Dolbeault theorem for sheaves of pseudoanalytic differential forms, an Hermitian structure on a holomorphic line bundle defined using pseudoanalytic functions

Author: irakli sikharulidze
Keywords: sheaf, pseudoanalytic, Dolbeault theorem, holomorphic, line bundle, connection, Hermitian structure
Annotation:

A generalization of the sheaf of holomorphic functions is introduced such that the component functions of so defined differential forms with respect to the basis induced by a local coordinate system satisfy Carleman-Bers-Vekua equations in each argument and so are pseudoanalytic functions; The question of whether the analogue of the Dolbeaut theorem holds for this sheaf is considered, the necessary and sufficient conditions for the sequence of sheaf homomorphisms whose kernels are the above-mentioned sheaves to constitute a complex are given and the question is reduced to the existence of a solution to a system of linear partial differential equations; in the second part a way of defining an Hermitian structure for a holomorphic line bundle equipped with a connection compatible with the complex structure is given such that the connection is Hermitian with respect to the so defined Hermitian structure.

Lecture files:

სამეცნიერო კვლევითი პროექტი 1 [ka]