Period integrals of hypersurfaces via tropical geometry

TL;DR

This paper computes the asymptotics of period integrals for complex hypersurfaces in toric varieties using tropical geometry, linking tropical cycles with classical integrals and exploring the Hodge structure at the limit.

Contribution

It applies tropical geometry methods to analyze period integrals of hypersurfaces, providing explicit asymptotics and connecting to Hodge structures, under unimodular triangulation assumptions.

Findings

Asymptotic formulas for period integrals derived

Explicit description of Hodge structures at the limit for d=1

Connection established between tropical cycles and classical integrals

Abstract

Let $\left\{ Z_t \right\}_t$ be a one-parameter family of complex hypersurfaces of dimension $d \geq 1$ in a toric variety. We compute asymptotics of period integrals for $\left\{ Z_t \right\}_t$ by applying the method of Abouzaid--Ganatra--Iritani--Sheridan, which uses tropical geometry. As integrands, we consider Poincar\'{e} residues of meromorphic $(d+1)$-forms on the ambient toric variety, which have poles along the hypersurface $Z_t$. The cycles over which we integrate them are spheres and tori which correspond to tropical $(0, d)$-cycles and $(d, 0)$-cycles on the tropicalization of $\left\{ Z_t \right\}_t$ respectively. In the case of $d=1$, we explicitly write down the polarized logarithmic Hodge structure of Kato--Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.