Uniqueness of the complex structure on Kähler manifolds of certain homotopy types
Uniqueness of the complex structure on Kähler manifolds of certain homotopy types
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DOI：
10.4310/jdg/1214445041
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发表时间：
1990
影响因子：
2.5
通讯作者：
A. Libgober;John W. Wood
中科院分区：
文献类型：
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作者：
A. Libgober;John W. Wood
本文证明了复射影空间和二次曲面的同伦类型支持唯一的Kahler型复结构。复射影空间上的结构引起了人们的广泛关注。Hirzebruch和Kodaira [14]， [11, p. 231]证明了具有CPn的同伦型和Pontryagin类的Kahler流形V解析等价于CPn，他们的附加假设cx(V) Φ - (n - \)x对于偶n后来被Yau的工作[31]所删除。这里x表示H（vz）的生成器，它是正的，因为它是V[13，§18.1]上某个Kahler度规的基本类。另一方面，我们知道对于每一个n > 2， CPn的同伦类型支持无穷多个由它们的Pontryagin类区分的不等价可微结构（参见Montgomery和Yang[25]或Wall[30]对于n > 3和Hsiang[15]对于n > 3）。此外，对于n - 3或4，这些光滑结构中的每一个都可以显示支持几乎复杂的结构。在§7中，我们用brumfield和Heaps的结果证明了n = 4的情况。本文的主要结果是当n < 6时，同伦CPn的其他光滑不支持Kahler结构。定理1。当n < 6时，等价于CPn的Kahler流形同伦解析等价于CPn。由Kodaira嵌入定理可知，任何具有Kahler结构的同伦复射影空间都是射影代数的，即解析等价于高维射影空间的非奇异子变[13，§18.1]。在[31]中，Yau应用了Kodaira的判据证明了一个等价于CP2的复流形同伦是代数的（因此是Kahler），并且证明了它是解析等价于CP2的。一个复杂的流形是否
In this note we show that the homotopy types of certain complex projective spaces and quadrics support a unique complex structure of Kahler type. Structures on complex projective space have attracted much attention. Hirzebruch and Kodaira [14], [11, p. 231] showed that a Kahler manifold V with the homotopy type and Pontryagin classes of CPn is analytically equivalent to CPn their additional assumption that cx(V) Φ —(n — \)x for even n was later removed by Yau's work [31]. Here x denotes the generator of H(V Z) which is positive in the sense that it is the fundamental class of some Kahler metric on V [13, §18.1]. On the other hand it is known that for every n > 2 the homotopy type of CPn supports infinitely many inequivalent differentiable structures distinguished by their Pontryagin classes (see Montgomery and Yang [25] or Wall [30] for n 3 and Hsiang [15] for n > 3). Moreover for n — 3 or 4 each of these smooth structures can be shown to support almost complex structures. In §7 we prove this for the case n = 4 by applying results of Brumfiel and Heaps. The main result of this paper is that for n < 6 these other smoothings of a homotopy CPn do not support a Kahler structure. Theorem 1. A Kahler manifold homotopy equivalent to CPn for n < 6 is analytically equivalent to CPn . It follows from the Kodaira embedding theorem that any homotopy complex projective space with a Kahler structure is projective algebraic, i.e., is analytically equivalent to a nonsingular subvariety of a higher dimensional projective space [13, §18.1]. In [31], Yau applied a criterion of Kodaira to show that a complex manifold homotopy equivalent to CP2 is algebraic (hence Kahler) and showed moreover that it is analytically equivalent to CP2. It is still an open question whether a complex manifold