$\Bbb C^n$、复流形和凸多胞体中的实二次曲面
Real quadrics in $\Bbb C^n$, complex manifolds and convex polytopes
摘要：
本文研究了一类非Kähler紧复流形的拓扑结构，推广了Hopf和Calabi-Eckmann流形的拓扑结构。这些流形与$\Bbb C^n$中的实二次曲面的特殊系统是同构的，这些系统对于实环面$(\Bbb S^1)^n$到$\Bbb C^n$上的自然作用是不变的。商空间是一个简单的凸多胞体。这个问题减少，从而研究的拓扑结构的某些真正的代数集，并可以处理使用组合的结果对凸多胞体。我们证明了这些紧致复流形的同调群可以具有任意的挠率，从而使得它们的拓扑极其丰富。通过引入与简单凸多胞体的某些变换相关的全纯等变初等运算，我们还解决了一个相关的跨壁问题。最后，作为一个很好的结果，我们得到了仿射非Kähler紧复流形在其同调群中可以有任意量的挠率，与Kähler情形相反.
英文摘要：
In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics in $\Bbb C^n$ which are invariant with respect to the natural action of the real torus $(\Bbb S^1)^n$ onto $\Bbb C^n$. The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.
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