Holomorphic 2-forms and Vanishing Theorems for Gromov-Witten Invariants

On a compact Kähler manifold X with a holomorphic 2-form α, there is an almost complex structure associated with α. We show how this implies vanishing theorems for the Gromov-Witten invariants ofX . This extends the approach, used in [LP] for Kähler surfaces, to higher dimensions. Let X be a Kähler surface with a non-zero holomorphic 2-form α. Then α is a section of the canonical bundle and its zero locus Zα, with multiplicity, is a canonical divisor. We showed in [L] that the real 2-form Re(α) determines a (non-integrable) almost complex structure Jα that has the following remarkable “Image Localization Property” : if a Jα-holomorphic map f : C → X represents a non-zero (1,1) class, then f is in fact holomorphic and its image lies in Zα. As shown in [LP], this property together with Gromov Convergence Theorem leads to : Theorem 1 ([LP]) Let X be a Kähler surface with a non-zero holomorphic 2-form α. Then, any class A with non-trivial Gromov-Witten invariant GWg,k(X,A) is represented by a stable holomorphic map f : C → X whose image lies in the canonical divisor Zα. This paper extends Theorem 1 to higher dimensions. The principle is the same: perturbing the Kähler structure to a non-integrable almost complex structure Jα forces the holomorphic maps to satisfy certain geometric conditions determined by α. This gives constraints on the Gromov-Witten invariants. Specifically, let X be a compact Kähler manifold with a non-zero holomorphic 2-form α. Then the real part of α defines an endomorphism Kα of TX and an almost complex structure Jα, just as in the surface case (see (2.1) and (2.2)). These geometric structures lead, naturally and easily, to our main theorem : Theorem 2 Let X be a compact Kähler manifold with a non-zero holomorphic 2-form α. Then any class A with non-trivial Gromov-Witten invariant GWg,k(X,A) is represented by a stable holomorphic map f : C → X satisfying the equation Kαdf = 0. This theorem follows from Theorem 3.1, which is more suitable for applications. It generalizes Theorem 1 since when X is a surface the kernel of the endomorphism Kα is trivial on X \ Zα (see Example 3.5). The equation Kαdf = 0 is a geometric fact about holomorphic maps that directly implies numerous vanishing results about Gromov-Witten invariants (see Section 3).