Abstract: Dolbeault cohomology of holomorphic vector bundles over compact K\"ahler manifolds have many nice properties (such as Kodaira vanishing theorem, holomorphic Morse inequalities, Guillemin-Sternberg "quantization commutes with reduction" property) which in general don't hold for non-compact manifolds. In the talk I will present a new normalization construction of cohomology of equivariant vector bundles over a non-compact Kahler manifold endowed with an action of a compact Lie group $G$. The new cohomology, which is called the "background cohomology", has many properties of Dolbeault cohomology of a compact manifold. It is also related to the index theory for non-compact $G$-manifolds, which was developed several years ago by P.-E. Paradan and myself.
This talk will be accessible to graduate students.