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Period integrals of hypersurfaces via tropical geometry

Yuto Yamamoto

TL;DR

This work generalizes tropical methods for period asymptotics from Calabi–Yau ambient hypersurfaces to arbitrary toric hypersurfaces by studying Poincaré residues with higher-order poles. It constructs sphere and torus cycles from the tropicalization and derives explicit asymptotic formulas for period integrals in terms of Newton polytope data, valuations, and the gamma class, revealing leading affine-tropical contributions and subleading gamma-terms. For curves (d=1) it provides a complete description of the polarized logarithmic Hodge structure at the limit, including monodromy and the limit filtration. The results bridge tropical geometry, Hodge theory, and toric degenerations, with concrete examples and applications to the tropicalized B-model/A-model correspondence in non-Calabi–Yau settings.

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 Poincaré 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.

Period integrals of hypersurfaces via tropical geometry

TL;DR

This work generalizes tropical methods for period asymptotics from Calabi–Yau ambient hypersurfaces to arbitrary toric hypersurfaces by studying Poincaré residues with higher-order poles. It constructs sphere and torus cycles from the tropicalization and derives explicit asymptotic formulas for period integrals in terms of Newton polytope data, valuations, and the gamma class, revealing leading affine-tropical contributions and subleading gamma-terms. For curves (d=1) it provides a complete description of the polarized logarithmic Hodge structure at the limit, including monodromy and the limit filtration. The results bridge tropical geometry, Hodge theory, and toric degenerations, with concrete examples and applications to the tropicalized B-model/A-model correspondence in non-Calabi–Yau settings.

Abstract

Let be a one-parameter family of complex hypersurfaces of dimension in a toric variety. We compute asymptotics of period integrals for by applying the method of Abouzaid--Ganatra--Iritani--Sheridan, which uses tropical geometry. As integrands, we consider Poincaré residues of meromorphic -forms on the ambient toric variety, which have poles along the hypersurface . The cycles over which we integrate them are spheres and tori which correspond to tropical -cycles and -cycles on the tropicalization of respectively. In the case of , 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.
Paper Structure (17 sections, 26 theorems, 225 equations, 4 figures)

This paper contains 17 sections, 26 theorems, 225 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Logarithmic Hodge theory
  4. Related work
  5. Organization of this article
  6. Forms having poles along hypersurfaces
  7. Construction of sphere cycles
  8. Proof of th:main1
  9. Local computation of the period integral
  10. The Duistermaat--Heckman theorem/complex volumes of polytopes
  11. The sum of local integrals
  12. The Dirichlet integral
  13. The volume of $S_t^{w}$
  14. Proof of th:main2
  15. Leading terms of periods
  16. ...and 2 more sections

Key Result

Theorem 1.1.1

(MR2079993) There exists a stratified torus fibration $p \colon \mathring{Z}_t \to V \left( \operatorname{trop} \left( f \right) \right)$ satisfying the following:

Figures (4)

  • Figure 1: The triangulation $\mathscr{T}$ of $\Delta$ and the tropicalization $V(\operatorname{trop} (f))$
  • Figure 2: The set $N_\kappa^w$ with $w=0$ and its partitions
  • Figure 3: The tropicalization $V(\operatorname{trop} (f))$ and the regions $S_t^{0, q, K} \cap \left\{ x=x_0 \right\}$
  • Figure 4: The curve $Z_t$ and the cycles $T_{t}^{\sigma_0}, T_t^{\sigma_1}, C_{t}^0, C_t^{e_1}$

Theorems & Definitions (53)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Corollary 1.2.1
  • Theorem 2.0.1
  • Remark 2.0.2
  • Example 3.0.1
  • Lemma 3.0.2
  • proof
  • Remark 3.0.3
  • ...and 43 more