Compact Self-Dual Manifolds with Torus Actions
Compact Self-Dual Manifolds with Torus Actions
复制标题
带环面动作的紧凑型自双歧管
DOI：
10.4310/jdg/1090340879
复制
发表时间：
2000
影响因子：
2.5
通讯作者：
A. Fujiki
中科院分区：
文献类型：
--
作者：
A. Fujiki
我们证明，具有二环面平滑作用和非零欧拉特征的紧致自对偶四流形必然微分同胚于复射影平面副本的连通和，而且自对偶结构同构于乔伊斯在[11]中构造的结构之一。这肯定地解决了乔伊斯[11]的猜想。我们的证明方法是通过复杂的几何技术来表明，相关的扭量空间是一个紧复三重，具有代数二环面的诱导全纯作用，具有非常特殊的结构，并且确实由某个不变量决定，该不变量最终与乔伊斯构造自对偶流形相关的不变量确定。
We show that a compact self-dual four-manifold with a smooth action of a two-torus and with non-zero Euler characterestic is necessarily diffeomorphic to a connected sum of copies of complex projective planes, and furthermore the self-dual structure is isomorphic to one of those constructed by Joyce in [11]. This settles a conjecture of Joyce [11] affirmatively. Our method of proof is to show, by complex geometric techniques, that the associated twistor space, which is a compact complex threefold with the induced holomorphic action of algebraic two-torus, has a very special structure and is indeed determined by a certain invariant which is eventually identified with the invariant associated with the Joyce’s construction of his self-dual manifolds.