On symplectic invariants of algebraic varieties coming from crepant contractions

Summary

Introduction

Let X be a smooth projective variety over the complex numbers; in standard notation, N₁(X) denotes the finite dimensional vector space of 1-cycles on X modulo numerical equivalence. N₁(X) contains the cone NE(X) of effective 1-cycles, generated by the classes of curves in X. In some geometric situations, the effective cone of X provides important deformation theoretic invariants of the manifold. For example, Wisniewski [Ws2] observed that the halfcone NE(X) ∩ (K_Xz < 0) is invariant in smooth families of projective varieties. Recently, however, Ruan has used the theory of Gromov-Witten invariants to show that, at least in dimension two and three, Mori extremal rays provide invariants of algebraic manifolds in a stronger sense [Ru]. To describe his results, let Y denote a complete C^∞ manifold of real dimension 2n, and let Jo and J₀ be integrable complex structures on Y such that Y₀ = (Y, J₀) and Y₁ = (Y, J₁) are projective varieties. Then we say that Y₀ and Y₁ are symplectically deformation equivalent if there exists a family ω_t of symplectic forms on Y such that ω₀ is a Hodge form on Y₀ (that is, i/2n times the curvature form of an ample line bundle) and ω₁ a Hodge form on Y₁. There is a family of almost complex structures J_t joining J₀ and J₁, such that J_t is compatible with ω_t for every t. If n ≤ 3, Ruan proved that every Mori extremal ray has a nonzero Gromov-Witten invariant, and so is common to Y₀ and Y₁. Thus Mori extremal rays are symplectic deformation invariants of low dimensional projective manifolds.