{"title":"L^2$ 消失定理和科拉尔猜想","authors":"Ya Deng, Botong Wang","doi":"arxiv-2409.11399","DOIUrl":null,"url":null,"abstract":"In 1995, Koll\\'ar conjectured that a complex projective $n$-fold $X$ with\ngenerically large fundamental group has Euler characteristic $\\chi(X, K_X)\\geq\n0$. In this paper, we confirm the conjecture assuming $X$ has linear\nfundamental group, i.e., there exists an almost faithful representation\n$\\pi_1(X)\\to {\\rm GL}_N(\\mathbb{C})$. We deduce the conjecture by proving a\nstronger $L^2$ vanishing theorem: for the universal cover $\\widetilde{X}$ of\nsuch $X$, its $L^2$-Dolbeaut cohomology $H_{(2)}^{n,q}(\\widetilde{X})=0$ for\n$q\\neq 0$. The main ingredients of the proof are techniques from the linear\nShafarevich conjecture along with some analytic methods.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"197 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"$L^2$-vanishing theorem and a conjecture of Kollár\",\"authors\":\"Ya Deng, Botong Wang\",\"doi\":\"arxiv-2409.11399\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1995, Koll\\\\'ar conjectured that a complex projective $n$-fold $X$ with\\ngenerically large fundamental group has Euler characteristic $\\\\chi(X, K_X)\\\\geq\\n0$. In this paper, we confirm the conjecture assuming $X$ has linear\\nfundamental group, i.e., there exists an almost faithful representation\\n$\\\\pi_1(X)\\\\to {\\\\rm GL}_N(\\\\mathbb{C})$. We deduce the conjecture by proving a\\nstronger $L^2$ vanishing theorem: for the universal cover $\\\\widetilde{X}$ of\\nsuch $X$, its $L^2$-Dolbeaut cohomology $H_{(2)}^{n,q}(\\\\widetilde{X})=0$ for\\n$q\\\\neq 0$. The main ingredients of the proof are techniques from the linear\\nShafarevich conjecture along with some analytic methods.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"197 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"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-2409.11399\",\"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 - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11399","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
$L^2$-vanishing theorem and a conjecture of Kollár
In 1995, Koll\'ar conjectured that a complex projective $n$-fold $X$ with
generically large fundamental group has Euler characteristic $\chi(X, K_X)\geq
0$. In this paper, we confirm the conjecture assuming $X$ has linear
fundamental group, i.e., there exists an almost faithful representation
$\pi_1(X)\to {\rm GL}_N(\mathbb{C})$. We deduce the conjecture by proving a
stronger $L^2$ vanishing theorem: for the universal cover $\widetilde{X}$ of
such $X$, its $L^2$-Dolbeaut cohomology $H_{(2)}^{n,q}(\widetilde{X})=0$ for
$q\neq 0$. The main ingredients of the proof are techniques from the linear
Shafarevich conjecture along with some analytic methods.