Quantum cohomology and toric minimal model programs

For any compact symplectic toric orbifold Y, we show that the quantum cohomology QH(Y) is isomorphic to a formal polynomial ring modulo the quantum Stanley-Reisner ideal introduced by Batyrev at a canonical bulk deformation. This generalizes results of Givental, Iritani and Fukaya-Oh-Ohta-Ono for toric manifolds and Coates-Lee-Corti-Tseng for weighted projective spaces. In the language of Landau-Ginzburg potentials, we identify QH(Y) with the ring of functions on the subset Crit_+(W) of the critical locus Crit(W) of an explicit potential W consisting of critical points mapping to the interior of the moment polytope, as in the manifold case. Our proof uses algebro-geometric virtual fundamental classes, a quantum version of Kirwan surjectivity, and an equality of dimensions deduced using a toric minimal model program (tmmp). The existence of a Batyrev presentation implies that the quantum cohomology of Y is generically semisimple. This is related by a conjecture of Dubrovin, to the existence of a full exceptional collection in the derived category of Y proved by Kawamata, also using tmmps. Finally we discuss a connection with Hamiltonian non-displaceability. Any tmmp for Y with generic symplectic class defines a splitting of the quantum cohomology QH(Y) with summands indexed by transitions in the tmmp, and each summand corresponds a collection of Hamiltonian non-displaceable Lagrangian tori in Y. In particular the existence of infinitely many tmmps can produce open families of Hamiltonian non-displaceable Lagrangians, such as in the examples in Wilson-Woodward.