Hodge numbers of Landau-Ginzburg models

TL;DR

This paper explores the computation of Hodge numbers for Landau-Ginzburg models, providing a method using mixed Hodge theory and establishing a mirror symmetry relation for Fano threefolds.

Contribution

It introduces a concrete recipe for calculating Hodge numbers of LG mirrors of Fano threefolds and proves a mirror symmetry relation for Gorenstein toric Fano threefolds.

Findings

Hodge numbers can be computed via mixed Hodge theory.

A concrete recipe for LG mirror Hodge numbers of Fano threefolds.

Mirror symmetry relation for crepant resolutions of Gorenstein toric Fano threefolds.

Abstract

We study the Hodge numbers of Landau-Ginzburg models as defined by Katzarkov, Kontsevich and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau-Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold $X$ there is a natural LG mirror $(Y,\mathsf{w})$ so that $h^{p,q}(X) = f^{3-q,p}(Y,\mathsf{w})$.