Triangular de Rham cohomology of compact Kähler manifolds (Q2759300)

The authors study the de Rham 1-cohomology \(H^1_{DR} (M,G)\) of the smooth manifold \(M\) with values in a Lie group \(G\). If \(G\) is non-Abelian, \(H^1_{DR} (M,G)\) does not admit a natural structure of group. It is the quotient of the set of flat connections on the trivial principal bundle \(M\times G\) by the gauge equivalence. The authors consider the case where \(M\) is a compact Kähler manifold and \(G\) is a soluble complex linear algebraic group of a special class which contains the Borel subgroups of all complex classical groups. They obtain a description of \(H^1_{DR} (M,G)\) in terms of the 1-cohomology of \(M\) with values in the (Abelian) sheaves of flat Lie algebra bundles having as the fibre the Lie algebra of \(G\). Equivalently, they obtain some harmonic differential forms on \(M\) representing this cohomology.