{"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}
引用次数: 0
Abstract
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.