On the existence of hermitian‐yang‐mills connections in stable vector bundles
On the existence of hermitian‐yang‐mills connections in stable vector bundles
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DOI：
10.1002/cpa.3160390714
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发表时间：
1986
影响因子：
3
通讯作者：
Karen K. Uhlenbeck;S. Yau
中科院分区：
文献类型：
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作者：
Karen K. Uhlenbeck;S. Yau
杨-米尔斯方程是由理论物理学家提出的，现在被认为是粒子理论的基本组成部分。在过去的十年中，这些方程在两个不同的数学领域变得重要。早期观察到，Penrose的扭量形式主义和Atiyah-Singer指数理论可以很好地用于描述特殊解，称为反自对偶解;参见[29]，[3]。利用代数几何的技巧，Atiyah-Drinfeld-Hitchin-Manin [2]能够用CP 3上的全纯丛来描述S4（或W 4）上的所有此类解。最近，唐纳森[81]用稳定向量丛得到了一个更简单的描述。Taubes [25]、[26]使用偏微分方程的技巧来构造任意四维流形上的解。这些结果和紧性定理的第一作者被纳入一般理论的西蒙唐纳森，以获得美丽和壮观的结果微分结构上的四个流形;见[lo]-[ll]。这是一个观察，基本上可以追溯到杨的形式主义，使用C而不是R，Taubes解决方案产生了全纯向量丛在Kiihler曲面;见[30]。全纯向量丛理论是代数几何的核心。稳定向量丛的概念是由Mumford在对丛的模的研究中引入的[18]。这一主题一直追求的几个数学家，特别是竹本，霍罗克斯，吉泽克尔，丸山和巴特。Bogomolov [4]的一个重要贡献是证明了：对一个投射簇M，若Pic（M）= Z，E是M上的稳定丛，则不等式k-1（c ~ 2-Tc：）u ~（2 0）成立。这里k= rank E，c和c2是第一和第二陈类，w是E稳定的极化。Bogomolov的工作启发了Miyaoka [17]关于不等式3c，2 c的工作：对于一般类型的代数曲面。同时，第二作者也独立地给出了这个不等式的证明。虽然Miyaoka和Bogomolov的方法基本上是基于代数几何，但第二作者的方法使用偏微分方程的技术，
The Yang-Mills equations were introduced by theoretical physicists and are now accepted as a basic ingredient in particle theory. In the past decade these equations have become important in mathematics in two separate areas. It was observed early on that the twistor formalism of Penrose and the Atiyah-Singer index theory can be employed to good purpose in order to describe special solutions, called anti-selfdual solutions; see [29],[3]. Using techniques of algebraic geometry, Atiyah-Drinfeld-Hitchin-Manin [2] were able to describe all such solutions over S4 (or W4) in terms of holomorphic bundles on CP3. More recently a simpler description using stable vector bundles was obtained by Donaldson [81. The techniques of partial differential equations were used by Taubes [25],[26] to construct solutions on arbitrary four-manifolds. These results and compactness theorems of the first author were incorporated into a general theory of Simon Donaldson to obtain beautiful and spectacular results on the differential structures on four-manifolds; see [lo]-[ll]. It is an observation dating essentially back to Yang’s formalism of using C instead of R that the Taubes solutions give rise to holomorphic vector bundles over Kiihler surfaces; see [30]. The theory of holomorphic vector bundles is central to algebraic geometry. The concept of a stable vector bundle was introduced by Mumford in hs study of moduli for bundles [18]. This theme has been pursued by several mathematicians, notably Takemoto, Horrocks, Gieseker, Maruyama and Barth. An important contribution was made by Bogomolov [4], who showed that, for a projective variety M with Pic (M)= Z and E a stable bundle on M, the inequality k-1 (c2-Tc:) u 2 0 is valid. Here k= rank E, c, and c2 are the first and second Chern classes, and w is a polarization for which E is stable.Bogomolov’s work inspired the work of Miyaoka [17] on the inequality 3c, 2 c: for algebraic surfaces of general type. Independently, the second author also gave a proof of this inequality at the same time. While the methods of Miyaoka and Bogomolov are essentially based on algebraic geometry, the method of the second author used the techniques of partial differential equations to