Log Hodge groups on a toric Calabi-Yau degeneration

TL;DR

This paper develops a spectral sequence framework for computing logarithmic Hodge groups on toric Calabi-Yau spaces, explicitly relates it to tropical data, and explores mirror symmetry dualities in low dimensions.

Contribution

It introduces a spectral sequence for logarithmic Hodge groups, computes its initial term explicitly, and proves its degeneration, advancing understanding of Hodge theory in toric Calabi-Yau degenerations.

Findings

Spectral sequence for logarithmic Hodge groups established.

Explicit computation of E_1 term using tropical data.

Mirror symmetry duality proven in dimensions two and four.

Abstract

We give a spectral sequence to compute the logarithmic Hodge groups on a hypersurface type toric log Calabi-Yau space, compute its E_1 term explicitly in terms of tropical degeneration data and Jacobian rings and prove its degeneration at E_2 under mild assumptions. We prove the basechange of the affine Hodge groups and deduce it for the logarithmic Hodge groups in low dimensions. As an application, we prove a mirror symmetry duality in dimension two and four involving the usual Hodge numbers, the stringy Hodge numbers and the affine Hodge numbers.