Localization of Bott-Chern classes and Hermitian residues
Localization of Bott-Chern classes and Hermitian residues
复制标题
Bott-Chern 类和 Hermitian 留数的本地化
DOI：
10.1112/jlms.12273
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Mauricio Correa Jr and Tatsuo Suwa
中科院分区：
文献类型：
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作者：
Mauricio Correa Jr and Tatsuo Suwa
我们发展了Č的ECH-伯特-陈上同调理论，并在此背景下自然地提出了相对的伯特-陈上同调。事实上，Bott-Chern上同源有两个亲戚，他们都来自一个单一的复合体。因此，我们统一地研究了这三个上同调，得到了一个涉及这三个上同调的长的精确序列。然后，我们研究了相对Bott-Chern上同调中特征类的局部化问题。为此，我们在我们的框架中定义了CUP乘积和积分，并讨论了局部和全局对偶态射。在回顾了一些关于联络的资料之后，我们给出了一个与我们的局部化有关的消失定理。在此基础上，我们证明了允许厄米连接与奇异分布的非奇异部分的作用相容的向量丛的留数定理。作为一个典型的例子，我们讨论了分布在不变子流形的法丛上的作用(所谓的Camacho-Sad作用)，并给出了一个具体的例子。
We develop a theory of Čech‐Bott‐Chern cohomology and in this context we naturally come up with the relative Bott‐Chern cohomology. In fact, Bott‐Chern cohomology has two relatives and they all arise from a single complex. Thus, we study these three cohomologies in a unified way and obtain a long exact sequence involving the three. We then study the localization problem of characteristic classes in the relative Bott‐Chern cohomology. For this, we define the cup product and integration in our framework and we discuss local and global duality morphisms. After reviewing some materials on connections, we give a vanishing theorem relevant to our localization. With these, we prove a residue theorem for vector bundles admitting a Hermitian connection compatible with an action of the non‐singular part of a singular distribution. As a typical case, we discuss the action of a distribution on the normal bundle of an invariant submanifold (the so‐called Camacho–Sad action) and give a specific example.