Residues of holomorphic actions on coherent sheaves and sections of vector bundles

Kam Cheung Kong, Purdue University

Abstract

This thesis mainly deals with the residues of holomorphic actions on coherent sheaves and the residues of sections of holomorphic vector bundles. In both the situations, the existence of residues is a consequence of the vanishing lemma which is, in its turns, a consequence of some special connection outside the singular sets. We explain the general theory, prove some properties of the residue numbers and study some examples. For a holomorphic vector field defined on the total space of a line bundle over a compact 1-manifold, if the vector field vanishes to a fixed order on the zero section of the bundle; then we can calculate a residue formula which is in terms of the infinitesimal direction of X. The concept of the infinitesimal direction is generalized to a bundle map. We study the relation between the infinitesimal bundle map and the splitting of exact sequence of holomorphic vector bundles.

Degree

Ph.D.

Advisors

Yue-Lin, Purdue University.

Subject Area

Mathematics

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