Noncommutative complex differential geometry

TL;DR

This paper develops foundational concepts in noncommutative complex differential geometry, defining analogues of classical structures and comparing them to existing noncommutative algebraic geometry frameworks.

Contribution

It introduces noncommutative analogues of complex geometric structures and explores their basic properties, bridging differential and algebraic perspectives.

Findings

Defined noncommutative almost complex structures

Analyzed properties of noncommutative holomorphic curvature

Compared noncommutative differential structures with algebraic geometry

Abstract

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential graded algebra. This is compared to current ideas on noncommutative algebraic geometry.