82(2009) 641-690
ON THE GENUS-ONE GROMOV-WITTEN INVARIANTS OF COMPLETE INTERSECTIONS
Jun Li & Aleksey Zinger
Abstract
We state and prove a long-elusive relation between genus-one Gromov-Witten of a complete intersection and twisted Gromov- Witten invariants of the ambient projective space. As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component M01,k(Pn, d) of the moduli space of stable genus-one holomorphic maps into Pn have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way,J-holomorphic maps. This extension is used to show that these cones are the correct genus-one analogues of the vector bundles relating genus- zero Gromov-Witten invariants of a complete intersection to those of the ambient projective space. A relationship for higher-genus invariants is conjectured as well.
Contents
1. Introduction 642
1.1. Gromov-Witten invariants and complete intersections 642 1.2. Cones of holomorphic bundle sections 647
1.3. Some special cases 651
2. Hyperplane property for genus-one GW-invariants 653
2.1. Review of definitions 653
2.2. Statement and proof of hyperplane property 657
3. Ingredients in proof of Theorem 1.3 659
3.1. Notation: genus-zero maps 659
3.2. Notation: genus-one maps 661
3.3. Topology 663
3.4. The structure of the moduli spaceM01,k(X, A;J, ν) 666 3.5. The structure of the cone V1,kA 668
The first author was partially supported by NSF grant DMS-0601002; the second author was partially by an NSF Postdoctoral Fellowship.
Received 03/17/2008.
641
4. Proof of Proposition 3.9 670
4.1. Outline 670
4.2. Proof of Lemma 4.1 675
4.3. Proof of Lemma 4.2 680
References 688
1. Introduction
1.1. Gromov-Witten invariants and complete intersections.
Gromov-Witten invariants of symplectic manifolds have been a subject of much research over the past two decades. A great deal of atten- tion has been devoted in particular to Calabi-Yau manifolds. These manifolds play a prominent role in theoretical physics, and as a result physicists have made a number of important predictions concerning CY- manifolds. Some of these predictions have been verified mathematically;
others have not.
IfY is a compact K¨ahler submanifold of the complex projective space Pn, one could try to compute GW-invariants of Y by relating them to GW-invariants of Pn. For example, supposeY is a hypersurface in Pn of degree a. In other words, if γ−→Pn is the tautological line bundle and L=γ∗⊗a−→Pn, then
Y =s−1(0),
for somes∈H0(Pn;L) such thatsis transverse to the zero set. Ifg,k, and d are nonnegative integers, let Mg,k(Pn, d) and Mg,k(Y, d) denote the moduli spaces of stable J0-holomorphic degree-dmaps from genus- g Riemann surfaces with k marked points to Pn and Y, respectively.
These moduli spaces determine the genus-g degree-d GW-invariants of Pn and Y.
By definition, the moduli space Mg,k(Y, d) is a subset of the moduli space Mg,k(Pn, d). In fact,
(1.1) Mg,k(Y, d) =
[C, u]∈Mg,k(Pn, d) :s◦u= 0∈H0 C;u∗L . Here [C, u] denotes the equivalence class of the holomorphic map u: C −→Pn from a genus-g curve C with k marked points. The relation- ship (1.1) can be restated more globally as follows. Let
πdg,k:Ug,k(Pn, d)−→Mg,k(Pn, d) be the universal family and let
evdg,k:Ug,k(Pn, d)−→Pn
be the natural evaluation map. In other words, the fiber of πg,kd over [C, u] is the curveC with kmarked points, while
evdg,k [C, u;z]
=u(z) if z∈C.
We define a section sdg,k of the sheafπg,k∗d evd∗g,kL−→Mg,k(Pn, d) by sdg,k [C, u]
= [s◦u].
By (1.1), Mg,k(Y, d) is the zero set of this section.
The previous paragraph suggests that it should be possible to relate the genus-gdegree-dGW-invariants of the hypersurfaceY to the moduli space Mg,k(Pn, d) in general and to the sheaf
πdg,k∗evd∗g,kL−→Mg,k(Pn, d) in particular. In fact, it can be shown that
GWY0,k(d;ψ)≡ ψ,
M0,k(Y, d)vir
=
ψ·e π0,k∗d evd∗0,kL ,
M0,k(Pn, d) (1.2)
for all ψ∈H∗(M0,k(Pn, d);Q); this was observed early on by Beauville [2], for example. The moduli space M0,k(Pn, d) is a smooth orbivariety and
(1.3) π0,k∗d evd∗0,kL−→M0,k(Pn, d)
is a locally free sheaf, i.e. a vector bundle. The right-hand side of (1.2) can be computed via the classical Atiyah-Bott localization theorem [1], though the complexity of this computation increases rapidly with the degree d.
A hyperplane property, i.e. a relationship such as (1.2), for positive- genus GW-invariants has been elusive since the early days of the Gro- mov-Witten theory. Ifg >0, the sheaf
πdg,k∗evd∗g,kL−→Mg,k(Pn, d)
is not locally free and need not define an euler class. Thus, the right- hand side of (1.2) may not even make sense if 0 is replaced by g >0.
Instead one might try to generalize (1.2) as GWYg,k(d;ψ)≡
ψ,
Mg,k(Y, d)vir
=?
ψ·e R0πg,k∗d evd∗g,kL−R1πdg,k∗evd∗g,kL ,
Mg,k(Pn, d)vir , (1.4)
where Riπdg,k∗evd∗g,kL−→Mg,k(Pn, d) is the i-th direct image sheaf. The right-hand side of (1.4) can be computed via the virtual localization theorem of Graber-Pandharipande [10]. However,
N1(d)≡GWY1,0(d; 1)
6
=
e R0πd1,0∗evd∗1,0L−R1π1,0∗d evd∗1,0L ,
M1(P4, d)vir , according to Graber-Pandharipade [11] and Katz [12] for a quintic threefold Y⊂P4.
In this paper we prove a hyperplane property for genus-one GW- invariants. We denote by
M01,k(Pn, d)⊂M1,k(Pn, d)
the closure inM1,k(Pn, d), either in the stable-map or Zariski topology, of the subspace
M01,k(Pn, d) =
[C, u]∈M1,k(Pn, d) :C is smooth . If Y⊂Pn is a hypersurface as above, let
M01,k(Y, d) =M1,k(Y, d)∩M01,k(Pn, d).
Since M01,k(Pn, d) is a unidimensional orbi-variety, it carries a funda- mental class. By [28, Corollary 1.6],M01,k(Y, d) carries a virtual funda- mental class. It can be used to definereducedgenus-one Gromov-Witten invariants:
GW0;Y1,k(d;ψ) ≡ ψ,
M01,k(Y, d)vir
∈Q,
where ψ is a tautological (cohomology) class on M01,k(Y, d); see below.
We show in this paper that the reduced genus-one GW-invariants satisfy a natural analogue of (1.2).
Theorem 1.1. Suppose dandaare positive integers,k is a nonneg- ative integer, L=γ∗⊗a−→Pn,
πd1,k:U1,k(Pn, d)−→M01,k(Pn, d) and evd1,k:U1,k(Pn, d)−→Pn are the universal family and the natural evaluation map, respectively. If Y⊂Pn is a smooth degree-a hypersurface, then
(1.5) GW0;Y1,k(d;ψ) =
ψ·e(π1,k∗d evd∗1,kL),
M01,k(Pn, d) for every tautological class ψ on M1,k(Pn, d).
The tautological classes onM1,k(Pn, d) are certain natural cohomology classes. They include all geometric classes defined in Subsection 1.3.
We describe the space of all cohomology classesψto which Theorem 1.1 applies in Subsection 2.2.
Implicit in the statement of Theorem 1.1 is that the euler class of the sheaf
(1.6) π1,k∗d evd∗1,kL−→M01,k(Pn, d)
is well-defined, even though it is not locally free. This is the case by [27, Theorem 1.1].
The right-hand side of (1.5) should in principle be computable via lo- calization directly. However, since the space M01,k(Pn, d) is not smooth
and the sheaf (1.6) is not locally free, the classical localization the- orem [1] is not immediately applicable. A desingularization of the space M01,k(Pn, d), i.e. a smooth orbivariety fM01,k(Pn, d) and a map
˜
π:fM01,k(Pn, d)−→M01,k(Pn, d),
which is biholomorphic onto M01,k(Pn, d), is constructed in [22]. This desingularization of M01,k(Pn, d) comes with a desingularization of the sheaf (1.6), i.e. a vector bundle
V˜1,kd −→fM01,k(Pn, d) s.t. π˜∗V˜1,kd =π1,k∗d evd∗1,kL.
In particular,
ψ·e π1,k∗d evd∗1,kL ,
M01,k(Pn, d)
=
˜
π∗ψ·e( ˜V1,kd ),fM01,k(Pn, d) . (1.7)
Since a group action onPn induces actions onMf01,k(Pn, d) and on ˜V1,kd , the classical localization theorem is directly applicable to the right-hand side of (1.7), for a natural cohomology class ψ.
By itself, Theorem 1.1 does not provide a way of computing the stan- dard genus-one GW-invariants of Y. However, the reduced genus-one GW-invariants capture the contribution of M01,k(Y, d) to the standard genus-one GW-invariants. Thus, the difference between the two invari- ants is completely determined by the genus-zero invariants of Y; see [31, Subsection 1.2]. We give explicit formulas in some special cases in Subsection 1.3 below.
Remark 1: Theorem 1.1 generalizes to arbitrary smooth complete intersections in projective spaces. More precisely, if
L=γ∗⊗a1⊕. . .⊕γ∗⊗am −→Pn,
with a1, . . . , am∈Z+, s∈H0(Pn;L) is transverse to the zero set in L, and Y=s−1(0), then
(1.8) GW0;Y1,k(d;ψ) =
ψ·e(π1,k∗d evd∗1,kL),
M01,k(Pn, d) , for every geometric cohomology class ψ onM1,k(Pn, d).
Remark 2: In turn, Remark 1 generalizes as follows. Suppose (X, ω, J) is a compact almost K¨ahler manifold,
A∈H2(X;Z)∗≡H2(X;Z)−{0},
(L,∇)−→X is a complex vector bundle with connection, andsis a ∇- holomorphic section of L; see Subsections 1.2 and 2.2 for terminology.
If J is genus-one A-regular in the sense of [26, Definition 1.4], s is
transverse to the zero set inL, and (L,∇) splits into line bundles that are (ω, A)-positive in the sense of Definition 1.2 below, then
GW0;Y1,k(A;ψ) =
ψ·e(V1,kA ),
M01,k(X, A;J)vir
≡
ψ,PDM0
1,k(X,A;J)e(V1,kA ) (1.9) ,
where Y=s−1(0), ψis a tautological class, and the cone V1,kA −→M01,k(X, A;J)
is the geometric analogue of the sheaf πd1,k∗evd∗1,kL. It consists of ∇- holomorphic sections of the vector bundleL, as defined in Subsection 1.2 below. By Corollary 1.4, the Poincare dual of its euler class is well defined as long as (L,∇) is a direct sum of (ω, A)-positive line bundles.
Theorem 1.1 and Remarks 1 and 2 have a natural, but rather specu- lative, generalization to higher-genus invariants. Suppose that the main component
M0g,k(X, A;J) ⊂Mg,k(X, A;J)
is well defined and carries a virtual fundamental class. If so, it deter- mines reduced genus-g GW-invariants GW0;Yg,k(A;ψ). Suppose further that (the Poincare dual of) the euler of the cone
Vg,kA −→M0g,k(X, A;J)
corresponding to the vector bundle (L,∇)−→X is well defined. If con- structions of these objects are direct generalizations of the correspond- ing constructions in Subsection 1.2 and in [26]-[28], then the proof of Theorem 1.1 can be generalized to show that
(1.10) GW0;Yg,k(A;ψ) =
ψ·e(Vg,kd ),
M0g,k(X, A;J)vir ,
provided appropriate generalizations of the assumptions in Remark 2 hold. Along with an equally speculative generalization of [28, Theo- rem 1.1] stated in [28, Subsection 1.2], (1.10) would, if true, provide an algorithm for computing arbitrary-genus GW-invariants of complete intersections.
From the point of view of algebraic geometry as described by Behrend- Fantechi [3] and Li-Tian [16], the genus-g degree-dGromov-Witten in- variant GWYg,k(d;ψ) is the evaluation of ψ on the virtual fundamental class [Mg,k(Y, d)]vir. Using the more concrete point of view of symplectic topology as described by Fukaya-Ono [6] and Li-Tian [15], GWYg,k(d;ψ) can be interpreted as the euler class of a vector bundle, albeit of an infinite-rank vector bundle over a space of the “same” dimension. As in the finite-dimensional case, this euler class is the number of zeros, counted with appropriate multiplicities, of a transverse (multivalued, generic) section. It is shown by Li-Tian [17] and Siebert [21] that the two approaches are equivalent. In this paper, we take the latter point
of view. Similarly, we view the euler class of the sheaf (1.6) as the zero set of a generic section of its geometric analogue V1,kd defined in Subsection 1.2.
Theorem 1.1 and Remark 1 are special cases of Remark 2, which is the same as Theorem 2.3. It is proved in Subsection 2.2 by showing that the zero sets of two bundle sections whose cardinalities are the two expressions in (1.9) are the same set. In fact, Theorem 2.3, just like its genus-zero analogue, follows easily from definitions of the two sides in (1.9), once it is established that these definitions are well-posed.
1.2. Cones of holomorphic bundle sections.Let (X, ω, J) be a compact almost K¨ahler manifold. In other words, (X, ω) is a symplectic manifold and J is an almost complex structure onX tamed by ω, i.e.
ω(v, Jv)>0 ∀ v∈T X−X.
Ifg, k are nonnegative integers andA∈H2(X;Z), let Mg,k(X, A;J) de- note the moduli space of (equivalence classes of) stable J-holomorphic maps from genus-gRiemann surfaces withkmarked points in the homol- ogy class A. Let M01,k(X, A;J) be the main component of the moduli space M1,k(X, A;J) described by [26, Definition 1.1]; see also Defi- nition 2.2 below. This closed subspace of M1,k(X, A;J) contains the subspace M01,k(X, A;J) consisting of the stable maps [Σ, u] such that the domain Σ is a smooth Riemann surface. If J is sufficiently regular, M01,k(X, A;J) is the closure of M01,k(X, A;J) inM1,k(X, A;J).
SupposeL−→X is a complex line bundle and∇is a connection inL.
If (Σ, j) is a Riemann surface andu: Σ−→X is a smooth map, let
∇u: Γ(Σ;u∗L)−→Γ Σ;T∗Σ⊗u∗L
be the pull-back of ∇by u. Ifb= (Σ, j;u), we define the corresponding
∂-operator by¯
∂¯∇,b: Γ(Σ;u∗L)−→Γ Σ; Λ0,1i,jT∗Σ⊗u∗L ,
∂¯∇,bξ = 1
2 ∇uξ+i∇uξ◦j , (1.11)
where iis the complex multiplication in the bundleu∗Land Λ0,1i,jT∗Σ⊗u∗L=
η∈Hom(TΣ, u∗L) :η◦j =−iη .
The kernel of ¯∂∇,b is necessarily a finite-dimensional complex vector space.
We denote by X1,k(X, A) the space of all degree-A stable smooth maps from genus-one Riemann surfaces with k marked points into X and by
V1,kA −→X1,k(X, A)
the cone, or the bundle of (orbi-)vector spaces, such that V1,kA
[b]= ker ¯∂∇,b
Aut(b) ∀ [b]∈X1,k(X, A).
The spaces X1,k(X, A) and V1,kA have natural topologies; see Subsec- tion 2.1 below. By [27, Theorem 1.1], if (X, ω, J) is the complex pro- jective space (Pn, ω0, J0) with its standard K¨ahler structure and (L,∇) is a positive power of the hyperplane line bundle, i.e. the dual of the tautological line bundle,γ∗with its standard connection, then the euler class of
V1,kA −→M01,k(X, A;J)
and its Poincare dual are well defined. By [27, Theorem 1.2], this is also the case if J is an almost complex structure onPn sufficiently close to J0.
The argument in [27] easily generalizes to all (X, ω, J), (L,∇), and A such that (L,∇) is a split positive vector bundle with connection and J satisfies a certain regularity condition. This regularity condition, which is described by [26, Definition 1.4], implies that M01,k(X, A;J) has the expected topological structure of a unidimensional orbivariety.
In this paper, we show that the Poincare dual of the euler classofV1,kA over M01,k(X, A;J) is well defined without any condition on J, as long as (L,∇) satisfies the requirement of Definition 1.2; see Corollary 1.4 below.
Definition 1.2. Suppose (X, ω) is a symplectic manifold and A∈ H2(X;Z). A complex line bundleL−→X is (ω, A)-positive if
hc1(L), Bi>0 ∀B∈H2(X;Z)∗ s.t. B=A orhω, Bi<hω, Ai. We note that V1,kA −→M01,k(X, A;J) is not a vector bundle, as the fibers ofV1,kA are of two possible dimensions. In [27, Subsections 1.2,1.3], the Poincare dual of the euler class of V1,kA is defined as the zero set of a generic multisection ϕ of V1,kA over M01,k(X, A;J). This zero set determines a homology class in M01,k(X, A;J) if ϕ is sufficiently regu- lar. In [27, Section 3], it is shown that V1,kA contains a vector subbun- dle of a sufficiently high rank over a neighborhood of every stratum of M01,k(X, A;J). The existence of such subbundles implies that regular sections of V1,kA exist; see [27, Subsection 3.1].
If J does not satisfy the regularity condition of [26, Definition 1.4], the moduli space M01,k(X, A;J) itself need not carry a fundamental class. In this case, we cannot define the Poincare dual of the euler class of V1,kA as the zero set of a section of V1,kA over M01,k(X, A;J). On the other hand, in [28], the definition of
M01,k(X, A;J) ⊂M1,k(X, A;J)
given in [26] is generalized to define the main componentM01,k(X, A;J, ν) of the moduli space M1,k(X, A;J, ν) of (J, ν)-holomorphic maps for an effectively supported perturbation ν of the ¯∂J-operator; see Defini- tions 2.1 and 2.2 below. By [28, Theorem 1.5], if ν is sufficiently small and generic, M01,k(X, A;J, ν) determines a rational homology class in a small neighborhood of M01,k(X, A;J) in X1,k(X, A). This rational homology class is independent of the choice of ν. We will define the Poincare dual of the euler class of V1,kA as the zero set of a generic mul- tisection of V1,kA over M01,k(X, A;J, ν). Tietze Extension Theorem will be used to show that V1,kA admits sections that are sufficiently nice for this purpose.
IfJ is an almost complex structure onXandJ≡(Jt)t∈[0,1]is a family of almost complex structures on X, we denote by
Ges1,k(X, A;J) and Ges1,k(X, A;J)
the spaces of effectively supported perturbations of the ¯∂J-operator on X1,k(X, A) and of effectively supported families of perturbations of the
∂¯Jt-operators onX1,k(X, A); see Subsection 2.1 for details. If
¯
ν≡(νt)t∈[0,1]∈Ges1,k(X, A;J), we put
M01,k(X, A;J , ν) =
(t, b)∈[0,1]×X1,k(X, A) :b∈M01,k(X, A;Jt, νt) . We denote by ¯Z+ the set of nonnegative integers. Let
dim1,k(X, A;L) = dim1,k(X, A)−2hc1(L), Ai
= 2 hc1(T X)−c1(L), Ai+k .
Theorem 1.3. Suppose (X, ω, J) is a compact almost K¨ahler mani- fold, A∈H2(X;Z)∗,k∈Z¯+,(L,∇)−→X is an(ω, A)-positive line bun- dle with connection, V1,kA −→X1,k(X, A) is the corresponding cone, and W is a neighborhood ofM01,k(X, A;J)inX1,k(X, A). Ifν∈Ges1,k(X, A;J) is sufficiently small and generic and ϕ is a generic multisection ofV1,kA
overM01,k(X, A;J, ν), thenϕ−1(0)determines a rational homology class inW. Furthermore, if J= (Jt)t∈[0,1] is a family ofω-tamed almost com- plex structures on X, such that J0=J and Jt is sufficiently close to J for all t, ν0 and ν1 are sufficiently small generic effectively supported perturbations of ∂¯J0 and ∂¯J1, and ϕ0 and ϕ1 are generic multisections of V1,kA over M01,k(X, A;J0, ν0) and M01,k(X, A;J1, ν1), then there exist homotopies
ν= (νt)t∈[0,1] ∈Ges1,k(X, A;J), Φ∈Γ M01,k(X, A;J , ν);V1,kA
between ν0 and ν1 and between ϕ0 andϕ1 such that Φ−1(0)determines a chain in W and
∂Φ−1(0) =ϕ−11 (0)−ϕ−10 (0).
Corollary 1.4. If(X, ω, J),A,k, and(L,∇)are as in Theorem 1.3, the cone V1,kA −→X1,k(X, A) corresponding to (L,∇) determines a well- defined homology class
PDM0
1,k(X,A;J)e(V1,kA )∈Hdim1,k(X,A;L) M01,k(X, A;J);Q).
This class is an invariant of (X, ω) and (L,∇).
As in [27], we will describe the local structure of the cone V1,kA . In contrast to [27], we will not construct a high-rank vector subbundle of V1,kA over a neighborhood of every stratum of M01,k(X, A;J, ν). Instead, we will use Tietze Extension Theorem to construct a sufficiently regular multisection ofV1,kA . Its zero set determines a homology class in a small neighborhood of M01,k(X, A;J) in the spaceX1,k(X, A).
For a generic ν, M01,k(X, A;J, ν) can be stratified by orbifolds Uα
of even dimensions; see Subsection 3.4 and Remark 1 at the end of Subsection 3.3. The main stratum of M01,k(X, A;J, ν),
M01,k(X, A;J, ν) ≡M01,k(X, A;J, ν)∩X01,k(X, A), is of dimension dim1,k(X, A), where
X01,k(X, A)⊂X1,k(X, A)
is the subspace of stable maps with smooth domains. In Subsection 3.5, we describe a subconeW1,kA ofV1,kA such thatW1,kA |Uα is a smooth vector bundle for every stratumUα. By analyzing the obstruction to extending holomorphic bundle sections from singular to smooth domains in Sec- tion 4, we show thatW1,kA is a regular obstruction-free cone in the sense of Definition 3.3. By Proposition 3.6, for a generic multisection ϕ of W1,kA ⊂V1,kA overM01,k(X, A;J, ν),ϕ|Uα is then transverse to the zero set inW1,kA |Uα. By the rank statements of Proposition 3.9, ϕ−1(0) is strat- ified by smooth orbifolds of even dimensions. Furthermore, the main stratum of ϕ−1(0) is of dimension dim1,k(X, A;L) and is contained in M01,k(X, A;J, ν). We can then choose an arbitrarily small neighborhood U of the boundary ofϕ−1(0) such that
Hl(U;Q) ={0} ∀l≥dim1,k(X, A;L)−1.
Since ϕ−1(0)−U is compact, via the pseudocycle construction of [19, Chapter 7] and [20, Section 1], ϕ−1(0) determines a homology class
ϕ−1(0)
∈Hdim1,k(X,A;L)(W, U;Q)
≈Hdim1,k(X,A;L)(W;Q);
see also [29]. The second part of Theorem 1.3 is a parametrized version of this construction. Corollary 1.4 is an immediate consequence of The- orem 1.3; see also Remark 2 in [28, Subsection 1.3] and the comments at the end of [28, Subsection 1.4].
The statement of Corollary 1.4 is not needed to show that the ex- pressions on the right-hand sides of (1.5) and (1.8) are well defined, as this is the case by [27, Theorem 1.1]. However, the detailed statement of Theorem 1.3 is useful for proving Theorem 1.1 and its generalizations in Remarks 1 and 2 whenever Y is not a Fano complete intersection. If Y is Fano, Theorem 1.1 can be obtained from [27] by working just with J-holomorphic, instead of (J, ν)-holomorphic, maps.
Remark: If L is a direct sum of (ω, A)-positive line bundles, the Poincare dual of the euler class of the corresponding cone is defined to be the intersection product of the Poincare duals of the euler classes of the cones corresponding to the component line bundles. The intersection product can be defined by intersecting pseudocycle representatives for the above homology classes; see [27, Subsection 1.2].
1.3. Some special cases.By [28, Proposition 3.1], the difference be- tween the standard and reduced genus-one invariants of a symplectic manifold (Y, ω) is a combination of the genus-zero invariants of Y. The exact form of this combination in general is determined in [31].
If (Y, ω, J) is an almost K¨ahler manifold, for each l= 1, . . . , k let evl:Mg,k(Y, A;J)−→Y,
Σ, y1, . . . , yk;u
−→u(yl),
be the evaluation map at thel-th marked point. We will call a cohomol- ogy class ψ on Mg,k(Y, A;J) geometric if ψ is a product of the classes ev∗lµl forµl∈H∗(Y;Z). By [28, Theorem 1.1], if A∈H2(Y;Z)∗, then
GWY1,k(A;ψ)−GW0;Y1,k(A;ψ)
=
(0, if dimRY= 4;
2−hc1(T Y),Ai
24 GWY0,k(A;ψ), if dimRY= 6, (1.12)
for every geometric cohomology class ψ onM1,k(Y, A;J).
In the rest of this subsection, we discuss some implications of The- orem 1.1 and Remarks 1 and 2, combined with (1.12), focusing on Calabi-Yau complete-intersection threefolds. We note that if Y is a Calabi-Yau threefold, then the expected dimension of the moduli space Mg,0(Y, A;J) is zero for every g and A.
With notation as in Theorem (1.3), if a= 5, Y is a quintic threefold.
It can be easily seen thatc1(T Y) = 0. Let Ng(d) = GWYg,0(d; 1).
Theorem 1.3 and equation (1.12) then give the following corollary.
d 1 2 3 4 h. . .i 0 2,87532 49,355,00081 952,691,384,375
256
N1(d) 2,87512 407,1258 243,388,750 9
366,163,353,125 16
n1(d) 0 0 609,250 3,721,431,625 Table 1. Low-degree GW-invariants of a quintic threefold.
Corollary 1.5. Suppose dis a positive integer,L=γ∗⊗5−→P4, and π1d:U1(P4, d)−→M01(P4, d) and evd1:U1(P4, d)−→P4
are the universal family and the natural evaluation map, respectively. If Y⊂P4 is a smooth quintic threefold,
(1.13) N1(d) = 1
12N0(d) +
e π1∗devd∗1 L ,
M01(P4, d) .
The middle number in (1.13) can be computed using (1.2). This has been done for every d in [5], [7], [9], [13], and [14]. As mentioned in Subsection 1.1, the last number in (1.13) can be computed, for each given d, via the classical localization theorem of [1]. Similarly to the genus-zero case, the complexity of computing the last term in (1.13) increases rapidly with the degree d, but this has been fully carried out in [30], finally confirming the genus-one prediction of [4] for a quintic threefold. A few low-degree values are shown in the second row of Ta- ble 1 (these numbers were obtained by a direct localization computation and predate the desingularization construction of [22] and the complete computation of [30]). The numbers n1(d) that appear in the last row of this table are defined by
N0(d) =X
k|d
n0(d/k) k3 ,
N1(d) = 1 12
X
k|d
n0(d/k)
k +X
k|d
σ(k)
k n1(d/k), σ(k) =X
r|k
r.
The numbersn0(d) and n1(d) are of importance in theoretical physics.
Conjecturally,ng(d) is a count ofJ-holomorphic degree-dgenus-gcurves inY for a generic almost complex structureJ on Y.
With notation as in Remark 1 in Subsection 1.1, if a1+. . .+am=n+1
and Y is a corresponding complete intersection, thenY is a Calabi-Yau threefold. Let
NgY(d) = GWYg,0(d; 1).
The identities in Remark 1 and in (1.12) then give N1Y(d) = 1
12N0Y(d) +
e π1∗d evd∗1 L ,
M01(Pn, d) .
Once again, both terms on the right-hand side are computable via (1.2) and the classical localization theorem.
In the more general case of Remark 2 in Subsection 1.1,Y is a Calabi- Yau threefold if
c1(L)−c1(T X) = 0 and dimRX−2rkCL= 6.
In such a case,
N1Y(A) = 1
12N0Y(A) +
e(V1,kA ),
M01(X, A;J) ,
whereV1,kA is the cone of “holomorphic sections” corresponding to (L,∇) and NgY(A) = GWYg,0(A; 1).
Two completely different approaches to computing positive-genus GW-invariants of complete intersections have been proposed by Gath- mann [8] and Maulik-Pandharipande [18]. Both approaches use degen- erations and relative Gromov-Witten invariants. The first approach can be used to compute the genus-one and -two GW-invariants of a quintic threefold. The latter can in principle be used to compute arbitrary- genus GW-invariants of a quintic threefold as well as of some other low-degree low-dimensional complete intersections. In contrast, Theo- rem 1.1 above and [28, Proposition 3.1] are at the present restricted to genus-one GW-invariants only, but are applicable to arbitrary complete intersections.
Acknowledgments. We would like to thank D. Maulik, R. Pand- haripande, G. Tian, and R. Vakil for a number of helpful discussions.
2. Hyperplane property for genus-one GW-invariants 2.1. Review of definitions.Suppose X is a compact manifold, A∈ H2(X;Z), and g, k∈Z¯+. Let Xg,k(X, A) denote the space of equiva- lence classes of stable smooth mapsu: Σ−→X from genus-g Riemann surfaces with k marked points, which may have simple nodes, to X of degree A, i.e.
u∗[Σ] =A∈H2(X;Z).
The spaces Xg,k(X, A) are topologized using Lp1-convergence on com- pact subsets of smooth points of the domain and certain convergence requirements near the nodes; see [15, Section 3]. Here and throughout the rest of the paper, p denotes a real number greater than two. The
spaces Xg,k(X, A) can be stratified by the smooth infinite-dimensional orbifolds XT(X) of stable maps from domains of the same geometric type and with the same degree distribution between the components of the domain; see Subsections 3.1 and 3.2. The closure of the main stratum, X0g,k(X, A), isXg,k(X, A).
If J is an almost complex structure on X, let Γ0,1g,k(X, A;J)−→Xg,k(X, A)
be the bundle of (T X, J)-valued (0,1)-forms. In other words, the fiber of Γ0,1g,k(X, A;J) over a point [b] = [Σ, j;u] inXg,k(X, A) is the space
Γ0,1g,k(X, A;J)
[b]= Γ0,1(b;J)
Aut(b), where Γ0,1(b;J) = Γ Σ; Λ0,1J,jT∗Σ⊗u∗T X
.
Herej is the complex structure on Σ, the domain of the smooth mapu.
The bundle Λ0,1J,jT∗Σ⊗u∗T X over Σ consists of (J, j)-antilinear homo- morphisms:
Λ0,1J,jT∗Σ⊗u∗T X =
η∈Hom(TΣ, u∗T X) :J◦η=−η◦j .
The total space of the bundle Γ0,1g,k(X, A;J) −→Xg,k(X, A) is topolo- gized usingLp-convergence on compact subsets of smooth points of the domain and certain convergence requirements near the nodes. The re- striction of Γ0,1g,k(X, A;J) to each stratum XT(X) is a smooth vector orbibundle of infinite rank.
We define a continuous section of the bundle Γ0,1g,k(X, A;J)−→Xg,k(X, A) by
∂¯J [Σ, j;u]
= ¯∂J,ju= 1
2 du+J◦du◦j .
By definition, the zero set of this section is Mg,k(X, A;J). The re- striction of ¯∂J to each stratum of Xg,k(X, A) is smooth. The section
∂¯J is Fredholm, i.e. the linearization of its restriction to every stra- tum XT(X) has finite-dimensional kernel and cokernel at every point of ¯∂J−1(0)∩XT(X). The index of the linearization of ¯∂J at an ele- ment of M0g,k(X, A;J) is the expected dimension of the moduli space Mg,k(X, A;J),
dimg,k(X, A)≡2 hc1(T X), Ai+ (1−g)(n−3) +k , where 2n= dimRX. This is the dimension of the cycle
Mg,k(X, A;J, ν)≡∂¯J+ν −1(0) for a small generic multivalued perturbation
ν ∈G0,1g,k(X, A;J)≡Γ Xg,k(X, A),Γ0,1g,k(X, A;J)
of ¯∂J, where G0,1g,k(X, A;J) is the space of all continuous multisections ν of Γ0,1g,k(X, A;J) such that the restriction of ν to each stratumXT(X) is smooth. (Our term multisection, or multivalued section, corresponds to the notion of locally liftable multi-section in [6, Section 3].) Since the moduli spaceMg,k(X, A;J) is compact, so isMg,k(X, A;J, ν) ifν is sufficiently small.
An element [Σ;u] of X1,k(X, A) is an equivalence class of pairs con- sisting of a prestable genus-one Riemann surface Σ and a smooth map u: Σ−→X. The prestable surface Σ is a union of the principal com- ponent(s) ΣP, which is either a smooth torus or a circle of spheres, and trees of rational bubble components, which together will be denoted by ΣB. Let
X{0}1,k(X, A) =
[Σ;u]∈X1,k(X, A) :u∗[ΣP]6= 0∈H2(X;Z) . Suppose
(2.1) [Σ;u]∈X1,k(X, A)−X{0}1,k(X, A),
i.e. the degree of u|ΣP is zero. Let χ0(Σ;u) be the set of components Σi of Σ such that for every bubble component Σh that lies between Σi
and ΣP, including Σi itself, the degree ofu|Σh is zero. The setχ0(Σ;u) includes the principal component(s) of Σ. We give an example of the set χ0(Σ;u) in Figure 1. In this figure, we show the domain Σ of the stable map (Σ;u) and shade the components of the domain on which the degree of the map uis not zero. Let
Σ0u= [
i∈χ0(Σ;u)
Σi.
Every bubble component Σi⊂ΣB is a sphere and has a distinguished singular point, which will be called theattaching node ofΣi. This is the node of Σithat lies either on ΣP or on a bubble Σhthat lies between Σi and ΣP. We denote by χ(Σ;u) the set of bubble components Σi such that the attaching node of Σi lies on Σ0u and the degree of u|Σi is not zero.
Definition 2.1. Suppose (X, ω) is a compact symplectic manifold and J≡(Jt)t∈[0,1] is a C1-continuous family of ω-tamed almost struc- tures on X. A continuous family of multisections ν≡(νt)t∈[0,1], with νt∈G0,11,k(X, A;Jt) for all t∈[0,1], is effectively supported if for every element
b≡[Σ;u]∈X1,k(X, A)−X{0}1,k(X, A)
there exists a neighborhood Wb of Σ0u in a semi-universal family of deformations for b such that
νt(Σ′;u′)Σ′∩Wb= 0 ∀[Σ′;u′]∈X1,k(X, A), t∈[0,1].
h0 h1
h2 h3 h4
h5
“tacnode”
χ0(Σ;u) ={h0, h3} χ(Σ;u) ={h1, h4, h5}
Figure 1. An illustration of Definition 2.2.
If b = [Σ;u] is an element of X1,k(X, A), a semi-universal universal family of deformations forbis a fibration
σb: ˜Ub−→∆b
such that ∆b/Aut(b) is a neighborhood of binX1,k(X, A) and the fiber of σb over a point [Σ′;u′] is Σ′. If J≡(Jt)t∈[0,1] is a continuous family of ω-tamed almost structures on X, we denote the space of effectively supported familiesν as in Definition 2.1 byGes1,k(X, A;J). Similarly, ifJ is an almost complex structure onX, letGes1,k(X, A;J) be the subspace of elements ν of G0,11,k(X, A;J) such that the family νt=ν is effectively supported.
Supposeν∈Ges1,k(X, A;J) and [Σ;u] is an element ofM1,k(X, A;J, ν) as in Definition 2.1. Since Σi⊂ΣB is a sphere, we can represent this element by a pair (Σ;u) such that the attaching node of every bubble component Σi⊂ΣB is the south pole, or the point ∞= (0,0,−1), of S2⊂R3. Let e∞= (1,0,0) be a nonzero tangent vector to S2 at the south pole. If i∈χ(Σ;u), we put
Di(Σ;u) =d
u|Σi ∞e∞∈Tu|Σ
i(∞)X.
We note thatu|Σ0u is a degree-zero holomorphic map and thus constant.
Thus,umaps the attaching nodes of all elements ofχ(Σ;u) to the same point in X.
Definition 2.2. Suppose (X, ω, J) is a compact almost K¨ahler man- ifold, A∈H2(X;Z)∗, andk∈Z¯+. If ν∈Ges1,k(X, A;J) is an effectively supported perturbation of the ¯∂J-operator, the main component of the space M1,k(X, A;J, ν) is the subset M01,k(X, A;J, ν) consisting of the elements [Σ;u] of M1,k(X, A;J, ν) such that
(a) the degree ofu|ΣP is not zero, or (b) the degree of u|ΣP is zero and
dimCSpan(C,J){Di(Σ;u) :i∈χ(Σ;u)}<|χ(Σ;u)|.
By [28, Theorem 1.4],M01,k(X, A;J, ν) is a compact space if ν is ef- fectively supported and sufficiently small. For a generic effectively sup- portedν,M01,k(X, A;J, ν) determines a homology class of the expected dimension in a small neighborhood ofM01,k(X, A;J) inX1,k(X, A) which is independent of ν andJ; see [28, Theorem 1.5, Corollary 1.6].
If X, A, g, and k are as above and (L,∇)−→X is a vector bundle with connection, we denote by
Γg,k(L, A)−→Xg,k(X, A)
the cone such that the fiber of Γg,k(L, A) over [b] = [Σ;u] in Xg,k(X, A) is the Banach space
Γg,k(L, A)
[b]= Γ(b;L)
Aut(b), where Γ(b;L) =Lp1(Σ;u∗L).
The topology on the total space of Γg,k(L, A) is defined analogously to the topology on Γg,k(T X, A) of [15, Section 3]. Let
Vg,kA =
[b, ξ]∈Γg,k(L, A) : [b]∈Xg,k(X, A); ξ∈ker ¯∂∇,b⊂Γg,k(b;L)
⊂Γg,k(L, A).
The cone Vg,kA −→Xg,k(X, A) inherits its topology from Γg,k(L, A).
2.2. Statement and proof of hyperplane property.We will call a cohomology class ψ on X1,k(X, A) tautologicalif there exists a vector bundle
W −→X1,k(X, A)
such that W|XT(X) is smooth for every stratum XT(X) of X1,k(X, A) and ψ=e(W).
If (X, J) is an almost complex manifold and (L,∇)−→X is a com- plex vector bundle with connection, we will call a section s of L ∇- holomorphic if
∂¯∇s≡ 1
2 ∇s+i∇s◦J
= 0.
Theorem 2.3. Suppose (X, ω, J) is a compact almost K¨ahler man- ifold, A∈ H2(X;Z)∗, k∈Z+, (L,∇)−→X is a complex vector bundle with connection, and s is a ∇-holomorphic section of L such that J is genus-one A-regular in the sense of [26, Definition 1.4], sis transverse to the zero set inL, and(L,∇)splits into(ω, A)-positive line bundles. If V1,kA −→X1,k(X, A) is the cone corresponding to (L,∇) and Y=s−1(0), (2.2) GW0;Y1,k(A;ψ) =
ψ,PDM0
1,k(X,A;J)e(V1,kA ) for every tautological class ψ on X1,k(X, A).
Proof. SinceJis genus-oneA-regular,M01,k(X, A;J) has the expected structure of a topological orbivariety. By a generalization of the proof of
the regularity statement of [26, Theorem 1.6] analogous to [28, Subsec- tion 2.5], for allν∈Ges1,k(X, A;J) sufficiently smallM01,k(X, A;J, ν) also has the expected structure of a topological orbivariety. In particular, it is stratified by smooth orbifolds of even dimensions as described in Subsection 3.4 below. We will call ν∈Ges1,k(X, A;J) (∇, s)-compatibleif
∇s|u◦ν(Σ;u) = 0 ∀[Σ;u]∈X1,k(X, A).
We note that ifν is (∇, s)-compatible, then the map (Σ;u)−→sA1,k(Σ;u)≡s◦u∈Γ(Σ;u∗L)
defines a continuous section of the coneV1,kA over M01,k(X, A;J).
Since the∇-holomorphic section sis transverse to the zero set inL, the (i, J)-linear map
∇s:T X −→L
is surjective along Y =s−1(0). Let Us be a small neighborhood of Y in X such that ∇s is surjective over Us. The kernel of ∇s over Us is then a complex subbundle of (T X, J)|Us, which restricts toT Y alongY. We denote this subbundle by ˜T Y. If ν∈Ges1,k(X, A;J) is such that for all [Σ;u]∈X1,k(X, A)
ν(Σ;u)
(∈Γ(Σ; Λ0,1J,jT∗Σ⊗T Y˜ ), ifu(Σ)⊂Us;
= 0, otherwise,
thenν is (∇, s)-compatible. Thus, every element νY∈Ges1,k(Y, A;J) can extended to a (∇, s)-compatible element ν of Ges1,k(X, A;J). Further- more, if νY is a small, then ν can also be chosen to be small.
For a small generic νY ∈Ges1,k(Y, A;J), M01,k(Y, A;J, νY) is stratified by smooth orbifolds of even dimensions so that the largest-dimensional stratum is M01,k(Y, A;J, νY) and
dimM01,k(Y, A;J, νY) = dim1,k(Y, A).
Let ν be an extension of νY to a small (∇, s)-compatible element of Ges1,k(X, A;J). Suppose
W −→X1,k(X, A)
is a complex vector bundle of rank dim1,k(Y, A)/2 as in the first para- graph of this subsection. Choose a section f ofW overM01,k(X, A;J, ν) such that f|Uα is transverse to the zero set in W|Uα for every stratum Uα ofM01,k(X, A;J, ν) and of M01,k(Y, A;J, νY). Then,
f−1(0)∩M01,k(Y, A;J, νY)⊂M01,k(Y, A;J, νY) and GW0;Y1,k(A;ψ) = ±f−1(0)∩M01,k(Y, A;J, νY). (2.3)
On the other hand, since ν is (∇, s)-compatible, sA1,k is a section of V1,kA −→M01,k(X, A;J, ν).
Furthermore,
sA1,k −1(0) =M01,k(X, A;J, ν)∩X1,k(Y, A)≡M01,k(Y, A;J, νY) =⇒ f−1(0)∩
sA1,k −1(0) =f−1(0)∩M01,k(Y, A;J, νY)
⊂M01,k(X, A;J, ν).
(2.4)
Note that if [b] = [Σ;u]∈M01,k(Y, A;J, νY), ker∇sA1,k|b =
ξ∈kerDJ,ν;b:∇s|u◦ξ= 0
= kerDJ,ν;b∩Γ(Σ;u∗T Y) = kerDJ|Y,νY;b, (2.5)
where DJ,ν;b and DJ|Y,νY;b are the linearizations of the sections ¯∂J+ν and ¯∂J|Y+νY at b. The second equality above is immediate from the transversality of s. By (2.5),
dimRIm∇sA1,k|(Σ;u) = dim kerDJ,ν;b−dim kerDJ|Y,νY;b
= dim1,k(X, A)−dim1,k(Y, A)
= 2hc1(L), Ai= dimRV1,kA |[b]. (2.6)
The second equality above follows from our assumption that the opera- torsDJ,ν;bandDJ|Y,νY;bare surjective; the last equality is a consequence of the (ω,L)-positivity assumption. By (2.6), sA1,k is transverse to the zero set in V1,kA along M01,k(X, A;J, ν). Sincef is transverse to the zero set in W alongM01,k(X, A;J, νY), it then follows from (2.4) that
ψ,PDM0
1,k(X,A;J)e(V1,kA )
≡ ±f−1(0)∩
sA1,k −1(0)
= ±f−1(0)∩M01,k(Y, A;J, νY). (2.7)
Theorem 2.3 follows from (2.3) and (2.7). q.e.d.
3. Ingredients in proof of Theorem 1.3
3.1. Notation: genus-zero maps.In this subsection we describe our detailed notation for bubble maps from genus-zero Riemann surfaces and for related objects. In general, moduli spaces of stable maps can stratified by the dual graph. However, in the present situation, it is more convenient to make use of linearly ordered sets.
Definition 3.1. (1) A finite nonempty partially ordered set I is a linearly ordered set if for all i1, i2, h∈I such thati1, i2< h, either i1≤i2 ori2≤i1.
(2) A linearly ordered set I is a rooted tree ifI has a unique minimal element, i.e. there exists ˆ0∈I such that ˆ0≤ifor all i∈I.
If I is a linearly ordered set, let ˆI be the subset of the non-minimal elements of I. For every h∈I, denote byˆ ιh∈I the largest element ofI which is smaller than h, i.e. ιh= max
i∈I :i < h .
We identifyCwithS2−{∞}via the stereographic projection mapping the origin in Cto the north pole, or the point (0,0,1), inS2. LetM be a finite set. Agenus-zero X-valued bubble map with M-marked pointsis a tuple
b= M, I;x,(j, y), u , where I is a rooted tree, and
x: ˆI−→C=S2−{∞}, j:M−→I, y:M−→C, and u:I−→C∞(S2;X) (3.1)
are maps such that uh(∞) =uιh(xh) for all h∈Iˆ. We associate such a tuple with Riemann surface
Σb= G
i∈I
Σb,i.
∼, where
Σb,i={i}×S2, (h,∞)∼(ιh, xh) ∀h∈I,ˆ (3.2)
with marked points
yl(b)≡(jl, yl)∈Σb,jl and y0(b)≡(ˆ0,∞)∈Σb,ˆ0,
and continuous mapub: Σb−→X, given by ub|Σb,i=ui for alli∈I. The general structure of bubble maps is described by tuplesT = (M, I;j, A), where
Ai =ui∗[S2]∈H2(X;Z) ∀i∈I.
We call such tuples bubble types. Let ˜XT(X) be the space of all bubble maps of typeT. For l∈{0}⊔M, let
evl: ˜XT(X)−→X
be the evaluation map corresponding to the marked point yl. With notation as above, suppose
b≡ M, I;x,(j, y), u
∈X˜T(X).
In particular,I is a linearly ordered set with minimal element ˆ0 and the special marked point is the point
y0(b) = (ˆ0,∞)∈Σb,ˆ0.
Let χ0(b) be the set of components Σb,i of Σb such that for every com- ponent Σb,h that lies between Σi and Σb,ˆ0, including Σb,i and Σb,ˆ0, the degree ofu|Σb,h is zero. The setχ0(b) is empty if and only if the degree
