{"title":"Stable vector bundles on fibered threefolds over a surface","authors":"Tohru Nakashima","doi":"10.1007/s10711-024-00946-8","DOIUrl":null,"url":null,"abstract":"<p>Let <i>X</i> be a smooth projective threefold and let <i>H</i> be an ample line bundle on <i>X</i>. We investigate the existence of vector bundles on <i>X</i> which are <span>\\(\\mu \\)</span>-stable with respect to an ample divisor <span>\\(H_{\\epsilon }=H+\\epsilon D\\)</span> for sufficiiently small <span>\\(\\epsilon >0\\)</span> where <i>D</i> is a divisor with <span>\\(D\\cdot H^2=0\\)</span>. In particular, when <i>X</i> is a Fano conic bundle over a rational surface, we show that there exists a family <span>\\(\\{E_n\\}\\)</span> of <span>\\(H_{\\epsilon }\\)</span>-stable vector bundles with <span>\\(c_1(E_n)=0\\)</span> and <span>\\(c_2(E_n)\\cdot H\\)</span> becomes arbitrarily large as <i>n</i> goes to infinity.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-024-00946-8","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
Let X be a smooth projective threefold and let H be an ample line bundle on X. We investigate the existence of vector bundles on X which are \(\mu \)-stable with respect to an ample divisor \(H_{\epsilon }=H+\epsilon D\) for sufficiiently small \(\epsilon >0\) where D is a divisor with \(D\cdot H^2=0\). In particular, when X is a Fano conic bundle over a rational surface, we show that there exists a family \(\{E_n\}\) of \(H_{\epsilon }\)-stable vector bundles with \(c_1(E_n)=0\) and \(c_2(E_n)\cdot H\) becomes arbitrarily large as n goes to infinity.
让 X 是光滑的投影三褶,让 H 是 X 上的充裕线束。我们研究了 X 上向量束的存在性,这些向量束在足够小的\(\epsilon >0\)(其中 D 是具有\(D\cdot H^2=0\) 的充裕分部时,相对于充裕分部\(H_{\epsilon }=H+\epsilon D\) 是稳定的。)特别是,当X是一个有理面上的法诺圆锥束时,我们证明存在一个\(\{E_n\}\)\(H_{epsilon }\)-stable vector bundles的族,其\(c_1(E_n)=0\)和\(c_2(E_n)\cdot H\) 随着n的无穷大而变得任意大。