摘要翻译：
本文的第一个目的是解决非代数流形上全纯丛理论中的几个基本问题：例如，当Gauduchon度映射是拓扑不变量时，或者当参数流形是紧的时，我们证明了族的稳定性和半稳定性是Zariski开性质。其次，我们证明了对于K\“Ahler流形上泛型稳定的丛族，Petersson-Weil形式在该族的整个参数空间上扩展为闭正电流。这个推广定理使用了由Donaldson发展的Yang-Mills理论中的经典工具（例如Donaldson泛函和全纯丛上Hermitian度量的热方程）。我们将这些结果应用于非K\“Ahlerian曲面$X$参数化的K\”Ahlerian流形$Y$上的丛族，证明了此类丛族必须满足非常严格的条件。这些结果对证明VII类曲面上曲线的存在性起到了重要作用。
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英文标题：
《Families of holomorphic bundles》
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作者：
Andrei Teleman
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Differential Geometry        微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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英文摘要：
  The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds: For instance we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second we show that, for a generically stable family of bundles over a K\"ahler manifold, the Petersson-Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang-Mills theory developed by Donaldson (e.g. the Donaldson functional and the heat equation for Hermitian metrics on a holomorphic bundle). We apply these results to study families of bundles over a K\"ahlerian manifold $Y$ parameterized by a non-K\"ahlerian surface $X$, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces.
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PDF链接：
https://arxiv.org/pdf/0704.2629