Periods of tropical Calabi--Yau hypersurfaces

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.

35 pages, 1 figure. v2: The title and the abstract have been changed. The main results are generalized to the case of Calabi--Yau hypersurfaces and formulated in terms of logarithmic Hodge theory; v3: revised following the suggestions by the referee