Integrability of Koszul connections on complex vector bundles over domains in ${\mathbf C}^n$

TL;DR

This paper investigates the existence and regularity of solutions to a complex matrix differential equation related to Koszul connections on vector bundles over domains in complex Euclidean space, under certain geometric conditions.

Contribution

It establishes existence and regularity results for solutions to the integrability equation on specific types of domains in ^n, extending previous understanding of complex vector bundle connections.

Findings

Solutions exist with sharp regularity on strongly pseudoconvex domains.

Solutions exist with sharp regularity on domains with at least 3 negative Levi eigenvalues.

The study provides conditions under which the integrability equation admits solutions.

Abstract

We study invertible matrix solutions $A$ to the equation $A^{-1}\overline\partial A=\omega^{(0,1)}$ on a small open subset $U$ of the closure $\overline M$ of a domain $M\subset{\mathbf C}^n$, where $\omega^{(0,1)}$ is a matrix of $(0,1)$ forms on $\overline M$ satisfying the formal integrable condition $\overline{\partial}\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)}$. For a $C^2$ domain $M$ that is either strongly pseudoconvex or has at least $3$ negative Levi eigenvalues at a boundary point contained in $U$, we obtain existence and sharp regularity of the solutions.