arXiv 1202.34222012-02-15math.SGmath.AG

射影空间辛丛上的环面结构

Toric Structures on Symplectic Bundles of Projective Spaces

Andrew Fanoe

备注：: 26 pages, one figure

摘要：

最近，Karshon，Kessler和Pinsonnault，Borisov和McDuff的工作表明，一个给定的辛流形$(M,ω)$有有限个不同的环面结构。此外，McDuff还证明了两个射影空间的乘积$\bC P^r\times \bC P^s$具有任何给定的辛形式，只要$r,s\geq 2$具有唯一的环面结构.相反，乘积$\bC P^r \times \bC P^1$可以被赋予无穷多个不同的复曲面结构，尽管其中只有有限数量的复曲面结构与每个给定的辛形式$ω$相容。在本文中，我们推广了这些结果考虑可能的环面辛流形$(M,ω)$与$\dim H^2(M)=2$的环面结构。特别地，对于某些$r,s$，所有这样的流形都是$\bC P^s$上的$\bC P^r$丛。我们表明，有一个独特的复曲面结构，如果$r<s$，也是，如果$r,s\geq 2$，那么$M$有至多1000个不同的复曲面结构，是兼容的任何辛结构的$M$。因此，在这种情况下，有限性结果并不依赖于固定的辛结构。我们还将给出$(M,ω)$具有唯一复曲面结构的其他示例，例如$(M,ω)$是单调的情况。

英文摘要：

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,ω)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\bC P^r\times \bC P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product $\bC P^r \times \bC P^1$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $ω$. In this paper we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,ω)$ with $\dim H^2(M)=2$. In particular, all such manifolds are $\bC P^r$ bundles over $\bC P^s$ for some $r,s$. We show that there is a unique toric structure if $r<s$, and also that if $r,s\geq 2$ then $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,ω)$ has a unique toric structure, such as the case where $(M,ω)$ is monotone.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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