{"title":"我们仿射黎曼曲面","authors":"Richard Cushman","doi":"arxiv-2407.06332","DOIUrl":null,"url":null,"abstract":"We show that the universal covering space of a connected component of a\nregular level set of a smooth complex valued function on ${\\mathbb{C}}^2$,\nwhich is a smooth affine Riemann surface, is ${\\mathbb{R}}^2$. This implies\nthat the orbit space of the action of the covering group on ${\\mathbb{R}}^2$ is\nthe original affine Riemann surface.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On affine Riemann surfaces\",\"authors\":\"Richard Cushman\",\"doi\":\"arxiv-2407.06332\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the universal covering space of a connected component of a\\nregular level set of a smooth complex valued function on ${\\\\mathbb{C}}^2$,\\nwhich is a smooth affine Riemann surface, is ${\\\\mathbb{R}}^2$. This implies\\nthat the orbit space of the action of the covering group on ${\\\\mathbb{R}}^2$ is\\nthe original affine Riemann surface.\",\"PeriodicalId\":501155,\"journal\":{\"name\":\"arXiv - MATH - Symplectic Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Symplectic Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.06332\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Symplectic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.06332","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We show that the universal covering space of a connected component of a
regular level set of a smooth complex valued function on ${\mathbb{C}}^2$,
which is a smooth affine Riemann surface, is ${\mathbb{R}}^2$. This implies
that the orbit space of the action of the covering group on ${\mathbb{R}}^2$ is
the original affine Riemann surface.
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