Elliptic genera, torus manifolds and multi-fans
Elliptic genera, torus manifolds and multi-fans
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椭圆属、环面流形和多扇形
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
M. Masuda
中科院分区：
文献类型：
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作者：
A. Hattori;M. Masuda
Witten-Bott-Taubes-Hirzebruch的刚性定理告诉我们，如果圈群作用在闭的几乎复（或更一般的酉）流形上，其第一个陈类可被大于1的正整数N整除，则其水平N的等变椭圆亏格是刚性的。应用这一点到一个非奇异紧环面簇，我们看到，它的椭圆亏格的水平N是刚性的，如果它的第一陈类是由N整除。但是，利用希泽布鲁赫的消失定理，我们还可以证明亏格实际上是消失的。本文将把这个结果推广到环面流形。事实上，一个非奇异的完全多重扇是与一个环面流形相联系的，并且对于非奇异的完全多重扇，可以公式化并证明其N级椭圆亏格的刚性和消失。给出了第一类可被大正整数整除的紧非奇异环面簇的一些应用。
The rigidity theorem of Witten-Bott-Taubes-Hirzebruch tells us that, if the circle group acts on a closed almost complex (or more generally unitary) manifold whose first Chern class is divisible by a positive integer N greater than 1, then its equivariant elliptic genus of level N is rigid. Applying this to a non-singular compact toric variety, we see that its elliptic genus of level N is rigid if its first Chern class is divisible by N. But, using a vanishing theorem of Hirzebruch, we can show moreover that the genus actually vanishes. In this note we shall extend this result to torus manifolds. In fact, a non-singular complete multi-fan is associated with a torus manifold, and rigidity and vanishing of elliptic genus of level N can be formulated and proved for non-singular complete multi-fans. Some applications on compact non-singular toric varieties with the first Chern class divisible by a large positive integer are given.