Periods of tropical Calabi--Yau hypersurfaces

TL;DR

This paper studies the extension of Hodge structures associated with toric Calabi-Yau hypersurfaces, linking them to integral affine structures in the context of the Gross--Siebert program.

Contribution

It describes the polarized logarithmic Hodge structure of toric Calabi-Yau hypersurfaces in terms of the dual intersection complex's affine structure.

Findings

Description of PLH via integral affine structure

Extension of Hodge structures to the whole disk

Connection to the Gross--Siebert program

Abstract

We consider the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi--Yau hypersurfaces over a punctured disk $D \setminus \{ 0\}$. It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato--Usui on the whole disk $D$. By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross--Siebert program.