Mirror Symmetry for Concavex Vector Bundles on Projective Spaces

Let V + = ⊕i∈IO(ki) and V − = ⊕j∈JO(−lj) be vector bundles on P s with ki and lj positive integers. Suppose X ι →֒ Ps is the zero locus of a generic section of V + and Y is a projective manifold such that X j →֒ Y with normal bundle NX/Y = ι ∗(V −). The relations between Gromov-Witten theories of X and Y are studied here by means of a suitably defined equivariant Gromov-Witten theory in Ps. We apply mirror symmetry to the latter to evaluate the gravitational descendants of Y supported in X. Section 2 is a collection of definitions and techniques that will be used throughout this paper. In section 3, using an idea from Kontsevich, we introduce a modified equivariant Gromov-Witten theory in Ps corresponding to V = V + ⊕ V −. The corresponding D-module structure ([6],[15],[29]) is computed in section 4. It is generated by a single function J̃V . In general, the equivariant quantum product does not have a nonequivariant limit. It is shown in Lemma 4.1.1 that the generator J̃V does have a limit JV which takes values in H∗Pm[[q, t]]. It is this limit that plays a crucial role in this work. Let Y be a smooth, projective manifold. The generator JY of the pure Dmodule structure of Y encodes one-pointed gravitational descendents of Y . It takes values in the completion of H∗Y along the semigroup (Mori cone) of the rational curves of Y . The pullback map j∗ : H∗Y → H∗X extends to a map between the respective completions. In Theorem 4.2.2 we describe