Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact K¨ahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress.In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C_1~n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that,under bounded curvature conditions, C_1~n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on K¨ahler manifolds with minimal volume growth.     

Classes de Chern Des Ensembles Analytiques

Let V be a compact complex analytic subset of a non-singular holomorphic manifold M. Assume that V has pure complex dimension n. Denote by V₀ its regular part, and by [V] its fundamental class in H_2n(V; ). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*) (V) of the virtual tangent bundle T_vir(V):=[TM|_V - N_V] in the K-theory K⁰(V). This has applications

Kähler–de Rham Cohomology and Chern Classes

Constraints on the Chern classes of a vector bundle on a possibly singular algebraic variety are found, which are stronger than the obvious Hodge theoretic constraints. This is done by showing that these lift to Chern classes in the hypercohomology of the complex of Kähler differentials.     

非定域变换的广义Ward恒等式

基于高阶微商奇异拉氏量系统相空间Green函数的生成泛函,导出了该系统在定域和非定域变换下的广义正则Ward恒等式.对规范不变系统,从位形空间生成泛函出发,导出了该系统在定域、非定域和整体变换下的广义Ward恒等式.用于高阶微商非Abel(Chern-Simons CS)理论,无需作出生成泛函中对正则动量的路径积分,即可导出正规顶角的某些关系.此外还给出了BRS变换下的Ward-Takahashi恒等式.