Localization theory of Atiyah classes and its applications

(1)关于Atiyah类的局部化理论，与M.Abate，F.Bracci和F.Tovena合作，我们建立了以下基本理论：(1)适用于局部化理论的Atiyah类的简单定义，(2)Cech-Dolbeault上同调理论，(3)引入复解析Thom类，(4)用Atiyah形式证明Bott型消失定理。(2)关于向量丛同态的退化轨迹问题，与T.Ohmoto合作，我们开始尝试证明Thom-Porteous公式位于退化局部。这是通过构造Chern的Schur多项式的普适局部化来实现的。它是向量丛的Thom类的广泛推广。

(1) Concerning the localization theory of Atiyah classes, with collaboration of M. Abate, F. Bracci and F. Tovena, we established the following fundamental theories : (1) a simple definition of Atiyah classes suitable for the localization theory,(2) Cech-Dolbeault cohomology theory,(3) introduction of the complex analytic Thom class,(4) proof of a Bott type vanishing theorem in terms of Atiyah forms.(2) Concerning the degeneracy loci problem of a homomorphism of vector bundles, with collaboration of T. Ohmoto, we started to try to prove the Thom-Porteous formula localized at the degeneracy loci. This is done by constructing a universal localization of a Schur polynomial of Chern. It is a vast generalization of the Thom class of a vector bundle.