Non-affine Landau-Ginzburg models and intersection cohomology

TL;DR

This paper constructs Landau-Ginzburg models for certain complete intersections in toric manifolds, establishing a mirror symmetry that links quantum cohomology to intersection cohomology D-modules and exploring their Hodge properties.

Contribution

It introduces a novel construction of Landau-Ginzburg models for complete intersections in toric manifolds and demonstrates a mirror symmetry relating quantum cohomology to intersection cohomology D-modules.

Findings

Quantum D-module of the ambient cohomology is isomorphic to an intersection cohomology D-module.

Establishment of Hodge properties for these differential systems.

Partial compactifications of Laurent polynomial families serve as models for mirror symmetry.

Abstract

We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.