants for g = 0 and X homogeneous are computed as an integral over the spaceof stable maps [M0,n(X,β)], which is a smooth Deligne-Mumford stack of theexpected dimension. (We will look at this space and stable maps in section 1.)For g > 0 and X non-homogeneous, [Mg,n(X,β)] is a singular Deligne-Mumfordstack with many components of a larger than expected dimension, and we can’tjust do usual intersection theory. We must define a virtual fundamental class— defining virtual classes is our way of making spaces “act smooth.” This willoccupy section 3.

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There are two techniques for computing general Gromov-Witten invariants:degeneration and localization. Localization is the focus of this paper and mycurrent research. It is a technique that allows explicit computation of Gromov-Witten invariants for spaces X with nice torus action. (By nice, we mean thatthe torus-fixed points and torus-invariant one-dimensional orbits on X must beisolated.) We will prove the virtual localization formula in section 4, so thatwe can apply the technique to situations like the higher genus multiple coverformula. In the proof of the higher genus multiple cover formula, we will seethat localization is also used fruitfully as a technique for uncovering relationsbetween integrals. These relations can be exploited to find helpful generatingseries, 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 understandMg,n(X,β). Mg,n(X,β) is the spaceof stable maps toX of irreducible, genus g, n-pointed curves, for which the imagetakes values in the homology class β on X . Mg,n(X,β) is its compactification,described by Kontsevich in [Kon95].

1.1 Stable maps

What is a stable map? The points of Mg,n(X,β) are triples (C, pi, µ),where C is a genus g complex curve with n distinct nonsingular marked pointsp1, . . . , pn, and a map µ : C → X , such that µ∗([C]) = β. We require that Cbe projective, connected, reduced, and (at worst) nodal. A map (C, pi, µ) isstable if for every component E ⊂ C,

1. If E ∼= P1 and µ maps E to a point in X , then E contains at least threespecial points.

2. If E is an elliptic component (arithmetic genus 1) and µ maps E to a pointin X , then E contains at least one special point.

Special point here means marked or nodal point. These requirements ensurethat the data (C, pi, µ) has finite automorphism group. Contrast stability ofmaps with Deligne-Mumford stability of curves: for a stable map, only thosecomponents which contract to a point in X need be stable in the sense of stablecurves.

Families of stable maps A family of pointed maps (π : C → S, pi, µ)is stable if each map on a geometric fiber (Cs, pi(s), µ) of π is stable. WhenX = Pr, stability can be expressed in terms of ωC/S, the relative dualizing sheaf.A flat family of maps (π : C → S, pi, µ) is stable if and only if ωC/S(p1 + · · ·+pn) ⊗ µ∗(OPr(3)) is π-relatively ample.

2

The moduli space Define a contravariant functor, Mg,n(X,β), from the cat-egory of complex algebraic schemes to sets. Then Mg,n(X,β)(S) is the set ofisomorphism 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 aproper Deligne-Mumford stack ([Kon95], [BM96]) and Mg,n(X,β) is the pro-jective, coarse moduli space of Mg,n(X,β) (Theorem 1, [FP97]). In particular,M0,n(X,β) is an orbifold if X is homogeneous.

The points of the boundary of M0,n(X,β) correspond to reducible domaincurves. The boundary locus of M0,n(X,β) is a divisor with normal crossingsingularities when X is homogeneous (but only “virtually so” when X is nothomogeneous). For Mg,n(X,β) with g ≥ 1, there are two types of boundarydivisors: virtual divisors ∆ξ corresponding to stable splittings

ξ = (g1 + g2 = g,A1 ∪A2 = [n], β1 + β2 = β),

each of whose points gives a map with reducible domain curve, and the addi-tional divisor ∆0 whose points give maps with irreducible nodal domain curve.A recurring theme in considering Mg,n(X,β) is that taking “virtual” classes(here, divisors) allows us to treat things as in some sense smooth: what thismeans will be discussed later.

In genus zero, fundamental relations between Gromov-Witten invariantscome from linear equivalences between boundary components in M0,n(X,β).These relations come from looking at partitions of four marked points, and withsome manipulation give us associativity of the quantum product. Associativityof the quantum cohomology ring of P2 gives a recursive formula for the numberof degree d rational plane curves passing through 3d− 1 general points in P2.

Universal family As stacks,Mg,n+1(X,β) is the universal family overMg,n(X,β),by the forgetful functor, πn+1 : Mg,n+1(X,β) →Mg,n(X,β).

2 Gromov-Witten invariants

In “ideal” cases, Gromov-Witten invariants count the number of curves of agiven genus and degree with marked points pi mapping to fixed algebraic cyclesγi = [Vi] on a nonsingular projective algebraic variety W . However, only g = 0and X = G/P a homogeneous space is “ideal.” This characterization is not veryuseful beyond initial intuition and enumerative geometry on projective space,but it shows the enumerative beginnings of the Gromov-Witten invariant. Wewill use it later in calculating the number of lines passing through two points inP2.

However, Gromov-Witten invariants don’t always have obvious enumerativesignificance. For example, consider the degree d maps from rational curvesto a P1 embedded with normal bundle OP1(−1) ⊕ OP1(−1) into a Calabi-Yauthreefold X . Maps factoring through a d-fold cover of P1 contribute 1/d3 tothe Gromov-Witten invariant of X — but this is not an integer! Any time the

3

same class β can be “multiply covered,” the contribution to the Gromov-Witteninvariant will be a rational number. This will be explored in the section on themultiple cover formula.

Definition, with caveats It seems premature to attempt to define Gromov-Witten invariants; a definition that holds for Mg,n(X,β) requires the virtualfundamental class, which will be defined over the next few pages. However, thereader would feel frustrated if I didn’t define it until the end!

Definition 1. Let ej : Mg,n(X,β) → X be the evaluation map taking [C, pi, µ] ∈Mg,n(X,β) to µ(pj). Let γ1, . . . , γn be classes in A∗(X). Then the Gromov-Witten invariant 〈γ1, . . . , γn〉Xg,n,β is defined by the integral

〈γ1, . . . , γn〉X0,n,β =

∫

[Mg,n(X,β)]vir

e∗1(γ1) ∪ · · · ∪ e∗n(γn).

Notice that this will be a non-zero number only if the sum of the codimen-sions of the γi is equal to the (virtual) dimension of Mg,n(X,β).

3 Virtual Fundamental Class

Example: Vector bundles Consider a vector bundle π : E → X of rank rover a variety X of dimension n. A section s of E has an associated subschemeZ(s) on X , its zero locus. The dimension of Z(s) is n − r when s is a regularsection, but there are other possibilities: what if s is the zero section? ThenZ(s) = X and has dimension n. What if s is some other section, which doesn’tintersect 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 (heren − r) which behaves like the fundamental class of Z in all situations. This isthe virtual fundamental class.

In the case of the vector bundle E of rank r, let’s define the virtual funda-mental class. Let I be the ideal sheaf of Z(s) in X . Then the normal cone ofZ in X is defined as

CZX = Spec(⊕∞k=0Ik/Ik+1).

The normal cone will have pure dimension n, equal to the dimension of thevariety. Notice that if Z is regularly embedded, the normal cone is equal to thetotal space of the normal bundle.

CZX is an affine cone over Z, and it embeds into the bundle E|Z : the mapO(E∗) → I is a surjection, inducing

⊕kSymk(O(E∗)/IO(E∗)) → ⊕kIk/Ik+1,

which in turn induces our desired embedding when Spec is applied. Thus[CZX ] ∈ An(E|Z), the Chow group. Apply the Gysin morphism s∗ inducedby the zero section of E to get s∗[CZX ] ∈ An−r(Z), a class in the Chow groupof our “expected dimension.” One last lemma finishes the picture:

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Lemma 1. Lemma 7.1.5, [CK99] If i : Z → X is the inclusion, then i∗(s∗[CZX ]) ∈

An−r(X) is the Euler class cr(E) ∩ [X ] of E.

As well as being a fine motivational example, this is actually directly appli-cable to calculation of Gromov-Witten invariants, as we’ll see in the examplesbelow.

Moduli space of stable maps Only rarely does the dimension of the mod-uli space match the expected dimension. In the unobstructed situation, forinstance, Mg,n(X,β) has dimension

(dimX − 3)(1 − g) + n+

∫

β

c1(TX).

This expected dimension is correct for g = 0 and X convex, but in most situa-tions Mg,n(X,β) has components of higher dimension.

In order to construct the virtual fundamental class of the moduli space of sta-ble maps, a trip through obstruction theory is necessary. In Fulton’s expositionof intersection theory for schemes [Ful], intersection products are constructedthrough deformation to the normal cone, and this theory carries over to stacksin many respects [Vis89]. We will need some additional tools, though.

Perfect obstruction theory For the purposes of this paper, full generalityis not necessary. We assume a scheme (or Deligne-Mumford stack) X admits aglobal 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 → E0], and

• a morphism φ from E• to the cotangent complex L•X of X , satisfyingthe 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 tork(E0) − rk(E−1).

In this situation we can use just the first two terms of the cotangent complex:

φ : E• → [I/I2 → ΩY |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.

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Cones Take the mapping cone of the sequence (1) to get the exact sequence

E−1 → E0 ⊕ I/I2 γ→ ΩY → 0.

(As Graber and Pandharipande point out, this sequence is exact if and only ifφ 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/I2) ×X E0 → C(Q) → 0.

Notice the dualization happening here.We know that CXY , the normal cone of X in Y , embeds naturally into

C(I/I2) as a closed subscheme. Thus we can define D = CXY ×X E0 ⊂C(I/I2)×X E0. In the terminology used by Behrend and Fantechi [BF97], D isa TY -cone. We let D/TY = Dvir. (By [CFK], it is in fact possible to choosean embedding nice enough that this quotient by TY is unnecessary in moduliproblems.)

Virtual fundamental class Define the virtual fundamental class from this.Let [X ]vir = s∗E1

[Dvir], where s∗E1is the Gysin map induced by the zero section

of the bundle E1. It is worth noting that the virtual fundamental class isindependent of the quasi-isomorphism class in the derived category of the perfectobstruction theory, but not of the obstruction theory chosen [BF97].

Comparison to classical intersection When i : X → Y is a codimensiond regular embedding, Y is smooth, and V is another smooth k-dimensionalvariety, we get a cartesian diagram

Wj−−−−→ V

y

g

y

f

Xi−−−−→ Y

.

The normal cone CWV is a closed subcone of g∗NXY , of pure dimension k,and Fulton [Ful] defines the intersection product X · V to be s∗[CWV ], wheres : W → g∗NXY is the zero section and s∗ is the Gysin map.

What would the perfect obstruction theory for W be?

E• = [g∗NXY∨ → j∗ΩV ]

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 Mg,n(X,β) Let Mg,n be the stack of n-pointedgenus g prestable curves (that is, they are at worst nodal but do not need tosatisfy the stabilization conditions). This is a smooth Artin stack. The functor

F : Mg,n(X,β) → Mg,n

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is a well-defined map of stacks which forgets the map and does not stabilizethe curves. We also have for Mg,n+1(X,β) the usual map π, which forgets then + 1-th marked point, and the evaluation map en+1 : Mg,n+1(X,β) → X .For Mg,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 theconstruction above with some modification — for instance, the definition of aperfect relative obstruction theory is the same as the absolute definition, replac-ing L•

X with L•X/Y [BF97] — we obtain the relative virtual fundamental class

[Mg,n(X,β)/Mg,n, Rπ∗e∗TX ] in the bivariant Chow group A∗(Mg,n(X,β) →

Mg,n), and we can define the virtual fundamental class of Mg,n(X,β) to be

[Mg,n(X,β)]vir = Mg,n ∩ [Mg,n(X,β)/Mg,n, Rπ∗e∗TX ]

in ArkE•+3g−3+n(Mg,n(X,β)). [CK99]

Cases of note, [Hor03] The case g = 0 and X convex gives

[M0,n(X,β)]vir = [M0,n(X,β)].

This is true for all cases for which the moduli space is unobstructed; thatis, all cases for which the obstruction space at a point of the moduli spaceOb(C, p1, . . . , pn, f) is zero. When X is convex, we have by definition thatH1(C, f∗TX) = 0, and so the relative obstruction theory is trivial. In such asituation the moduli space will be of the expected dimension.

If a moduli space is nonsingular but of unexpected dimension, the virtualfundamental class will be the Euler class (top Chern class) of the canonicalobstruction bundle Ob ∼= [H1(C, f∗TX)]. (Note that we denote bundles bytheir fibers.) Since the moduli space is nonsingular, the obstruction bundle isof constant rank, and is thus a vector bundle. As it is a vector bundle,

[Mg,n(X,β)]vir = ctop(Ob) ∩ [Mg,n(X,β)].

(Compare to Prop. 7.3 in [BF97].) This situation is of particular note becausethe space of maps to a line in a Calabi-Yau threefold is of this type, and this isexactly the situation examined in the section on the multiple cover formula.

A last important case is that of X a hypersurface and domain curves ofgenus zero. If X is a hypersurface of degree l in Pm, we get an inclusion

i : M0,n(X, d) →M0,n(Pm, d).

Compare dimensions:

dimM0,n(Pm, d) = d(m+ 1) +m− 3 + n

whilevdim M0,n(X, d) = d(m+ 1 − l) + (m− 1) − 3 + n.

We can look at i∗[M0,n(X, d)]vir as a cycle class on M0,n(P

m, d); it turns outthat the virtual fundamental class pushed forward is the top Chern class of arank dl + 1 vector bundle on M0,n(P

m, d).

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Look at the fibers over each point of the moduli space: dimH0(C, f∗OPm(l)) =dl+1. We can glue these together to give us the sought-after bundle, π∗f

∗OPm(l)):

Cf−−−−→ Pm

y

π

M0,n(Pm, d)

where π is the universal map. Since X is a degree l hypersurface, it is describedby the vanishing of a section s of OPm(l). There is an induced section s ofπ∗f

∗O(l) that vanishes precisely on the stable maps to X . This returns us toour motivating example, the zero locus of a section of a vector bundle on a space;by our first lemma, we have i∗[M0,n(X, d)]

vir = ctop(π∗f∗OPm(l))∩M0,n(P

m, d).These cases are not altogether representative, though. Consider the situ-

ation of our hypersurface with g > 0: if φ : Mg,n(Y, d) → Mg,n(X, d) forsome general X,Y , it may be that there is no α for which φ∗[Mg,n(Y, d)]

vir =α ∩ [Mg,n(X, d)]

vir . A better way to look at the construction of the virtualfundamental class is K-theoretically, or using dg-algebras [CFK].

4 Localization: Concepts

When we have a torus action, localization is a useful technique for simplifyingcomputation of Gromov-Witten invariants. Consider the action of the torusT = (C∗)n on a space Y . Atiyah and Bott’s Localization Theorem gives usan isomorphism between equivariant cohomology on Y and equivariant coho-mology on the components Zj of the fixed point locus of the torus action afterinverting the equivariant parameters. [CK99] Moreover, if Y is a nonsingularvariety, Bott’s formula gives us an explicit isomorphism mapping classes α inequvariant cohomology of Y to

∑

i∗j (α) ∪ eT (Nj)−1, which are sums of classes

in the equivariant cohomology of each torus-fixed Zj. We can use these ideasto shift the integral over [Mg,n(X,β)] in the formula for Gromov-Witten invari-ants to a sum of integrals over torus-fixed loci in the moduli space, indexed bygraphs Γ.

Equivariant cohomology: definitions Localization rests on the use ofequivariant cohomology. Let G be a compact connected Lie group. Let EG →BG be the principal G-bundle classifying G, and let XG = EG×GX . (EG×GXis the twisted product, just (EG×X)/ ∼, where (eg−1, gx) ∼ (e, x).) The equiv-ariant cohomology H∗

G(X) of X is defined by

H∗G(X) := H∗(XG).

The situation considered exclusively in this paper is that in which G = T =(C∗)n. In this case, BG = (P∞)n and EG = π∗

1S⊕· · ·⊕π∗nS, with πi : BG→ P∞

the projection to the ith factor and S the tautological bundle on CP∞, with

sheaf of sections OCP∞(−1) [CK99].

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In this case, if X is a point, then X ×T ET → BT is an isomorphism, andso H∗

T (pt) = H∗(BT) ∼= C[α1, . . . , αn], where αi = c1(π∗iOP∞(1)). (We can also

think of −αi as the weight by which T acts on each one-dimensional subspace.)More generally, if X has trivial T -action, H∗

T (X) = H∗(X) ⊗C C[α1, . . . , αn].Pullback on equivariant cohomology is defined, so we get a functor from

the category of topological spaces with torus actions and torus-equivariant mor-phisms to the category of C-algebras with algebra homomorphisms. This al-lows us to establish that H∗

T (X) is a H∗T (pt)-module, as we alway have a T -

equivariant map X → pt, giving us C[α1, · · · , αn] ∼= H∗T (pt) → H∗

T(X).

Atiyah-Bott Localization If X is a proper scheme or Deligne-Mumfordstack with a T -action, then the fixed point set XT is the disjoint union of afinite set of closed connected subspaces Zj. If X is smooth, so are the Zj (theo-rem by Iversen). Let ij : Zj → X be the inclusion. Then ij∗ : AT∗ (Zj) → AT∗ (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

⊕HT∗ (Zj) ⊗C[α] C(α) → HT

∗ (X) ⊗C[α] C(α).

For αj ∈ HT∗ (Zj), ⊕jij∗αj = α for α ∈ HT

∗ (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

is an explicit inverse to the map ⊕jij∗ above. It is ⊕ji∗j

eT (Nj).

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 under-lying a smooth stack with a T -action. If α ∈ H∗

T (X) ⊗RT , then

∫

X

α =∑

j

∫

(Zj)

(i∗j(α)

ajeT (Nj)),

where aj is the order of the group H occurring in a local chart at the genericpoint of Zj.

Kontsevich realized that this method could be applied to the smooth stackM0,n(P

r, d) [Kon95]. However, the moduli stacks of stable maps Mg,n(X, d)are not smooth for g > 0 (or X not convex). An extension of Bott’s formula tovirtual fundamental classes is needed to allow computation in these situations.

4.1 Virtual localization

Localization for virtual fundamental classes was proven in [GP99]. Followingtheir format, we will consider the basic case of virtual localization for zero lociof sections of vector bundles, and then proceed to the general case.

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The basic case starts with a nonsingular scheme Y endowed with a C∗ actionand 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(Nviri )

(2)

in HC∗

∗ (X) ⊗ Q(α), where the Xi are the connected components of the fixedpoint scheme and ι is the inclusion.

Start with the perfect obstruction theory on X , as this will give the virtualstructure on the Xi and allow the computation of Nvir

i . X has a natural perfectobstruction theory, as it is the zero locus of a C∗-invariant section of V . Wehave

E• = [V ∨ → ΩY ],

andL•X = [I/I2 → ΩY ].

The morphism φ between them comes from the natural map V ∨ → I/I2. Fol-lowing the construction laid out above, the virtual fundamental class of X iseref (V ), the refined Euler class of V .

Given the tools of equivariant cohomology, we can find the C∗-fixed obstruc-tion theory on Xi. Denote by Yi the components of the fixed locus of Y . EachYi is smooth, with a vector bundle V |Yi

and a section v|Yivanishing strictly

at Xi = X ∩ Yi. The C∗-action induces a C∗-action on V . On each Yi, Vdecomposes into a direct sum of eigensheaves:

Vi = V fi ⊕ V mi .

V f is the eigensheaf invariant under C∗, called the fixed part, and V m is the di-rect sum of eigensheaves on which C∗ acts nontrivially, the moving part. [GP99]Each Xi carries a perfect obstruction theory

[(V fi )∨ → ΩYi].

(This follows from Proposition 1 of [GP99].) Thus the virtual fundamental class

of Xi is eref (Vfi ) on Yi, where V fi denotes the fixed part of V restricted to Yi.

Last, we need the virtual normal bundle. It is the moving part of the com-plex E• = [TY → V ] for each connected component. The moving part ofTY is the normal bundle to Yi, while the moving part of V is simply V mi , soNviri = [NYi/Y → V mi ]. Take the Euler class of a complex using multiplicativity,

obtaining

e(Nviri ) =

e(NYi/Y )

e(V mi ).

One must check that e(V mi ) is invertible. It is, essentially because it has nofixed component.

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Let us return to equation (2). Use the expressions for e(Nviri ), [Xi]

vir, and[X ]vir to see that to establish (2), we must prove

eref (V ) = ι∗∑ eref (V

fi ) ∩ e(Vmi )

e(NYi/Y ). (3)

Usual localization for Y tells us

[Y ] = ι∗∑ [Yi]

e(NYi/Y ),

and intersecting both sides with eref (V ) gives

eref (V ) = ι∗∑ eref (V ) ∩ [Yi]

e(NYi/Y ).

Consider the numerators on the right-hand side: taking refined Euler class com-mutes with pullback, so eref (V ) ∩ [Yi] = eref (Vi). Vi splits on each componentYi into fixed and moving parts. The section v giving X is C∗-fixed, so it livesin V mi , which implies eref (Vi) = eref (V

fi ) ∩ e(Vmi ). 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. LetX be an arbitrary scheme admitting an equivariant embedding intoa nonsingular scheme Y . Compare our desired formula, (2), with the formulaobtained by intersecting the localization formula for Y with [X ]vir on bothsides. Then use Vistoli’s rational equivalence and a lemma on Gysin morphismsto show the desired equality.

Earlier, we constructed the virtual fundamental class using the exact se-quences

0 → TY → D → Dvir → 0 (4)

andD = CXY × E0. (5)

Dvir embeds as a closed subcone of E1 and [X ]vir = s∗E1[Dvir ]. Another way of

characterizing [X ]vir is by the fiber square

TY −−−−→ D

y

y

X0E1−−−−→ E1

,

using 0E1 to denote the zero section. Then [X ]vir = s∗TY 0!E1

[D]. We have theobvious C∗-fixed analogues of (4) and (5) for the embeddings Xi ⊂ Yi.

Localization for Y tells us that

[Y ] = ι∗∑ [Yi]

e(TYm)

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in AT∗ (Y )α. Take refined intersection product with [X ]vir gives

[X ]vir = ι∗∑ [X ]vir · [Yi]

e(TYm)

in AT∗ (X)α. Compare this with what we want, keeping in mind that the normalbundle to Xi is defined to be the moving part of the complex E•,i. Thus itsuffices to show

[X ]vir · [Yi]e(TYm)

=[Xi]

vir ∩ e(Em1 )

e(Em0 )(6)

in AT∗ (Xi)α.From the discussion of [X ]vir above, we can intersect [X ]vir and [Yi] to get

[X ]vir · [Yi] = ι!s∗TY 0!E1

[D]

= s∗TY 0!E1ι![D],

using commutativity of the intersection product for the second equality. UsingVistoli’s rational equivalence, we can continue a step further:

[X ]vir · [Yi] = s∗TY 0!E1

[Di × Em0 ]. (7)

All this occurs in AT∗ (Xi).(Vistoli’s rational equivalence [Vis89] in NYi

Y × ι∗CXY implies ι![CXY ] =[CXi

Yi] in A∗(ι∗CXY ). Translated to the C∗-equivariant case it gives the same

equation in AT∗ (ι∗CXY ). Pull back this relation to ι∗D = ι∗CXY × E0 to getι![D] = [Di × Em0 ]. )

Consider the TYi-action on ι∗D. It leaves Di × Em0 invariant, and sinceDi/TYi = Dvir

i , the class [Di×Em0 ] ∈ AT∗ (ι∗D) is the pullback of [Dviri ×Em0 ] ∈

AT∗ (ι∗D/TYi). Thus we can rewrite s∗TY 0!E1

[Di ×Em0 ] as s∗TYm0!E1

[Dviri ×Em0 ].

Now we need a lemma in order to rewrite [X ]vir · [Yi] in terms of Em0 and Em1rather than TY m. Consider the scheme-theoretic intersection 0−1

E1(Dvir

i ×Em0 ).It lies in TY m. The map

Dviri × Em0 → E1

is the product of inclusion Dviri ⊂ Ef1 and the natural map Em0 → Em1 . Thus

0−1E1

(Dviri × Em0 ) also lies in Em0 .

Lemma 2. [GP99] Let B0 and B1 be C∗-equivariant bundles on Xi. Let Z bea scheme equipped with two equivariant inclusions j0, j1 over Xi:

Z −−−−→ B1

y

y

B0 −−−−→ Xi

.

Let ζ ∈ AC∗

∗ (Z). Then

s∗B0j0∗(ζ) ∩ e(B1) = s∗B1

j1∗(ζ) ∩ e(B0) ∈ AC∗

∗ (Xi).

12

We can apply this lemma to the diagram

0−1E1

(Dviri × Em0 ) −−−−→ Em0

y

y

TY m −−−−→ Xi

using ζ = 0−1E1

[Dviri × Em0 ] to get

[X ]vir · [Yi] = s∗Em0

(0−1E1

[Dviri × Em0 ] · e(TY

m)

e(Em0 )(8)

since we can invert e(Em0 ).Consider the class 0−1

E1[Dvir

i ×Em0 ] to lie in AT∗ (Em0 ), and notice that it doesnot depend on the bundle map Em0 → Em1 . Thus we can assume the bundlemap is trivial. Notice that if we “divide by e(TY m)” on both sides of (8) wewill have the left side of (6), the identity we wanted to show. To finish, we mustshow

[Xi]vir ∩ e(Em1 ) = s∗Em

0(0!E1

[Dviri × Em0 ]).

This follows from the definition of [Xi]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 in-tegration over [Mg,n(X,β)] to integration over torus-fixed loci in Mg,n(X,β).The components of the torus-fixed loci can be indexed by graphs — trees withvertices and edges — when X has isolated torus-fixed points and isolated torus-invariant one-dimensional orbits. When this is true, we can transform a geomet-ric problem into a more combinatorial problem, calculating the contribution tothe Gromov-Witten invariant of each component and summing over the possiblegraphs. Projective space is an ideal example, and will be the case consideredhere.

Each graph Γ corresponds to a substack MΓ.Let (C, f, p1, . . . , pn) be a stable map. Construct the graphs as follows:

• Vertices: The only torus-fixed points of Pr are the points q0 = [1 : 0 : · · · :0], q1 = [0 : 1 : · · · : 0], through qr = [0 : 0 : · · · : 0 : 1]. For a map to be sta-ble and T -equivariant, all nodes, marked points, ramification points, andcontracted components of C must be mapped to T -fixed points. There isexactly one vertex of Γ for each connected component of f−1(q0, . . . , qr).Label vertices by the fixed point to which they correspond.

• Edges: The torus-invariant one-dimensional orbits in Pr are the coordinatelines lj . Each edge in a graph corresponds to a rational (non-contracted)component Ce of C mapped to a coordinate line lj. We label each edge ewith the degree d(e) of the map taking Ce to lj .

13

• 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 thecorresponding fixed point in Pr, vqi

, we must label vertices with the genus of thecorresponding contracted component, if any, and with the marked points. Wewill adopt the convention that the lack of a label for genus indicates a rationalor trivial contracted component.

Examples It is useful to actually try constructing such a graph. Consider theexample of degree two maps from rational curves to P1 with no marked points.Notice that we have two basic types of graphs, since P1 has only two torus-fixedpoints and the degree is restricted:

qi

2

qi qjqi qj

Figure 1: Figure 1

The graph on the left corresponds to a map from a curve with two inter-secting rational components, with the node mapping to qj . There can be nomore than two components of the source curve, since we require all contractedcomponents to be stable and here we have no marked points. The graph on theright corresponds to the double cover of P1, with ramification point mapped toqj .

Deformation/tangent/obstruction sequence As can be seen from Bott’sFormula above, localization requires the calculation of the value of e(Nvir

j ), theEuler class of the (possibly virtual) normal bundle for each component Zj ofthe fixed-point locus. (The virtual bundle is discussed below.)

DefineMΓ = Πv∈ΓMg(v),val(v)+n(v).

We must take into account A, the group of automorphisms acting on MΓ:

1 → ΠedgesZ/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 : MΓ/A→Mg,n(Pr, d).

[GP99] tells us that a C∗-fixed component of Mg,n(Pr, d) is supported onMΓ/A,and that through this we can relate Gromov-Witten invariants of Pr to integralsover moduli spaces of pointed curves.

[GP99] give us the following tangent-obstruction sequence on the substackMΓ/A:

14

0 → Ext0(ΩC(D),OC) → H0(C, f∗TPr) → T 1

→ Ext1(ΩC(D),OC) → H1(C, f∗TPr) → T 2 → 0 (10)

(Note again we label vector bundles by their fibers, and we use D to de-note the divisor

∑

i−pi on C.) This uses the canonical obstruction theory onMg,n(Pr, d), which will be denoted by E•. T 1 and T 2 are defined on MΓ/A bylooking at the restriction of the dual canonical perfect obstruction theory:

0 → T 1 → E0,Γ → E1,Γ → T 2 → 0.

(Check back to the non-virtual case: the definition above does give theusual normal bundle. We just work with a complex of one term! Also note, inparticular, that on projective space when g = 0, we have [Ni]

vir = [Ni].)Thus we can rewrite. Borrowing notation from Graber and Pandharipande,

letBi be the vector bundle making up the ith term of the deformation/obstructionsequence. We can see that

e(Nviri ) =

e(Bm2 )e(Bm4 )

e(Bm1 )e(Bm5 ).

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 torus-

invariant stable maps to projective space, and calculate e(NΓ). We can do itpiece by piece.

5 Localization: Computations

Infinitesimal automorphisms of (C, p1, . . . , pn) Ext0(ΩC(D),OC) is thespace of infinitesimal automorphisms of the pointed curve (C, p1, . . . , pn). Cconsists of contracted and non-contracted components; for non-contracted com-ponents Ce, a weight zero piece comes from the infinitesimal automorphism of Cefixing the two special points. This will cancel with a term from H0(C, f∗TPr).If a vertex v has genus zero and valence one, then the action of T will have

weightαi(v)−αi(v′)

de, which we can write ωF (v). Over all vertices, this term will

give us∏

val(v)=1n(v)=0

ωi(v).

If a vertex has genus zero and valence two, we can use flag notation to write thecontribution over all such vertices:

∏

val(v)=2n(v)=0

(ωF1(v) − ωF2(v)).

15

The space of deformations of (C, p1, . . . , pn) Look at Ext1(ΩC(D),OC).One can deform contracted components, but this will give only weight zero piecesof the bundle. Non-contracted components can’t be deformed on their own; theother deformations come from smoothings of nodes joining contracted to non-contracted components. Look at one vertex v corresponding to such a node: thespace of deformations of a node p at which components C1 and C2 intersect isisomorphic to Tp(C1)⊗Tp(C2) [HM98]. Consider the tangent spaces separately:for C1 the non-contracted component, then just as in the previous paragraph,the weight of the induced action of T will be ωF (v). The contracted component,C2, could be any kind of stable curve, so the tangent space may vary and theweight of the torus action is trivial. We must take into account the variationspossible for C2. Define the ψ-classes by ψF := c1(LF ) ∈ H2(Mg(v),val(v)+n(v)),where LF is the bundle whose fibers are the cotangent space to the curve atthe marked point associated to the flag. The final contribution to e(NΓ) of thebundle whose fibers are Ext1(ΩC(D),OC) is

∏

val(F )+n(F )+2g(v)>2

(ωF (v) − ψF (v))

where the conditions on the flag come from the fact that we’re looking at a nodeof valence at least two, with non-trivial contracted component (i.e., having eithermarked points or genus greater than zero).

Deformations and obstructions of f H0(C, f∗TPr) is the space of thefirst-order deformations of the map f , while H1(C, f∗TPr) can be seen as theobstructions to deformations of f . It is useful to look at these together. Con-tributions come from resolving all the nodes of C implicit in being associatedto a graph Γ. Consider the normalization sequence resolving these nodes:

0 → OC →⊕

vertices

OCv⊕

⊕

edges

OCe→

⊕

flags

OxF→ 0

Twist by f∗(TPr) and take cohomology, remembering that since non-contractedcomponents are rational, they have no higher cohomology:

0 → H0(f∗TPr) →⊕

vertices

H0(Cv, f∗TPr) ⊕

⊕

edges

H0(Ce, f∗TPr) →

→⊕

flags

Tpi(F )Pr → H1(f∗TPr) →

⊕

vertices

H1(Cv, f∗TPr) → 0. (11)

K-theoretically, we can write

H0 −H1 =⊕

vertices

H0(Cv, f∗TPr) ⊕

⊕

edges

H0(Ce, f∗TPr)

−⊕

flags

Tpi(F )Pr −

⊕

vertices

H1(Cv, f∗TPr).

16

Note that on the right-hand side, H0(Cv, f∗TPr) = Tqi(v)

Pr, as Cv is aconnected, contracted component (on Cv, f is constant). The weights of theinduced torus action are αi(v) −αj for all j 6= i(v). Likewise, the weights of theaction on Tpi(F )P

r are αi(F ) − αj for all j 6= i(F ). These contribute

∏

vertices

∏

j 6=i(v)(αi(v) − αj)∏

flags

∏

j 6=i(F )(αi(F ) − αj).

To find the contribution of H0(Ce, f∗TPr), we’ll need the Euler sequence

0 → OPr → O(1) ⊗ V → TPr → 0.

Pull it back to Ce and take cohomology:

0 → C → H0(Ce,O(de)) ⊗ V → H0(Ce, f∗TPr) → 0.

The weight of the action on C is trivial. Look at each piece in the middle term:the weights on V are just −α0, . . . ,−αr, while the weights on H0(O(de)) areadeαi(v) + b

deαi(v′) for a+ b = de and v, v′ the vertices of the edge e. The weights

of the middle term are the pairwise sums of these, adeαi(v) + b

deαi(v′) − αj .

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 cancelswith the trivial weight from the infinitesimal automorphisms of (C, p1, . . . , pn).In addition, when j = i(v), the product of weights

∏

1≤a≤de( ade

αj+de−ade

αi−αj)gives us de!

ddee

(αj − αi)de . Something similar happens when j = i(v′). Final

contribution: a product over edges of

(−1)de(de!)

2

d2dee

(αi − αj)∏

a+b=de

k 6=i,j

(a

deαi +

b

deαj − αk).

Finally, we look atH1(Cv , f∗TPr). Notice thatH1(Cv, f

∗TPr) = H1(Cv,OCv)⊗

Tqi(v)Pr, and that H1(Cv,OCv

) = E∨, where E = π∗ω is the Hodge bundle. (UseSerre duality to see this.) Tensoring with Tqi(v)

Pr gives r copies of E∨ twistedby the weights αi(v) − αj , for j 6= i(v). Take the top Chern class to get:

∏

j 6=i

c(αi(v)−αj)−1(E∨) · (αi(v) − αj)g(v),

where for a bundle Q of rank q,

ct(Q) = 1 + tc1(Q) + . . . tqcq(Q).

Putting it all together Combining all the above contributions,

e(NΓ) = evΓeFΓ e

eΓ (12)

17

eFΓ =∏

flags

∏

val(F )+n(F )+2g(v)>2

(ωF − ψF )∏

j 6=i(F )

1

αi(F ) − αj

evΓ =∏

vertices

∏

j 6=i

∏

val(v)+n(v)+2g(v)>2

1

c(αi(v)−αj)−1(E∨) · (αi(v) − αj)g(v)

∏

val(v)=1n(v)=0

1

ωi(v)

∏

val(v)=2n(v)=0

(ωF1(v)−ωF2(v))

eeΓ =∏

edges

(de!)2(αi(v) − αi(v′))

2de

(−1)ded2dee

∏

a+b=de

k 6=i,j

(a

deαi +

b

deαj − αk)

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 termsabove. Terms of the form αi − αj and variations thereof will be constant termsthat we can take out of the integral. Terms of the form 1

1−ψiwill have to be

expanded as formal power series. We will then be left to evaluate integrals ofthe form

∫

Mg,n

∏

i

ψai

i

∏

i

λi,

where λi = ci(E). These are called Hodge integrals. For g = 0, we can use aconsequence of the string equation:

∫

[M0,n]

ψa11 · · ·ψak

k =

(

n− 3

a1a2 . . . ak

)

,

where a1 + · · · + ak = n − 3, as λi = 0. (This is immediate from the stringequation. In higher genus, pure ψ-integrals can be evaluated recursively byusing Witten’s conjecture, or Kontsevich’s theorem.) General Hodge integralsare not always tractable and their evaluation is an active area of research.

Even when we can evaluate these integrals, virtual localization gives us onlyan algorithm for computing Gromov-Witten invariants. The number of graphsgrows quickly with degree, and the combinatorial expressions become difficultto evaluate. A primary goal is the simplification of computation, usually byjudicious choices of weights, so that graph sums can be evaluated in closedform.

Number of lines through two points in P2 We would like to computethe number of lines (genus 0, degree 1) through two points in P2. This answershould be known to the reader already. In fancier notation, we’re computing

〈h2h2〉P2

0,2,1P1

where h is the Poincare dual to the hyperplane class.First, consider the (C∗)3-fixed points of P2: let q0 = [1 : 0 : 0], q1 = [0 : 1 : 0],

q2 = [0 : 0 : 1]. Then notice that the torus-invariant one-dimensional orbits of

18

P2 are the coordinate lines joining the fixed points. To draw the graphs, wemust notice that we are dealing with two marked points and that our maps areof degree one. Thus in any graph, there is exactly one edge, and all graphscorrespond to one of the types in Figure 2.

qi qjqi qj

p2

p1

p2p1

Figure 2: Figure 2

There are twelve graphs, as one can see by considering the six combinationsof i and j.

Now we face a choice of linearizations. A linearization is a lifting of the torusaction on a space to a vector bundle on that space. Here, we’re looking at liftingsof the torus action from P2 to the line bundle O(1), and the lifting is uniquelydetermined by the weights [l0, l1, l2] of the fiber representations at the fixedpoints q0, q1, q2. The standard linearization lifts the three-torus action to theline bundle as (t0, t1, t2) · [x0 : x1 : x2] → (t0x0, t1x1, t2x2). h is the first Chernclass of O(1), and the cohomology class we get as a lift is the first equivariantChern class of O(1) with this action. To finish the computation, the sum oftwelve terms is required. From the four graph with vertices corresponding to q0and q1, we get the contributions

α20 · α2

0 + α21 · α2

1

(α0 − α1)2(α0 − α2)(α1 − α2)− α2

0 · α21 + α2

1 · α20

(α0 − α1)2(α0 − α2)(α1 − α2). (13)

Add to this the contributions from the graphs with the two other possiblecombinations of vertices to get a great big sum. With careful cancellation, wefind the result to be 1.

Do not despair. There are easier ways. Instead of the standard lift, we couldalso use (t0, t1, t2) · [x0 : x1 : x2] → [x0 : t−1

0 t1x1 : t−10 t2x2], lifting the class

h to h − α0. The contribution of any graph with a vertex corresponding to q0vanishes. This reduces our sum to four terms total:

(α1 − α0)4 + (α2 − α0)

4

(α1 − α2)2(α1 − α0)(α2 − α0)− (α1 − α0)

2(α2 − α0)2 + (α2 − α0)

2(α1 − α0)2

(α1 − α2)2(α1 − α0)(α2 − α0).

(14)

An even better linearization was suggested by Pandharipande: use the torusaction that lifts h2 to (h − α1)(h − α2) on one “factor,” while using the lift(h− α0)(h− α2) on the other. Then the only graph that contributes is the onein Figure 3.

19

p2p1

q0 q1

Figure 3:

Notice that the calculation is made very easy.

(α0 − α1)(α0 − α2) · (α1 − α0)(α1 − α2)

(α0 − α1)2(α0 − α2)(α1 − α2)= 1.

5.1 The multiple cover formula

In the section on Gromov-Witten invariants, it was observed that one reasonGromov-Witten invariants fail to be strictly enumerative even in “nice” situa-tion is because of multiple covers. Localization can be applied to computing thecontribution of these multiple covers. The genus zero case is relatively straight-forward, but in higher genus Hodge integrals appear and integration becomesmore difficult. Luckily, [FP00] established some very nice results for highergenus, illustrating once again the benefit of looking at generating series: it iseasier to compute answers all at once (all g ≥ 2, here) than to do it one case ata time.

Genus zero Manin computed the contribution of degree d multiple covers ofa rigid smooth rational curve C on a Calabi-Yau threefold V to be 1/d3, usingKontsevich’s formulas and clever summation. We will prove this here. LetC ∼= P1, with normal bundle N ∼= OP1(−1)⊕OP1(−1). Denote by Md,C(V ) thecomponent of M0,0(V, d[C]) consisting of stable degree d maps. It is connected,it is isomorphic to M0,0(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 Fd,

C(0, d) :=

∫

Md,C(V )

e(Fd).

The obstruction sheaf can be written more informatively as R1π∗µ∗N , where

π : M0,1(P1, d) → M0,0(P

1, d) is the universal stable map (forgetful functorwith stabilization) and µ : M0,1(P

1, d) → P1 is the evaluation map.Part of the computation is relatively mechanical; the contributions e(NΓ)

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 reducesthe number of graphs to consider. We must also calculate the contribution ofthe 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

loci of T ′ and T correspond, and we can lift the action to OP1(−1) ⊕OP1(−1)by letting t ∈ T ′ act by

t · (l(x0, x1),m(x0, x1)) = (l(t · x0, x1),m(x0, t−1 · x1)),

where l and m are homogeneous of degree −1 in (x0, x1).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, thedomain curve C of the stable map must have a node (or more). Let C split intocomponents Ci, with nodes ri. The normalization exact sequence, then, tells us

0 → f∗(N) → ⊕i(f |Ci)∗(N) → ⊕if∗(N)ri

→ 0.

The long exact sequence in cohomology, then, tells us that ⊕iH0(C, f∗(N)ri)

injects into H1(C, f∗N). The weights of the T ′-action depend on where thenodes get mapped: if f(ri) = q0, then a basis for OP1(−1) at q0 is 1/x0, inwhich case T ′ acts with weight zero on the first factor f∗OP1(−1). If f(ri) = q1,T ′ acts trivially on the second factor. Since weights are multiplicative, thecontribution of N vanishes over any graph with vertex of valence greater thanone [CK99].

This leaves us to consider only the case where Γ has one edge of degree d.Use Cech cohomology to calculate the contribution ofH1(P1, N), which will giveus the contribution of e(R1π∗µ

∗N). Since the map is to P1, use the usual opencover Ui = zi 6= 0, where (z0, z1) are our coordinates. A basis forH1(P1, f∗N)is given by Cech cocycles

( 1

zkozd−k1

, 0)

,(

0,1

zk0zd−k1

)

, 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 x0 and x1 in H0(P1,O(1)), it will act onz0 and z1 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)!)2h2d−2

d2d−2.

Now it remains to calculate the contribution from the normal bundle. Usingequation (12) from the previous section, we can compute contributions usingα0 = h and α1 = 0. We end up with

e(NΓ) =(−1)d(d!)2h2d−2

d2d−2.

It is easy to forget our last factor: the contribution of automorphisms of MΓ.The exact sequence (9) reminds us that we get a factor of d in the denominator.Including this, we get

(−1)d−1((d− 1)!)2h2d−2/d2d−2

d · (−1)d(d!)2h2d−2d2d−2=

1

d3.

Done!

21

Higher genus In the genus one case, physics predicts the contribution tobe 1/12d. Graber and Pandharipande computed this mathematically, usinglocalization. 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) =|B2g| · d2g−3

2g · (2g − 2)!= |χ(Mg)| ·

d2g−3

(2g − 3)!,

where B2g is the 2gth Bernoulli number and χ(Mg) = B2g/2g(2g − 2) is theHarer-Zagier formula for the orbifold Euler characteristic of Mg.

This closed-form equation follows from an intermediate result,

C(g, d) = d2g−3∑

g1+g2=gg1,g2≥0

bg1bg2 ,

where

bg =

1 g = 0∫

Mg,1ψ2g−2

1 λg g > 0,

which is obtained from a straightforward localization computation. Calculatingbg, though, uses generating series, and along the way establishes a number ofbeautiful identities. We will sketch the proof here, giving a tour of the paper.

Recall that we are focused on the integral

C(g, d) =

∫

[Mg,0(P1,d)]vir

e(R1π∗µ∗N),

just as in the genus zero case. As in the genus zero case, we can look at differentlinearizations in order to simplify computation of the integral. Manin’s trickinvolves choosing a localization so that only graphs with the vertex q0 contribute,for instance, and the sum over graphs reduces to a sum over partitions of d. Thisis how Graber and Pandharipande computed the g = 1 contribution, C(1, d) =1/12d, and how Manin originally computed C(0, d) = 1/d3. But as we saw inthe genus zero section, the linearization [0, 1], [−1, 0] results in the vanishing ofthe contribution of any graph Γ containing a vertex with valence greater thanone. Thus all contributing graphs have exactly one edge.

In this situation, the localization sum reduces to a sum over partitions g1 +g2 = 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 thegraph, and all other components will be contracted to one of the fixed points.The components contracted to q1 will have genus g1, while the componentsmapped to q2 will have genus g2. We need not worry about marked points. ForΓ such a graph, the contribution of e(NΓ) can be computed, noting that [FP00]

22

use the linearization with weight 1 on q1 and weight 0 on q2:

eFΓ = (1−ψ1)(−1−ψ2)(−1)(1)

evΓ = d2

c1(E∨)c−1(E∨(−1)g2

eeΓ = (d!)2

(−1)dd2d

The contributions of the normal sheaf, on the other hand, are calculated ina way similar to the genus zero case. The difference is that now we must takeinto account the fact that

H1(C,O(−1)) ∼= H1(Cg1 , L−1|Cg1) ⊕H1(Cg2 , L−1|Cg2

) ⊕H1(P1, µ∗O(−1)),

where Ln denotes the line bundle with (z0, z1, χ) ∼ (λz0, λz1, λnχ). (O(n)

is its associated sheaf of sections, which is why we’re looking at L−1.) Thecontribution of H1(P1, µ∗O(−1)) is exactly the same as in the genus zero case:

(−1)d−1((d − 1)!)2h2d−2

d2d−2.

Notice how many of these terms will cancel with the contribution of e(NΓ).The contribution of H1(Cgi

, L−1|Cgi), for i = 1, 2, is a bit different. Using

duality and definitions,

H1(Cgi, L−1|Cgi

) ∼= H0(Cgi, ωCgi

)∨ ⊗ L−1|qi∼= E∨ ⊗ L−1|qi

.

Taking the Euler class of H1(Cgi, L−1|Cgi

), then, is equivalent to

e(E∨ ⊗ L−1|qi) =

gi∑

j=0

cj(E∨)αgi−j ,

where α represents the linearization used.When the above contributions are combined, many terms cancel. To simplify

our notation, introduce the notation

Λ1(k) =∑g1i=0 k

iλg1−i ∈ A∗(Mg1,1)

Λ2(k) =∑g2i=0 k

iλg2−i ∈ A∗(Mg2,1)

for k ∈ Z. After cancellation and change of notation, using the linearization[1, 0], we are left with the integrand

Λ1(1)Λ1(0)Λ1(−1)Λ2(−1)Λ2(0)Λ2(1)

(1/d− ψ1)(1/d+ ψ2)d3.

23

We can use Mumford’s relationship Λi(1)Λi(−1) = (−1)gi to simplify. Then,integrating, we get

d2g−3

∫

[Mg1,1]vir

λg1ψ2g1−2

∫

[Mg2,1]vir

λg2ψ2g2−2. (15)

The λgicomes from Λgi

(0), in which all terms vanish except for cgi(E∨;

d2g−3 comes about because in expanding 11/d±ψi

we get a factor of d2gi − 3 as

the coefficient of ψ2g−2i .

To simplify this, we can use the notation bg =∫

[Mg,1]vir λgψ2g−2, with the

understanding that g ≥ 0, b0 = 1. Then we have

C(g, d) = d2g−3∑

g1+g2=g

bg1bg2 .

How to compute the right-hand side?The paper [FP00] generalizes the question. They find a generating function

for bg,∑

g≥0

bgt2g =

( t/2

sin(t/2)

)

(16)

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∑

i=0

t2gki∫

Mg,1

ψ2g−2+i1 λg−i. (17)

Then

F (t, k) = fk(t) = (t/2

sin(t/2))k+1

for all k ∈ Z.

Proof. A sketch of the proof:[FP00] define an intermediate function

fξ(t) = 1 +∑

g≥1

t2g∫

Mg,1

Λξ

1 − ψ1= 1 +

∑

g≥1

g∑

i=0

t2gξi∫

Mg,1

ψ2g−2+i1 λg−i (18)

for ξ ∈ Z. They prove that for ξ ∈ Z, fξ(t) = f0(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 ofR1π∗µ

∗OP (V ),and y the top Chern class of R1π∗µ

∗OP (v)(−1)). Use the linearizations [α, α]on OP (v) and [β, β + 1] on OP (V )(−1), α, β ∈ Z, and compute

∫

[Mg,0(P (v),1)]vir

x ∪ y = (−1)gIg(α, β)

24

where

Ig(α, β) =∑

∫

Xg1,g2

Λ1(−1)Λ1(−α)Λ1(β)

1 − ψ1

Λ2(−1)Λ2(α)Λ2(β + 1)

1 − ψ2. (19)

Similarly,∫

[Mg,0(P (v),1)]vir

y ∪ y = (−1)gJg(α, β)

for the linearizations [α, α+ 1], [β, β + 1] on each factor OP (V )(−1) and

Jg(α, β) =∑

∫

Xg1,g2

Λ1(−1)Λ1(−α)Λ1(β)

1 − ψ1

Λ2(−1)Λ2(α+ 1)Λ2(β + 1)

1 − ψ2. (20)

Thus Jg(α, β) = Jg(α′, β′), for all α, α′, β, β′ ∈ Z.

Since the value of the integral is the same regardless of linearization, we findthat

Ig(α, β) = Ig(α′, β′)Jg(α, β) = Jg(α

′, β′) (21)

for all α, α′, β, β′ ∈ Z.These integration formulas let us establish the trivial cases

1 +∑

g≥1

t2gIg(0, 0) = fo(it)

1 +∑

g≥1

t2gJg(0,−1) = f2o (it)

where i =√−1 gives alternating signs and we use Mumford’s relations to sim-

plify the integrands of Ig and Jg. This is the base case for induction. Therelations (21) allow us to claim

1 +∑

g≥1

t2gIg(ξ, 0) = fo(it)

1 +∑

g≥1

t2gJg(0, ξ) = f2o (it).

Define a new series for ξ ∈ Z, in order to proceed with induction:

gxi(t) = 1 +∑

g≥1

t2g∫

Mg,1

Λ(−1)Λ(0)Λ(−ξ)1 − ψ1

.

The integration formulas above give

1 +∑

g≥1

t2gIg(ξ, 0) = gξ(t)fξ(it)

1 +∑

g≥1

t2gJg(0, ξ) = gξ(t)fξ+1(it).

25

This, though, implies

gξ(t)fξ(it) = f0(it), gξ(t)fξ+1(it) = f20 (it),

so fξ(it) = fξ(it)f0(it) for all ξ ∈ Z. By induction, fξ(t) = f0(t)ξ+1.

Next, relations from classical curve theory are used to show

f−2(t) = 1 +∑

g≥1

t2g∫

Mg,1

ψ2g−21 (λg − 2ψ1λg−1 + · · · + (−2ψ1)

g) =sin(t/2)

t/2

(Proposition 4, [FP00]). Together these results combine to prove Theorem 2 in

the paper, which gave us F (t, k) =( t/2

sin(t/2)

)k+1.

Returning to our goal of computing d2g−3∑

g1+g2=g bg1bg2 , we can rewritet/2

sin(t/2) in terms of Bernoulli numbers using a “well-known” computation:

t/2

sin(t/2)=

it

eit − 1eit/2 =

it

eit/2 − 1− it

eit − 1= 2β(it/2)− β(it)

= 2 − 1

2it−

∑

g≥1

1

22g−1

|B2g|(2g)!

t2g − 1 − 1

2it−

∑

g≥1

|B2g|(2g)!

t2g

= 1 +∑

g≥1

22g−1 − 1

22g−1

|B2g|(2g)!

t2g.

Then by the definition of bg and one of the equations above, we have (forg ≥ 1)

bg =

∫

Mg,1

ψ2g−21 λg =

22g−1 − 1

22g−1

|B2g|(2g)!

.

Setting βg = (2 − 22g)B2g

(2g)! , we have the identities∑∞

g=0 βgx2g−1 = 1

sinh(x) and∑∞g=0

22gB2g

(2g)! x2g−1 = coth(x). Since (cothx)′ = −(sinhx)−2, coefficients can be

matched to find that

2βg +

g−1∑

h=1

βhβg−h = −22g

2g

B2g

(2g − 2)!

which, by some algebra, is equivalent to

g∑

h=0

bhbg−h =|B2g|2g

1

(2g − 2)!.

This proves Theorem 3 of [FP00], which is what we wanted:

C(g, d) =|B2g| · d2g−3

2g · (2g − 2)!= |χ(Mg)| ·

d2g − 3

(2g − 3)!.

26

6 Future work

Consider maps into P2 × P2, with the action of the two-dimensional diagonaltorus. Here we do not have isolated torus-equivariant one-dimensional orbits,although fixed points are isolated; recall that isolated orbits are a necessaryhypothesis for the localization equation proved above. [BCS] may have sometechniques of relevance to this problem.

This is a specific case of a much more general situation. My advisor, IonutCiocan-Fontanine, is looking at extending the abelian/non-abelian correspon-dence to higher-genus Gromov-Witten invariants. As part of this larger picture,one can attempt to express usual invariants of X//G, the GIT quotient of X forG a Lie group, in terms of twisted Gromov-Witten invariants of X//T , the GITquotient of X by T a torus in G. An example of this is X//G = Grass(k, n) andthe product of k terms X//T = Pn−1 × · · · × Pn−1, with (C∗)n acting on bothspaces. Torus-fixed points and torus-invariant curves on X//G will be isolated,but this is not true for X//T . Clearly, k = 2, n = 3 is the place to start! Thegoal is to show that contributions from these “messy” places cancel out, leavingonly contributions from the open set in X//T corresponding to G-stable points.Maps with image in this open set obey the hypotheses of the virtual localizationformula.

A broader question is this: how can we simplify computation of Gromov-Witten invariants in situations without a torus action? This is something to keepin mind over the next few years. The obvious first step is to learn about de-generation, the only other known technique for computation of Gromov-Witteninvariants.

References

[BCS] Tom Braden, Linda Chen, , and Frank Sottile. The Equivariant Chowrings of quot schemes.

[Beh97] K. Behrend. Gromov-Witten invariants in algebraic geometry. Invent.Math., 127(3):601–617, 1997.

[BF97] K. Behrend and B. Fantechi. The intrinsic normal cone, 1997.

[BM96] K. Behrend and Yu. Manin. Stacks of stable maps and Gromov-Witteninvariants. Duke Math. J., 85(1):1–60, 1996.

[CFK] Ionut Ciocan-Fontanine and Mikhail Kapranov. Virtual fundamentalclasses via dg-manifolds.

[CK99] D. Cox and S. Katz. Mirror symmetry and algebraic geometry, 1999.

[FP97] W. Fulton and R. Pandharipande. Notes on stable maps and quantumcohomology, 1997.

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[FP00] C. Faber and R. Pandharipande. Hodge integrals and Gromov-Wittentheory, 2000.

[Ful] William Fulton. Intersection theory.

[GP99] T. Graber and R. Pandharipande. Localization of virtual classes, 1999.

[HM98] Joe Harris and Ian Morrison. Moduli of Curves, volume 187 of GraduateTexts in Mathematics. Springer, 1998.

[Hor03] Kentaro Hori. Mirror Symmetry, volume 1 of Clay Mathematics Mono-graphs. American Mathematical Society, 2003.

[Kat06] Sheldon Katz. Enumerative Geometry and String Theory, volume 32 ofStudent Mathematical Library. American Mathematical Society, 2006.

[Kon95] Maxim Kontsevich. Enumeration of rational curves via torus actions,1995.

[Vis89] Angelo Vistoli. Intersection theory on algebraic stacks and on theirmoduli spaces, 1989.

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