%0 Journal Article %T Convolution Integrals and a Mirror Theorem from Toric Fiber Geometry %A Jeff Brown %J Advances in Pure Mathematics %P 637-684 %@ 2160-0384 %D 2019 %I Scientific Research Publishing %R 10.4236/apm.2019.99033 %X Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let La be convex line bundles over B, Aa smooth divisors of B arising as the zero loci of generic sections of La , and \"\" a particular fixed-point section of E. Further assume the {Aa} 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 {Aa}, the matrix used for the symplectic reduction description of the fiber of the toric fibration E¡úB, 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. %K Gromov-Witten Invariant %K Quantum Cohomology %K Fixed-Point Localization %K Birational Geometry %U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=95016