Quantum cohomology and toric minimal model programs
Abstract
We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental, Iritani, and Fukaya-Oh-Ohta-Ono for toric manifolds and Coates-Lee-Corti-Tseng for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity and an equality of dimensions deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each giving rise to a collection of Hamiltonian non-displaceable tori.
Cite
@article{arxiv.1207.3253,
title = {Quantum cohomology and toric minimal model programs},
author = {Eduardo Gonzalez and Chris Woodward},
journal= {arXiv preprint arXiv:1207.3253},
year = {2024}
}
Comments
50 pages, 9 figures. In this version a few lines on p.12 (definition of Jacobian ring) which were accidentally deleted are restored; thanks to Guangbo Xu for pointing out the missing definition