Triangular De Rham Cohomology of Compact Kähler Manifolds

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 principal bundle M × G by the so-called gauge equivalence. We consider the case when M is a compact Kähler 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 (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 g (the Lie algebra of G) or, equivalently, in terms of the harmonic forms on M representing this cohomology.