Quantum cohomology and toric minimal model programs

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 from [4] to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental [23], Iritani [34] and Fukaya-Oh-Ohta-Ono [21] for toric manifolds and Coates-Lee-Corti-Tseng [11] for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity (Theorem 2.6 below) and an equality of dimensions (Theorem 4.19 below) 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 of which gives rise to a collection of Hamiltonian non-displaceable Lagrangian tori.