Sasakian流形的基本Dolbeault上同调的刚性与消失
Rigidity and vanishing of basic Dolbeault cohomology of Sasakian manifolds
摘要：
Sasakian流形M的基本Dolbeault上同调群是其特征叶理F（Reeb流的轨道叶理）的不变量。我们展示了这种上同调的一些基本性质，这些性质对它的计算是有用的。在文章的第一部分中，我们证明了基本Hodge数，即基本Dolbeault上同调群的维数，只依赖于其基础CR结构的同构类。等价地，我们证明了基本Hodge数在I型变形下是不变的。这一结果减少了他们的计算准正规的情况下。在第二部分中，我们给出了Carrell-Lieberman定理的一个基本版本，该定理将（M，F）的基本Dolbeault上同调与（C，F）的基本Dolbeault上同调联系起来，其中C是F的闭叶的并。作为一种特殊情况，如果F只有100个闭叶，则当p不等于q时，第（p，q）基本Hodge数为零。结合这两个结果，我们证明了：如果M允许一个无处为零的CR向量场，且该CR向量场具有n个闭轨道，则当p不等于q时，（p，q）阶基本Hodge数为零.作为这些结果的应用，我们计算齐次Sasakian流形的变形的基本Hodge数。
英文摘要：
The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, the dimensions of the basic Dolbeault cohomology groups, only depend on the isomorphism class of the underlying CR structure. Equivalently, we show that the basic Hodge numbers are invariant under deformations of type I. This result reduces their computation to the quasi-regular case. In the second part, we show a basic version of the Carrell-Lieberman theorem relating the basic Dolbeault cohomology of (M,F) to that of (C,F), where C is the union of closed leaves of F. As a special case, if F has only finitely many closed leaves, then the (p,q)-th basic Hodge numbers are zero for p not equal to q. Combining the two results, we show that if M admits a nowhere vanishing CR vector field with finitely many closed orbits, then the (p,q)-th basic Hodge numbers are zero for p not equal to q. As an application of these results, we compute then the basic Hodge numbers for deformations of homogeneous Sasakian manifolds.
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