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.

中文翻译：

---

Bott-Chern类和Hermitian残基的本地化

我们发展了Čech‐Bott‐Chern同调理论，在这种情况下，我们自然会提出相对的Bott‐Chern同调论。实际上，Bott-Chern同调有两个亲戚，它们都来自一个单一的复合体。因此，我们以统一的方式研究了这三个同调，并获得了涉及这三个的长序列。然后，我们研究相对Bott-Chern同调学中特征类的定位问题。为此，我们在框架中定义杯子产品和集成，并讨论局部和全局对偶态射影。在复习了有关连接的一些资料后，我们给出了与我们的本地化有关的消失定理。借助这些，我们证明了向量束的残差定理，该向量束允许厄米连接与奇异分布的非奇异部分的作用兼容。作为典型案例