GW invariants relative to normal crossing divisors

Eleny-Nicoleta Ionel
2015 Advances in Mathematics  
In this paper we introduce a notion of symplectic normal crossing divisor V and define the GW invariant of a symplectic manifold X relative such a divisor. Our definition includes normal crossing divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [IP2], which covered the case V was smooth. The main step is the construction of a compact
more » ... moduli space of relatively stable maps into the pair (X, V ) in the case V is a symplectic normal crossing divisor in X. Research supported in part by the NSF grants DMS-0605003 and DMS-0905738. 1 Symplectic normal crossing divisors In this section we define a notion of symplectic normal crossing divisors, generalizing that from algebraic geometry, and encoding the geometrical information required for the analysis of [IP2] and [IP3] to extend after appropriate modifications. Clearly the local model of such divisor V should be the union of k coordinate planes in C n , where the number of planes may vary from point to point. But we also need a local model for the symplectic form ω and the tamed almost complex structure J near such divisor, and we require that each branch of V is both ω-symplectic and J-holomorphic. This will allow us to define the order of contact of J-holomorphic curves to V . We also need a good description of the normal directions to the divisor, as these are the directions in which the manifold X will be rescaled when components of the J-holomorphic curves fall into V . So we keep track of both the normal bundle to each branch of V and its inclusion into X describing the neighborhood of that branch. Definition 1.1 (Local model). In C n , consider the union V of k ≥ 0 (distinct) coordinate hyperplanes H i = {x|x i = 0}, together with their normal direction N i defined by the projection π i : C n −→ H i , π i (x) = x i , and the inclusion ι : (N i , 0) → (C n , H i ). We say that these form a model for a normal crossing divisor in C n with respect to a pair (ω, J) if all the divisors H i are both ω-symplectic and Jholomorphic. Remark 1.2. There is a natural action of C * on the model induced by scaling by a factor of t −1 in the normal direction to each H i , for i = 1, . . . , k. This defines a rescaling map R t : C n → C n for t ∈ C * . By construction, the R t leaves the divisors H i invariant, but not pointwise, and may not preserve J. However, as t → 0, R * t J converges uniformly on compacts to a C * invariant limit J 0 which depends on the 1-jet of J along the divisor. Definition 1.3. Assume (X, ω, J) is a symplectic manifold with a tamed almost complex structure. V is called a normal crossing divisor in (X, ω, J) with normal bundle N if there exists a smooth manifold V with a complex line bundle π : N → V and an immersion ι : U V → X of some disk bundle U V of N V into X satisfying the following properties: (i) V is the image of the zero section V of N (ii) the restriction of ι * J to the fiber of N along the zero section induces the complex multiplication in the bundle N . (iii) at each point p ∈ X we can find local coordinates on X in which the configuration (X, π, ι, V ) becomes identified with one of the local models in Definition 1.1. Note that ι induces by pullback from X both a symplectic structure ω and an almost complex structure J on the total space of the disk bundle in N over on V , which serves as a global model of X near V . Its zero section V is both symplectic and J-holomorphic and serves as a smooth model of the divisor V . N is also a complex line bundle whose complex structure comes from the restriction of J along the zero section. Thus N also comes with a C * action which will be used to rescale X normal to V . Remark 1.4. We are not requiring J to be locally invariant under this C * action. We also are not imposing the condition that the branches are perpendicular with respect to ω or that the projections π i are J-holomorphic. We also allow transverse self intersections of various components of V . When each component of V is a submanifold of X the divisor is said to have simple normal crossing singularities. Any of these assumptions would simplify some of the arguments, but are not needed. In this paper we only work with J's compatible with V in the sense of Definition 3.2 of [IP2]. This is a condition on the normal 1-jet of J along V : This extra condition is needed to ensure that the stable map compactification has codimension 2 boundary strata, so it gives an invariant, independent of parameters. A priori, even 4 when V is smooth the relatively stable map compactification may have real codimension 1 boundary strata without this extra assumption. A symplectic normal crossing divisor could be defined locally in terms of an atlas of charts on (X, ω, J) compatible with V . The local models suffice to construct both a smooth resolution V of V by separating its local branches as well as the complex normal bundle N , in effect proving a tubular neighborhood theorem in this context. For simplicity of exposition, we decided instead to include the global existence of V and N as part of the definition of a normal crossing divisor. One could also define a notion of normal crossing in the smooth or even orbifold category. In this paper we insist that the normal bundle N carry a complex structure, which induces a local C * -action normal to V . Otherwise, one only has an R + action, which is the typical situation in SFT, leading to further complications. Example 1.5. A large class of examples is provided by algebraic geometry. Assume X is a smooth projective variety and V a smooth normal crossing divisor in X in this category (i.e. the normalization of V is a smooth projective variety). Then V is a symplectic normal crossing divisor for (X, J 0 , ω 0 ) where J 0 is the integrable complex structure and ω 0 the Kahler form. For example (a) X could be a Hirzebruch surface and V the union of the zero section, the infinity section and several fibers or (b) V could be the union of a section and a nodal fiber in an elliptic surface X. An important example of this type is when X is a toric manifold and V is its toric divisor, which is a case considered in mirror symmetry, see for example [Au2]. Example 1.6. A particular example to keep in mind is X = CP 2 with a degree 3 normal crossing divisor V . For example V could be a smooth elliptic curve, or V could be a nodal sphere, or finally V could be a union of 3 distinct lines. In the second case the resolution V is CP 1 with normal bundle O(7) while in the last case it is CP 1 CP 1 CP 1 , each component with normal bundle O(1). Of course, in a complex 1-parameter family, a smooth degree three curve can degenerate into either one of the other two cases. Another motivating example of this type comes from a smooth quintic 3-fold degenerating to a union of 5 hyperplanes in CP 4 . Remark 1.7. Another special case is X = M 0,n the Deligne-Mumford moduli space of stable genus 0 curves and V the union of all its boundary strata (i.e. the stratum of nodal curves). The usual description of each boundary stratum and of its normal bundle provides the required local models for a symplectic normal crossing divisor. This discussion can also be extended to the orbifold setting to cover the higher genus case M g,n and includes its smooth finite covers, the moduli space of Prym curves [Lo] or the moduli space of twisted G-covers [ACV]. Of course, there are many more symplectic examples besides those coming from algebraic geometry. Example 1.8. Assume V is a symplectic codimension two submanifold of (X, ω). The symplectic neighborhood theorem allows us to find a J and a model for the normal direction to V , so V is normal crossing divisor in (X, ω, J). Of course in this case V is a smooth divisor, so it has empty singular locus. One may have hoped that the union of several transversely intersecting codimension two symplectic submanifolds would similarly be a normal crossing divisor. Unfortunately, if the singular locus is not empty, that may not be the case: Example 1.9. Let V 1 be an exceptional divisor in a symplectic 4-manifold and V 2 a sufficiently small generic perturbation of it, thus still a symplectic submanifold, intersecting transversely V 1 . This configuration cannot be given the structure of a normal crossing divisor, simply because one cannot find a J which preserves both. If such a J existed, then all the intersections between V 1 and V 2 would be positive, contradicting the fact that exceptional divisors have negative self intersection. This example illustrates the fact that a normal crossing divisor is not a purely symplectic notion, but rather one also needs the existence of an almost complex structure J compatible with the crossings. The positivity of intersections of all branches is a necessary condition for such a J to exist in general. 5 Remark 1.10. One could ask what are the necessary and sufficient conditions for V inside a symplectic manifold (X, ω) to be a normal crossing divisor with respect to some J on X. Clearly V should be locally the transverse intersection of symplectic submanifolds, and the intersections should be positive. If we assume that the branches of V are orthogonal wrt ω, the existence of an ω-compatible J which is compatible with V is straightforward (see Appendix). In general, one might be able to use a homotopy argument to prove that positivity of intersections is the only obstruction to the existence of a J compatible with V and tamed by ω. We do not pursue this issue further in this paper. Example 1.11. Symplectic Lefschetz pencils or fibrations provide another source of symplectic normal crossing divisors. Assume X is a symplectic manifold which has a symplectic Lefschetz fibration with a symplectic section, for example one coming from Donaldson Theorem [Do2] where the section comes from blowing up the base locus. Gomph [GoS] showed that in this case there is an almost complex structure J compatible with this fibration. We could then take V the union of the section with a bunch of fibers, including possibly some singular fibers. Example 1.12. (Donaldson divisors) Assume V is a normal crossing divisor in (X, ω, J), J is ωcompatible and [ω] has rational coefficients. We can use Donaldson theorem [Do1] to obtain a smooth divisor D representing the Poincare dual of kω for k 0 sufficiently large, such that D is ε-J-holomorphic and η-transverse to V (see also [Au1] ). Choosing carefully the parameters η and ε, one can find a sufficiently small, ω-tamed deformation of J such that V ∪ D is also a normal crossing divisor (cf. [IP4]). Remark 1.13. The definition of a normal crossing divisor works well under taking products of symplectic manifolds with divisors in them. If V i is a normal crossing divisor in X i for i = 1, 2 then π −1 Remark 1.14. The definition of a normal crossing divisor also behaves well under symplectic sums. Assume U i ∪ V is a symplectic divisor in X i for i = 1, 2 such that the normal bundles of V in X i are dual. If U i intersect V in the same divisor W then Gomph's argument [Go] shows that the divisors U i glue to give a normal crossing divisor U 1 # W U 2 in the symplectic sum X 1 # V X 2 . Remark 1.15. (Stratifications associated to a normal crossing divisor) Any symplectic normal crossing divisor V in (X, ω, J) induces a stratification of X, whose closed stratum V k consists of those points X where at least k local branches of V meet. Each closed stratum has a smooth resolution V k → V k which comes with an induced (ω, J) and an intrinsic symplectic normal crossing divisor V k+1 over the lower depth stratum V k+1 , as described in §A.1. More precisely, for each point x ∈ X, its depth k(x) is the largest k such that x ∈ V k , or equivalently the cardinality of ι −1 (x), where ι : V X is the immersion parameterizing V . So a point in X \ V has depth 0 while points in the singular locus of V have depth at least 2. This defines an upper semicontinuous function depth : X → N whose level sets are the open strata where precisely k local branches of V meet. The fiber of V k → V k over a depth k point x is intrinsically ι −1 (x) and keeps track of the k local branches of V meeting at x. In fact, for any finite set I of order k, we get a resolution V I → V k ; its fiber at a depth k point x consists of bijections I → ι −1 (x), with a symmetric group S I action reordering I, see §A.1 for more details. Remark 1.16. A special case of symplectic normal crossing divisor V (with simple crossings) is the union of codimension 2 symplectic submanifolds which intersect orthogonally wrt ω, and whose local model matches that of toric divisors in a toric manifold. This is a case that fits in the exploded manifold set-up of Brett Parker (see Example 5.3 in the recent preprint [P]), so in principle one should be able to compare the relative invariants we construct in this paper with the exploded ones of [P]. It is unclear to us what is exactly the information that the exploded structure records in this case, and what is the precise relation between the two moduli spaces. But certainly the relatively stable map compactification we define in this paper seems to be quite different from the exploded one, so it is unclear whether they give the same invariants, even in the case when V is smooth. 6 Remark 1.17. In a related paper, Gross and Siebert define log GW invariants in the algebraic geometry setting [GrS]. If V is a normal crossing divisor in a smooth projective variety X, then it induces a log structure on X. However, even when V is a smooth divisor, Gross and Siebert explain that the stable log compactification they construct is quite different from the relatively stable map compactification constructed earlier in that context by J. Li [Li] (and which agrees with that of [IP2] in this case). So a priori, even when V is smooth, the usual relative GW invariants may be different from the log GW invariants of [GrS]. The authors mention that in that case at least there is a map from the moduli space of stable relative maps to that of stable log maps, which they claim could be used to prove that the invariants are the same. Presumably there is also a map from the relatively stable map compactification that we construct in this paper to the appropriate stable log compactification in the more general case when V is a normal crossing divisor in a smooth projective variety. In another paper [AC] Abramovich and Chen explain how, in the context of algebraic geometry, the construction of a log moduli space when V is a normal crossing divisor (with simple crossings) follows from the case when V is smooth by essentially functorial reasons. Again, it is unclear to us how exactly the two notions of log stable maps of [GrS] and [AC] are related in this case. Remark 1.18. One note about simple normal crossing vs general normal crossing: they do complicate the topology/combinatorics of the situation, but if set up carefully the analysis is unaffected. If the local model of X is holomorphic near V (as is the case in last two examples above), even if V did not have simple crossings, one could always blow up the singular locus W of V to get a total divisor π −1 (V ) = Bl(V ) ∪ E with simple normal crossing in Bl(X), where E is the exceptional divisor. Blowing up in the symplectic category is a more delicate issue, but when using the appropriate local model, one can always express (a symplectic deformation) of the original manifold (X, V ) as a symplectic sum of its blow up (Bl(X), Bl(V )) along the exceptional divisor E with a standard piece (P, V 0 ) involving the normal bundle of the blowup locus. Since we are blowing up the singular locus of V , the proper transform Bl(V ) intersects nontrivially the exceptional divisor E; the symplectic sum Bl(X)# E P = X then also glues Bl(V ) to the standard piece on the other side to produce V , as in Remark 1.14. So a posteriori, after proving a symplectic sum formula for the relative GW of normal crossing divisors passing through the neck of a symplectic sum, one could also express the relative GW invariants of the original pair (X, V ) as universal expressions in the relative GW invariants of its the blow up and those of the piece obtained from the normal bundle of the blow-up locus. For the rest of the paper, unless specifically mentioned otherwise, a (normal crossing) divisor V in X will always mean a symplectic normal crossing divisor for some tamed pair (ω, J) on X which is compatible with V to first order. We denote by J (X, V ) the space of such tamed pairs, described in §A.2. The restriction of (ω, J) to V pulls back to a tamed pair on the smooth resolution of each depth k stratum of V , and each stratum is itself a symplectic normal crossing divisor in the smooth resolution of the previous one, cf. §A.1. Basic facts about stratifications are reviewed in §A.3. Outline of the construction of M(X, V ) The construction of the moduli spaces M(X, V ) relative normal crossing divisors V is modeled on the construction of [IP2] and [IP3] for the case when V is smooth, which in turn is modeled on the classical construction of the absolute moduli space M(X). This section reviews some of the main steps involved in the construction, focusing on the global aspects of the theory, while the later sections focus on the local considerations required in making these statements precise and proving them. Roughly speaking, the construction of the moduli space and its GW invariants involves three main ingredients: Gromov compactness theorem, transversality and gluing. 2.1. Brief review of the absolute moduli space. Fix a closed, finite dimensional smooth symplectic manifold (X, ω, J) with a tamed almost complex structure and let M J (X) denote the moduli space of stable J-holomorphic maps into X. To construct GW invariants one needs to consider deformations of the structure, so let J (X) be the space of smooth tamed pairs (ω, J) on the target X, and M(X) → J (X) the 7 into a level m building, with their resolutions f : C → X m and deformations f μ,λ : C μ → X λ as well as the refined evaluation map. Remark 2.2. (Stability and automorphisms) To simplify the discussion in this paper, in all local analysis arguments we assume that all nontrivial components of the domains C are already stable, and were further decorated to kill their automorphism groups. For Gromov compactness-type arguments it suffices to add extra marked points. For transversality arguments, if the domains are already stable, one can pass to a finite cover of the Deligne-Mumford moduli space M, using instead the space M G → M of twisted G-covers for a suitably chosen finite group G (cf. [IP4]). At first the assumption that all (nontrivial) domain components have trivial automorphism group seems to be a very restrictive assumption, but it can always be achieved whenever one of the branches of V is a sufficiently positive Donaldson divisor. This not only provides a global way to stabilize all domains, but simultaneously simplifies the analysis required in proving transversality and gluing (at the expense of complicating the topology and combinatorics). Moreover, the general case follows by functorial considerations from this seemingly very special case (cf. [IP4]). Remark 2.3. In this paper, we only work with Gromov-type perturbations (J, ν) ∈ J V of the Jholomorphic map equation: These are global and functorial perturbations induced by deformations J ν of the product almost complex structure j × J on U × X ⊂ P N × X, where U → P N is a fixed embedding of the universal curve U → M. The projection (J, ν) → J defines a fibration with fiber ν ∈ V = Hom 0,1 (T U , T X) and a zero section: When the domain C has trivial automorphisms the graph F is an embedding, thus somewhere injective, which is often sufficient for the standard transversality argument in [MS] to extend. On the other hand, Gromov-type perturbations vanish on all unstable components of C (thus on trivial components), so these are always J-holomorphic. Under the assumption in Remark 2.2, Gromov-type perturbations provide enough transversality at nontrivial components to be able to deal with the potential lack of transversality at the trivial ones. In the context of relative moduli spaces, we work only with V -compatible parameters (J, ν) ∈ J V(X, V ) for the equation (2.5), see §A.2. If we let (X , V ) = (U × X, U × V ), this condition is equivalent to requiring that the almost complex structure J ν on X be compatible with the divisor V to first order. The situation extends to a level m building, where the corresponding almost complex structure J ν on X m is required to be compatible with the total divisor, and to its deformations X → B where the almost complex structure is required to be compatible with the fibration. So in most of the arguments below we switch back and forth between f and its graph F , converting statements about a (J, ν)-holomorphic map f : C → X into statements about its J ν -holomorphic graph F : C → U × X. The Relative Moduli Space M(X, V ) Assume (X, ω, J) is a smooth symplectic manifold with a normal crossing divisor V , and restrict to the subspace J (X, V ) of V -compatible almost complex structures on X. For transversality purposes we also need to turn on Gromov-type perturbations (J, ν) ∈ J V(X, V ). The definition of M(X, V ) takes several stages. In this section we describe the main piece M(X, V ) → J V(X, V ) (3.1) consisting of stable (J, ν)-holomorphic maps f : C → X into X without any components or nodes in V , such that all the points in f −1 (V ) are marked, and each marked point x of C comes decorated by a 10
doi:10.1016/j.aim.2015.04.027 fatcat:75et6d7pwvfz3mieeidcsyq3ym