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.