Convolution Integrals and a Mirror Theorem from Toric Fiber Geometry

Abstract

Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) K ä hler base B . Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let L a be convex line bundles over B , A a smooth divisors of B arising as the zero loci of generic sections of L a , and a particular fixed-point section of E . Further assume the { A a } to be mutually disjoint. The manifold is a new manifold with tautological line bundles over new projective spaces in the geometry, where previously there was a simpler vector bundle in the given local geometry (Section 1.5). Thus, we compute genus-0 Gromov-Witten invariants of in terms of genus-0 Gromov-Witten invariants of B and of { A a } , the matrix used for the symplectic reduction description of the fiber of the toric fibration E → B , and the restriction maps . The proofs utilize the fixed-point localization technique describing the geometry of and its genus-0 Gromov-Witten theory, as well as the Quantum Lefschetz theorem relating the genus-0 Gromov-Witten theory of A with that of B .