Stable vector bundles on fibered threefolds over a surface

Pub Date : 2024-09-09 DOI:10.1007/s10711-024-00946-8
Tohru Nakashima
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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.

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曲面上纤维三折上的稳定向量束
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。
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