A Mirror Theorem for T-Equivariant Blowups

TL;DR

This paper establishes a mirror theorem for T-equivariant blowups of toric fibrations, linking Gromov-Witten invariants of the blowup to those of the base and divisors, advancing understanding in symplectic geometry.

Contribution

It provides a new mirror theorem for T-equivariant blowups of toric fibrations, explicitly relating their Gromov-Witten invariants to those of the base and divisors.

Findings

Computed genus-0 Gromov-Witten invariants of blowups in terms of base and divisor invariants.

Established formulas involving symplectic reduction data and restriction maps.

Extended mirror symmetry techniques to a broader class of toric fibrations.

Abstract

Let E be a toric fibration arising from symplectic reduction of a direct sum of line bundles over (almost-) K\"ahler 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:B\to E a particular fixed-point section of E. Further assume the \{A_a\} to be mutually disjoint. We compute genus-0 Gromov--Witten invariants of the blowup of E along \a(\coprod_a A_a) 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\to B, and the restriction maps i_{A_a}^*:H^*(B)\to H^*(A_a).