arXiv subjects

Yuto Yamamoto

Publications and source records attributed to Yuto Yamamoto.

7 recordsLinked to original sources

arXiv2026

Lifts of cycles in tropical hypersurfaces and the Gamma conjecture

For a complex hypersurface of dimension $d \geq 1$ in a toric variety, we construct lifts of tropical $(p, q)$-cycles with $p+q=d$ in the associated tropical hypersurface. The tropical cycles we consider are described by Minkowski weights, and their lifts are realized as topological cycles admitting a torus fibration structure over the tropical cycles. The intersection numbers of these lifted cycles are computed in terms of tropical intersection theory. We further derive the asymptotic formulas for the period integrals of the lifts in the tropical limit, which are closely related to the mirror symmetric Gamma conjecture. Throughout the paper, we assume that the tropicalization is dual to a unimodular triangulation of the Newton polytope.

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arXiv2025

Period integrals of hypersurfaces via tropical geometry

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.

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arXiv2024

Toric degenerations of Calabi--Yau complete intersections and metric SYZ conjecture

We consider a toric degeneration $\mathcal{X}$ of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. For the toric degeneration $\mathcal{X}$, we study the real Monge--Ampère equation corresponding to the non-archimedean Monge--Ampère equation that yields the non-archimedean Calabi--Yau metric. Our main theorem describes the real Monge--Ampère equation in terms of tropical geometry and proves the metric SYZ conjecture for the toric degeneration $\mathcal{X}$ supposing the existence of its solution.

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arXiv2024

Tropical contractions to integral affine manifolds with singularities

We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The other one is an integral affine manifold with singularities, which arises as the dual intersection complex of the toric degeneration. In this article, we show that the latter is contained in the former as a subset, and construct an integral affine contraction map from the former to the latter. We also show that the contraction preserves tropical cohomology groups, and sends the eigenwave to the radiance obstruction.

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arXiv2024

Non-archimedean SYZ fibrations via tropical contractions

We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. The author showed in his previous work that there exists an integral affine contraction map called a tropical contraction, from the tropical variety obtained as its tropicalization to the dual intersection complex of the toric degeneration. In this article, we prove that the dual intersection complex is isomorphic to the essential skeleton of the Berkovich analytification as piecewise integral affine manifolds, and the composition of the tropicalization map and the tropical contraction is an affinoid torus fibration with a discriminant of codimension $2$, which induces the same integral affine structure as the one coming from the toric degeneration. This is a generalization of an earlier work by Pille-Schneider for a specific degeneration of Calabi--Yau hypersurfaces in projective spaces.

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arXiv2020

Periods of tropical Calabi--Yau hypersurfaces

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.

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arXiv2016

Geometric Monodromy around the Tropical Limit

Let $\{V_q\}_{q}$ be a complex one-parameter family of smooth hypersurfaces in a toric variety. In this paper, we give a concrete description of the monodromy transformation of $\{V_q\}_q$ around $q=\infty$ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin.

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