Mirror Symmetry via Logarithmic Degeneration Data II

TL;DR

This paper advances the understanding of mirror symmetry by computing the cohomology of log Calabi-Yau spaces associated with affine manifolds with singularities, confirming expected Hodge number exchanges and describing monodromy explicitly.

Contribution

It provides a method to calculate the cohomology of Calabi-Yau varieties from affine manifolds with singularities, linking log geometry, toric degenerations, and mirror symmetry.

Findings

Cohomology groups of Calabi-Yau are described via sheaf cohomology on affine manifolds.

Mirror symmetry via Legendre duality exchanges Hodge numbers as expected.

Explicit description of monodromy of smoothings in the log geometric setting.

Abstract

This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.