Period integrals of hypersurfaces via tropical geometry
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915254038953984 |
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| author | Yamamoto, Yuto |
| author_facet | Yamamoto, Yuto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_00814 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Period integrals of hypersurfaces via tropical geometry Yamamoto, Yuto Algebraic 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. |
| title | Period integrals of hypersurfaces via tropical geometry |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2205.00814 |