Triangular de Rham Cohomology of Compact Kahler Manifolds

TL;DR

This paper investigates the de Rham 1-cohomology of compact Kähler manifolds with values in certain solvable complex Lie groups, providing a description in terms of harmonic forms and flat sections.

Contribution

It offers a novel description of the de Rham 1-cohomology for specific Lie groups on compact Kähler manifolds, linking it to harmonic forms and flat sections.

Findings

Cohomology described via harmonic forms.

Applicable to solvable complex algebraic groups.

Provides explicit cohomology characterization.

Abstract

We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principle bundle $M\times G$ by the so-called gauge equivalence. We consider the case when M is a compact K\"ahler manifold and G is a solvable complex linear algebraic group of a special class which contains the Borel subgroups of all complex classical groups and, in particular, the group $T_n(\Bbb C)$ of all triangular matrices. In this case, we get a description of the set H^1_{DR}(M,G) in terms of the 1-cohomology of M with values in the (abelian) sheaves of flat sections of certain flat Lie algebra bundles with fibre $\frak g$ (the Lie algebra of G) or, equivalently, in terms of the harmonic forms on M representing this cohomology.