用户: 广濑亚纪/NOTE2/Misha Verbitsky 0403430

1Introduction

设 $M$ 是紧复流形, $\pi :M\to X$ 是 $T$-主丛, $T$ 是椭圆曲线. 在本文中 $X$ 是 Kähelr 流形.

2Principal elliptic fibrations

Positive elliptic fibrations: definition and examples

定义 2.1. Fibration $\pi :M\to X$ is called positive if the pullback ${\pi }^{\ast }{\omega }_X$ is exact, for some Kahler form ${\omega }_X$ on $X$.

例 2.2. Regular Vaisman manifolds are principal elliptic fibrations.

Ample line bundle; 商空间与主丛. 主丛约化与商群.

注 2.3. A special case of the above example is a regular Hopf manifold $M\cong S^{2n-1}\times S^1$. Regular Hopf manifold is obtained if one applies the construction of Example 2.2 to $X=\mathbb{C}{P}^{n-1},L=\mathcal{O}(1)$.

注 2.4. 在 [1, Topology of the total space] 中, 但那里 $M$ 是底空间而 $X$ 是全空间.

例 2.5. 同 Example 2.2 类似

Preferred Hermitian metrics

Blanchard’s Theorem 出自 [1, Theorem 1.7], 表达了当底空间是 Kahler 流形时, 全空间是 Kahler 当且仅当 $\delta =0$.

Riemannian submersion 是关于主丛和 Riemann 度量的. 对于 Hermite 流形, 由于复线性型可由实部决定 (因为已经给出复结构), 故 Riemannian submersion 同样诱导了 Hermitian metric 的等距同构, 并且这个直和也是关于 Hermitian metric 的正交分解.

定义 2.6. Let $T$ be an elliptic curve, and $\pi :M\to X$ a positive principal $T$-fibration. Consider a Kahler metric $g_X$ on $X$, such that the pullback ${\pi }^{\ast }{\omega }_X$ of the corresponding Kahler form is exact. A Hermitian metric $g$ on $M$ is called preferred if $g$ is $T$-invariant, and the projection $\pi :(M,g)\to (X,g_X)$ is a Riemannian submersion.

Preferred Hermitian metrics 的存在性

3Stable bundle on Hermitian manifolds

Gauduchon metrics and stability

定义 3.1. Let $(M,g)$ be a Hermitian manifold, and $\omega \in {\Lambda }^{1,1}(M)$ its Hermitian form. We say that $g$ is a Gauduchon metric if $\partial \bar{\partial }({\omega }^{{\dim }_{\mathbb{C}}M-1})=0$.

这里的问题是, ${\partial }^{\ast }{\bar{\partial }}^{\ast }\omega =0$ 与 $\partial \bar{\partial }({\omega }^{{\dim }_{\mathbb{C}}M-1})=0$ 的等价性.

The metrics $g,g^{\prime }$ on $M$ are called conformally eauivalent if $g=f{g}^{\prime }$ for some positive $f\in C^{\infty }(M)$.

定理 3.2. Let $(M,g)$ be a compact Hermitian manifold. Then there exists a unique Gauduchon metric $g^{\prime }$ which is conformally equivalent to $g$.

证明. This is [2, Theorem 1.2.4].

$\square$

定义 3.3. Let $M$ be a compact complex manifold equipped with a Gauduchon metric, and $\omega \in {\Lambda }^{1,1}(M)$ the corresponding Hermitian form. Consider a torsion-free coherent sheaf $F$ on $M$. Denote by $\det F$ its determinant bundle. Pick a Hermitian metric $v$ on $\det F$, and lete $\Theta$ be the curvature of the associated Chern connection. We define the degree of $F$ as follows: $$\mathrm{deg}F={\int }_M\Theta \wedge {\omega }^{{\dim }_{\mathbb{C}}M-1}$$This notion is independent from the choice of the metric $v$.

Kahler 流形上就是由第一陈类定义的度. 非 Kahler 时不是拓扑不变量. 具体见 [2].

定义 3.4. Let $F$ be a non-zero torsion-free coherent sheaf on $M$. Then $\mathrm{slope}(F)$ is defined as $$\mathrm{slope}(F)=\frac{\mathrm{deg}F}{\mathrm{rank}F}$$The sheaf $F$ is called

•	stable if for all subsheaves $F^{\prime }$, we have $\mathrm{slope}(F^{\prime })<\mathrm{slope}(F)$;
•	semistable if for all subsheaves $F^{\prime }$, we have $\mathrm{slope}(F^{\prime })\le \mathrm{slope}(F)$;
•	polystable if $F$ can be represented as a direct sum of stable coherent sheaves with the same slope.

注 3.5. Jordan-Holder filtratoins; Harder-Narasimhan filtrations. 具体见 [2].

Kobayashi-Hitchin correspondence

定义 3.6. Let $B$ be a holomorphic Hermitian vector bundle on a Hermitian manifold $M$, and $\Theta \in {\Lambda }^{1,1}(M)\otimes \mathrm{End}(B)$ the curvature of its Chern connection $\Delta$. Let $\Lambda$ be dual Lefschetz operator. The connection $\Delta$ is called Hermitian-Einstein (or Yang-Mills) if $\Lambda (\Theta )=\text{const}\cdot {\mathrm{Id}}_B$.

定理 3.7 (Kobayashi-Hitchin correspondence). Let $B$ be a holomorphic vector bundle on a compact complex manifold equipped with a Gauduchon metric. Then $B$ admits a Hermitian-Einstein connection $\Delta$ if and only if $B$ is polystable. Moreover, the Hermitian-Einstein connection is unique.

4Hermitian-Einstein bundles on positive principal elliptic fibrations

Preferred metrics are Gauduchon

命题 4.1. Let $\pi :M\to X$ be a positive elliptic fibration, and $g$ a preferred Hermitian metric. Then $g$ is Gauduchon.

证明. $\omega$ 是 Hermitian metric $g$ 的基本形式.

$\omega$ 的分解实际上是它与 ${\bigwedge }^{2}TM$ 到以下三个子空间的投影的复合: $$\stackrel{2}{\bigwedge }{T}_{\pi }M,T_{\pi }M\wedge T_{\pi }{M}^{\perp },\stackrel{2}{\bigwedge }{T}_{\pi }{M}^{\perp }$$与第一个复合就是 ${\omega }_T$, 与第三个复合就是 ${\omega }_0$. 至于第二个, 由于 $T_{\pi }M\oplus T_{\pi }{M}^{\perp }$ 是正交分解, 故 $\omega$ 在这为零. 这样 $\omega ={\omega }_T+{\omega }_0$.

$\square$

Primitive forms on positive principal elliptic fibrations

这里似乎有 typo, $B_0$ 应为 $B$.

命题 4.2. Let $\pi :M\to X$ be a positive principal elliptic fibration, $n=\dim M\ge 3$, equipped with a preferred Hermitian metric, $B$ a Hermitian bundle with connection, and $\Theta \in {\Lambda }^{1,1}(M)\otimes \mathrm{End}(B)$ a closed $(1,1)$-form. Assume that $\Theta$ is primitive, that is $\Lambda \Theta =0$. Then $\Theta (v,\cdot )=0$ (这是个 1-形式) for any vertical tangent vector $v\in T_{\pi }M$.

证明. Hermitian connection 就是 metric connection.

$\square$

The Lubke-type positivity for Hermitian-Einstein bundles

定理 4.3. Let $\pi :M\to X$, ${\dim }_{\mathbb{C}}M=n$ be a positive principal elliptic fibration, equipped with a preferred Hermitian metric, $(B,\nabla )$ a Hermitian-Einstein bundle on $M$, and $\Theta \in {\Lambda }^{1,1}(M)\otimes \mathrm{End}(B)$ its curvature. Assume that $n\ge 3$. Then $\Theta (v,\cdot )=0$ for any vertical tangent vector $v\in T_{\pi }M$.

Complex subvarieties in positive principal elliptic fibrations

Equivariance of stable bundles

$G$-Equivariant bundle 是纤维丛 $\pi :M\to X$ 使得:

•	$M,X$ 是 $G$-空间.
•	$\pi$ 保持 $G$ 的作用.

命题 4.4. Let $T$ be an elliptic curve, and $\pi ::M\to X$, ${\dim }_{\mathbb{C}}M\ge 3$ a positive principal $T$-fibration, equipped with a preferred Hermitian metric. The universal covering $\tilde{T}$ acts on $M$ in a standard way. Consider a stable bundle $B$ on $M$. Then $B$ is equipped with a natural holomorphic $\tilde{T}$-equivariant structure.

证明. Hermitian-Einstein connection 的存在性来自于 Kobayashi-Hitchin correspondence Theorem 3.7.

$\tau$ 是 $M$ 上的垂直向量场.

$\square$

5Line bundles on principal elliptic fibrations

平坦联络; monodromy of connection 是什么?

$L$ 的全纯自同构群可视为 ${\mathbb{C}}^{\ast }$, 则 $\Gamma$ 的作用是 ${\mathbb{C}}^{\ast }$ 的子群 $U(1)$ (因为 $\Gamma \cong {\mathbb{Z}}^2$ ?)

命题 5.1. Let $T$ be an elliptic curve and $\pi :M\to X$ a positive principal $T$-fibration, $\dim M\ge 3$. Consider any character $\chi :\Gamma \to U(1)$. Then there exists a holomorphic line bundle $L$ on $M$ such that the correspondence ${\chi }_L:\Gamma \to U(1)$ is equal to $\chi$.

6Structure theorem for stable bundles

Equivariant $\tilde{T}$-action and the stable bundles

定理 6.1. Let $\pi :M\to X,{\dim }_{\mathbb{C}}M\ge 3$ be a positive principal elliptic fibration equipped a preferred Hermitian metric, and $B$ a stable holomorphic bundle on $M$. Then $B\cong L\otimes {\pi }^{\ast }{B}_0$, where $L$ is a line bundle on $M$ and $B_0$ a stable bundle on $X$.

证明. $B$ 稳定则它的自同构都是单位映射的倍乘?

$\square$

[1]	T.Höfer (1993) : Remarks on torus principal bundles
[2]	M.Lubke, A.Teleman(1995): The Kobayashi-Hitchin Correspondence.