{"title":"Uniqueness of irreducible desingularization of singularities associated to negative vector bundles","authors":"Fusheng Deng, Yinji Li, Qunhuan Liu, Xiangyu Zhou","doi":"arxiv-2409.09402","DOIUrl":null,"url":null,"abstract":"We prove that the irreducible desingularization of a singularity given by the\nGrauert blow down of a negative holomorphic vector bundle over a compact\ncomplex manifold is unique up to isomorphism, and as an application, we show\nthat two negative line bundles over compact complex manifolds are isomorphic if\nand only if their Grauert blow downs have isomorphic germs near the\nsingularities. We also show that there is a unique way to modify a submanifold\nof a complex manifold to a hypersurface, namely, the blow up of the ambient\nmanifold along the submanifold.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09402","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that the irreducible desingularization of a singularity given by the
Grauert blow down of a negative holomorphic vector bundle over a compact
complex manifold is unique up to isomorphism, and as an application, we show
that two negative line bundles over compact complex manifolds are isomorphic if
and only if their Grauert blow downs have isomorphic germs near the
singularities. We also show that there is a unique way to modify a submanifold
of a complex manifold to a hypersurface, namely, the blow up of the ambient
manifold along the submanifold.