On the chern numbers of certain complex and almost complex manifolds.
On the chern numbers of certain complex and almost complex manifolds.
复制标题
关于某些复数和准复流形的陈数。
DOI：
10.1073/pnas.55.6.1624
复制
发表时间：
1966
影响因子：
11.1
通讯作者：
A. V. D. Ven
中科院分区：
文献类型：
--
作者：
A. V. D. Ven
设V是一个紧的、定向的2n维（C-）可微流形。如果V容许一个与其可微结构相容的复结构，则V的切丛的结构群可以自然地从GL（2n，R）限制到GL（n，C）。这给了V拓扑性质的一个必要条件，这导致了Ch. Ehresmann和H。Hopf在20年前引入了几乎复结构的概念，确切地说是从GL（2n，R）到GL（n，C）的可微流形的切丛的结构群的限制。几乎复流形是具有几乎复结构的可微流形。复流形当然是几乎复流形，并且多年来已经知道一个条件，即Eckmann-Frohlicher条件，该条件对于C-y几乎复结构可积（即由复结构诱导）是必要且充分的。然而，一旦在一个可微流形上存在一个几乎复结构，通过这个结构的微小变化，就可以得到无穷多个其他的结构，并且对于一个给定的几乎复结构，总是有一个同伦的可积的结构是可能的，尽管不太可能。特别是，没有一个例子是已知的一个紧凑的，可微的流形，承认几乎复杂的结构，但没有复杂的结构。现在开始，我们想指出，许多例子紧4维可微流形，承认几乎复杂的结构，但没有复杂的结构，可以得到的一个定理相结合的CH。W. Wu（在一个公式由于F. Hirzebruch和H. Hopf），并推广了K.科代拉。这些最后的结果，因此，主要定理的公告，依赖于一个基本的和迄今为止不可避免的方式在阿蒂亚辛格黎曼罗克定理。正如将在本说明的最后解释的那样，为了获得具体的例子，除了Ehresmann-Wu和Atiyah-Singer的定理之外，我们只需要一些关于复曲面的非常一般的事实。但我们可以得到更多。为了正确地解释这一点，我们将回忆紧几乎复流形V2 '的陈数的定义。设[V]是V的基本圈，ci是V的第i个Chern类，il，. . .，为非负整数，i1 + 2 i2+。. . + m 4 = n.然后你就可以... U cnin C H2 n（V，Z），并且这个上同调类在[V]上的值是一个整数. 7 r（n）整数，对于所有的i1，i2，.. .在I1 + 2 I2+中。. . + J. Milnorl已经确定了7 r（n）整数的集合作为不一定连通的V的陈数出现。特别是，它们与作为复形甚至相同维数的射影代数流形的陈数出现的集合相同，但也不一定连通。米尔诺的方法不允许我们决定哪一组
Let V be a compact, oriented 2n-dimensional (C--)differentiable manifold. If V admits a complex structure, compatible with its differentiable structure, the structural group of the tangent bundle of V can be restricted in a natural way from GL(2n,R) to GL(n,C). This gives for V a necessary condition of topological nature, which led Ch. Ehresmann and H. Hopf some 20 years ago to the concept of an almost complex structure, exactly meaning a restriction of the structural group of the tangent bundle of a differentiable manifold from GL(2n,R) to GL(n,C). An almost complex manifold is a differentiable manifold, provided with an almost complex structure. A complex manifold is of course an almost complex manifold, and for several years a condition has been known, the Eckmann-Frohlicher condition, which is necessary and sufficient for a C-y almost complex structure to be integrable, that is, to be induced by a complex structure. However, once there exists on a differentiable manifold an almost complex structure, by a slight change of this structure, an infinity of others can be obtained, and it would have been possible, though not likely, that to a given almost complex structure there is always a homotopic one, which is integrable. In particular, no example was known of a compact, differentiable manifold, admitting almost complex structures, but no complex structure whatsoever. Now to begin with, we would like to point out here that many examples of compact 4-dimensional differentiable manifolds, admitting almost complex structures, but no complex structure, can be obtained by combining a theorem of Ch. Ehresmann and W. W. Wu (in a formulation due to F. Hirzebruch and H. Hopf) with certain results of K. Kodaira. These last results-and consequently the main theorems of this announcement-depend in an essential and up to now unavoidable way on the Atiyah-Singer Riemann-Roch theorem. As will be explained at the end of this note, to obtain specific examples we need, apart from the theorems of Ehresmann-Wu and Atiyah-Singer, only some very general facts about complex surfaces. But we can get slightly more. To explain this properly, we shall recall the definition of the Chern numbers of a compact almost complex manifold V2'. Let [V] be the fundamental cycle of V, ci the ith Chern class of V, and il,. . . , in nonnegative integers with i1 + 2i2 + . . . + ni4 = n. Then clil U c2 J2 U ... U cnin C H2n (V,Z), and the value of this cohomology class on [V] is an integer. The 7r(n) integers, obtained this way for all i1, i2, .. . in with il + 2i2 + . . . + nin = n are the Chern numbers of V. J. Milnorl has determined the sets of 7r(n) integers occurring as the Chern numbers of a not necessarily connected V. It turns out in particular that they are the same as those occurring as the Chern numbers of the complex or even the projective algebraic manifolds of the same dimension, but again not necessarily connected. Milnor's methods do not allow us to decide which set