Towards Mirror Symmetry for Varieties of General Type

TL;DR

This paper proposes a new framework for mirror symmetry in varieties of general type, linking Landau-Ginzburg models, Hodge number exchange, and categorical structures.

Contribution

It introduces a novel mirror symmetry theory for general type varieties using Landau-Ginzburg models and analyzes Hodge number exchanges via perverse sheaves.

Findings

Hodge numbers of hypersurfaces are exchanged with those of the critical locus.

The critical locus carries a perverse sheaf of vanishing cycles.

Hochschild homology explains the Hodge number exchange.

Abstract

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus.