齐性空间的环面亏格及其纤维化
Toric genera of homogeneous spaces and their fibrations
摘要：
本文的目的是进一步研究紧致齐性空间的泛环面亏格及其齐性纤维化。考虑具有正欧拉示性数的齐性空间。在极大环面$T^k$的规范作用下，这类空间具有许多稳定的等变复结构。由于这种情况下的环面作用只有孤立的不动点，因此可以有效地应用泛环面亏格的局部化公式。在此基础上，我们证明了在齐性空间上，仅用表示论就可以得到与Hirzebruch亏格的刚性和乘法性有关的著名拓扑结果.在这方面的齐性$SU$ -空间，我们证明了著名的结果刚性的Krichever属。我们还证明了对于一大类稳定的复齐性空间，任何由奇幂级数给出的$T^k$ -等变Hirzebruch亏格为零.与问题的乘法性，我们提供了稳定的复杂的$T^k$ -纤维化的泛环面属是扭曲乘法的建设。我们证明了它总是扭曲乘法几乎复杂的齐次纤维化，并描述这些纤维化，它是乘法。因此，这种纤维化的刚性和乘法之间的强关系的等变Hirzebruch属建立。还考虑了基不允许任何稳定复结构的纤维化的泛环面亏格。这里我们计算泛环面亏格的主要例子是四元数射影空间上的齐次纤维化。
英文摘要：
The aim of this paper is to study further the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus $T^k$. As the torus action in this case has only isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In that context for homogeneous $SU$-spaces we prove the well known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any $T^k$-equivariant Hirzebruch genus given by an odd power series vanishes. Related to the problem of multiplicativity we provide construction of stable complex $T^k$-fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces.
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