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Quantum cohomology and toric minimal model programs

Eduardo Gonzalez, Chris Woodward

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.

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.

Paper Structure

This paper contains 8 sections, 38 theorems, 98 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Quantum Kirwan surjectivity for toric orbifolds
  3. Quantum Stanley-Reisner ring and Jacobian ring
  4. Dimensional equality via a toric minimal model program
  5. Invariance of quantum cohomology under weighted toric flops
  6. Minimal models and non-displaceable Lagrangians
  7. Generic tmmp runnings
  8. Non-generic tmmp runnings

Key Result

Theorem 1.4

(Definition and properties of the quantum Kirwan map)

Figures (8)

  • Figure 2: A polytope whose dual polytope has too much volume
  • Figure 3: Values of the tropical moment map on the critical locus for a family of toric surfaces
  • Figure 4: Polytopes for a toric minimal model program
  • Figure 5: Fans for a toric minimal model program
  • Figure 6: Flipping simplices for a toric minimal model program
  • ...and 3 more figures

Theorems & Definitions (107)

  • Example 1.2
  • Remark 1.3
  • Theorem 1.4
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.10
  • Example 1.11
  • Definition 1.12
  • Definition 1.13
  • ...and 97 more