De Rham and Dolbeault cohomology of solvmanifolds with local systems

Abstract

Let G be a simply connected solvable Lie group with a lattice and the Lie algebra g and a representation � : G ! GL(V�) whose restriction on the nilradical is unipotent. Consider the flat bundle Egiven by �. By using many characters {�} of G and many flat line bundles {E�} over G/, we show that an isomorphism M {�} H � (g,VV�) � M {E�} H � (G/ , EE�) This isomorphism is a generalization of the well-known fact:If G is nilpotent andis unipotent then, the isomorphism H � (g, V�) � H � (G/ ,E�) holds. By this result, we construct an explicit finite dimensional cochain complex which compute the cohomology H � (G/ ,E�) of solvmanifolds even if the isomorphism H � (g,V�) � H � (G/ ,E�) does not hold. For Dolbeault co- homology of complex parallelizable solvmanifolds, we also prove an analogue of the above isomorphism result which is a generalization of computations of Dolbeault cohomology of complex parallelizable nilmanifolds. By this iso- morphism, we construct an explicit finite dimensional cochain complex which compute the Dolbeault cohomology of complex parallelizable solvmanifolds.