{"title":"A homology theory for tropical cycles on integral affine manifolds and a perfect pairing","authors":"Helge Ruddat","doi":"10.2140/gt.2021.25.3079","DOIUrl":null,"url":null,"abstract":"We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\\mathbb{Q}$ in degree one when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi-Yau degenerations. We also give an intersection theoretic application for Strominger-Yau-Zaslow fibrations. The treatment of the cap product and Poincare-Lefschetz by simplicial methods for constructible sheaves might be of independent interest.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"129 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"15","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometry & Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/gt.2021.25.3079","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 15
Abstract
We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\mathbb{Q}$ in degree one when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi-Yau degenerations. We also give an intersection theoretic application for Strominger-Yau-Zaslow fibrations. The treatment of the cap product and Poincare-Lefschetz by simplicial methods for constructible sheaves might be of independent interest.
引入了具有奇异点的积分仿射流形上热带旋的同调和上同调的帽积对。我们证明了在$\mathbb{Q}$上，当流形有最坏的单奇点时，配对是完美的。通过与Siebert的合作，该配对计算周期积分，其完备性暗示了典型Calabi-Yau退化的通用性。我们还给出了strominger - you - zaslow振动的交理论应用。用简化的方法处理可施工轮轴的帽积和Poincare-Lefschetz可能是一个独立的研究方向。
期刊介绍：
Geometry and Topology is a fully refereed journal covering all of geometry and topology, broadly understood. G&T is published in electronic and print formats by Mathematical Sciences Publishers.
The purpose of Geometry & Topology is the advancement of mathematics. Editors evaluate submitted papers strictly on the basis of scientific merit, without regard to authors" nationality, country of residence, institutional affiliation, sex, ethnic origin, or political views.
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