On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature
On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature
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DOI：
10.4310/jdg/1214438680
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发表时间：
1984
影响因子：
2.5
通讯作者：
S. Bando
中科院分区：
文献类型：
--
作者：
S. Bando
在Mori[5]和Siu&Yau[8]解决了Frankel猜想之后，自然会考虑非负对分曲率的紧致Kaehler流形的分类。在这个方向上，已经有一些前人的工作，例如Siu[7]对超二次曲面的刻画，Howard，Smyth和Wu[3]，[9]关于非负对分曲率的Kaehler流形的分裂定理。除了这些一般的维度研究外，Howard&Smyth[2]还得到了一个低维结果，即非负曲率的二维紧致Kaehler流形的完全分类。本文从这个方向出发，考虑了三维情形，得到了一些结果，结合Howard，Smyth和Wu，[2]，[3]和[9]的上述结果，解决了具有非负对分曲率的三维紧致Kaehler流形的分类问题。也就是说，我们的目标是下列定理。定理3.设M是具有非负υe对分曲率的三维紧致Kaehler流形。如果M有拟正υe Ricci曲率，则M是双全纯到下列之一：P3，Q1XP2，PXPXP1。作者要感谢邱士泰教授，在他的建议下完成了这项工作。
After the solution of Frankel conjecture by Mori [5] and Siu & Yau [8], it is natural to consider the classification of compact Kaehler manifolds of nonnegative bisectional curvature. In this direction there are some previous works, for example, the characterization of hyperquadrics by Siu [7], and the splitting theorem of Kaehler manifolds of nonnegative bisectional curvature by Howard, Smyth, and Wu [3], [9]. Besides these general dimensional studies there is a low dimensional result by Howard & Smyth [2] that is the complete classification of two-dimensional compact Kaehler manifolds of nonnegative curvature. In this paper, proceeding in this direction, we consider the case of three-dimension and obtain some results which, combined together with the above results of Howard, Smyth and Wu, [2], [3] and [9], enable us to settle the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature. Namely our goal is the following theorem. Theorem 3. Let M be a three-dimensional compact Kaehler manifold of nonnegatiυe bisectional curvature. If M has quasipositiυe Ricci curvature, then M is biholomorphic to one of the following: P 3 , Q\ P 1 X P 2 , P X P X P 1 . The author would like to thank Professor S.-T. Yau, under whose advice this work was done.