Vanishing of the first Dolbeault cohomology group of line bundles on complete intersections in infinite -dimensional projective space
Abstract
This dissertation is concerned with the first Dolbeault cohomology groups of line bundles on a submanifold X of finite codimension in infinite-dimensional projective space. The ∂-equation on X is reduced to a number of ∂-equations on a projective subspace of finite codimension via the method of projection from a submanifold to a projective subspace. Since the first Dolbeault cohomology group of any line bundle on infinite-dimensional projective space vanishes we obtain local solutions of the ∂-equation on X. The local solutions yield a holomorphic Čech 1-cocycle on X of a special kind. We show that if X is a complete intersection (for example a hypersurface) in infinite-dimensional projective space that admits smooth partitions of unity then the Cousin problem can be solved for this cocycle. The conclusion is that the first Dolbeault cohomology group of any line bundle on a complete intersection vanishes provided that the ambient infinite-dimensional projective space admits smooth partitions of unity.
Degree
Ph.D.
Advisors
Lempert, Purdue University.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.