用户: 广濑亚纪/NOTE2/R.Moraru 0408439

1Introduction

2Preliminaries

3Holomorphic Vector Bundles on Hopf Surfaces

Topologically trivial holomorphic vector bundles

命题 3.1. On a Hopf surface $X$, topologically trivial holomorphic vector bundles of rank greater than $1$ are not simple.

由 [1, p21, Remark 1.1.7], 全纯向量丛 $(E,\delta )$ ($\delta$ 是全纯结构) 被称为 simple 若全纯自同态都是数乘, 即 $H^0(X,\mathrm{End}(E))=\mathbb{C}\cdot {\mathrm{id}}_E$.

推论 3.2. Any topologically trivial holomorphic vector bundle on a Hopf surface $X$ possesses a filtration by vector bundles.

注 3.3. [2, Structure Theorem 3.2]: Let $X$ be an arbitrary Hopf manifold of dimension $n\ge 3$ or a diagonal Hopf surface.

If $E$ is a holomorphic vector bundle or rank $r$ on $X$, then the following statements are equivalent:

•	${\pi }^{\ast }(E)\cong {\mathcal{O}}_W^r$
•	$E$ possesses a filtration by vector bundles $$0\subset E^1\subset \cdots \subset E^{r-1}\subset E^r=E$$with $E^i$ of rank $i$.

命题 3.4. Let $E$ be an extension of line bundles on $X$. Then, there exists an exact sequence $$0\to L_a\to E\to L_b\to 0$$with $a,b\in {\mathbb{C}}^{\ast }$. We have the following possibilities:

•	If $a=b{\mu }_1^{m_1}{\mu }_2^{m_2}$, for non-negative integres $m_1$ and $m_2$, then $E=L_a\oplus L_b$ or $E$ is the unique non-trivial extension;
•	If $a{b}^{-1}\ne {\mu }_1^{m_1}{\mu }_2^{m_2}$ for all integers $m_1,m_2\ge 0$, then $E=L_a\oplus L_b$.

Construction rank-2 vector bundles

•	Double covers: 任何向量丛都可以这么构造吗?
•	Serre construction.

Moduli spaces

non-filtrable bundles are automatically stable.

Filtrable rank-2 vector bundles

Consider a filtrable rank-2 vector bundle $E$ on $X$ with $c_2(E)=c_2>0$. It can therefore expressed as an extension of the form:$$0\to L_a\to E\to L_b\otimes I_Z\to 0$$

定理 3.5.

命题 3.6. Let $E$ be a stable filtrable rank-2 vector bundle on $X$ with determinant $\delta$ and a jump of multiplicity $1$ on $T_1$. Then, $E$ is uniquely determined by a triple $(a,\lambda ,p)$ such that $$(a,\lambda )\in D_{1,0}\times {\mathrm{Pic}}^1(T_1)$$

[1]	M.Lubke, A.Teleman(1995): The Kobayashi-Hitchin Correspondence.
[2]	D.Mall(1992) On holomorphic vector bundles on Hopf manifolds with pullback on the universal covering