Complex structures on Three-point space

TL;DR

This paper explores complex and Kähler structures within non-commutative geometry on a finite three-point space, classifying all almost complex structures and analyzing their properties.

Contribution

It classifies all almost complex structures on a non-commutative three-point space and examines their relation to complex and Kähler structures.

Findings

All almost complex structures are also complex structures.

None of the structures are Kähler in the defined sense.

Provides a classification of structures on this non-commutative manifold.

Abstract

We discuss notions of almost complex, complex and K\"{a}hler structures in the realm of non-commutative geometry and investigate them for a class of finite dimensional spectral triples on the three-point space. We classify all the almost complex structures on this non-commutative manifold, which also turn out to be complex structures, but none of them are K\"{a}hler in our sense.