de Rham and Dolbeault cohomology of solvmanifolds with local systems.

*(English)*Zbl 1314.17010There have been many attempts to generalize to solvmanifolds those well known cohomology isomorphism results valid for nilmanifolds. But success has not been accomplished except in some particular cases. This paper proposes an innovative type of cohomology isomorphism theorems for solvmanifolds which allow to compute their de Rham and Dolbeault cohomology in general.

K. Nomizu proved in [Ann. Math. (2) 59, 531–538 (1954; Zbl 0058.02202)] for simply connected nilpotent Lie groups \(N\) admitting a lattice \(\Gamma\) that there is a cohomology isomorphism \(H^*(\mathfrak n, V_\rho)\simeq H^*(N/\Gamma,E_\rho)\), whenever \(\rho\) is a unipontent representation of \(N\) on \(V_\rho\) and \(E_\rho\) is the flat bundle \(E_\rho=(N\times V_\rho)/\Gamma\).

For a solvable Lie group \(G\) and a representation \(\rho\) of \(G\) on \(V_\rho\), the isomorphism above does not hold in general so the computation of the cohomology \(H^*(G/\Gamma,E_\rho)\) is complicated. Several authors worked on sufficient conditions on \(G\) and \(\rho\) in order obtain the isomorphism \(H^*(\mathfrak g, V_\rho)\simeq H^*(G/\Gamma,E_\rho)\), as for instance when \(\rho\oplus Ad\) is triangular.

The innovative approach of this paper is to construct an explicit finite-dimensional cochain complex which allows to compute the de Rham cohomology \(H^*(G/\Gamma,E_\rho)\), even in the cases where the isomorphism above does not hold. The Doulbeault cohomology is also treated in this paper with a similar spirit.

K. Nomizu proved in [Ann. Math. (2) 59, 531–538 (1954; Zbl 0058.02202)] for simply connected nilpotent Lie groups \(N\) admitting a lattice \(\Gamma\) that there is a cohomology isomorphism \(H^*(\mathfrak n, V_\rho)\simeq H^*(N/\Gamma,E_\rho)\), whenever \(\rho\) is a unipontent representation of \(N\) on \(V_\rho\) and \(E_\rho\) is the flat bundle \(E_\rho=(N\times V_\rho)/\Gamma\).

For a solvable Lie group \(G\) and a representation \(\rho\) of \(G\) on \(V_\rho\), the isomorphism above does not hold in general so the computation of the cohomology \(H^*(G/\Gamma,E_\rho)\) is complicated. Several authors worked on sufficient conditions on \(G\) and \(\rho\) in order obtain the isomorphism \(H^*(\mathfrak g, V_\rho)\simeq H^*(G/\Gamma,E_\rho)\), as for instance when \(\rho\oplus Ad\) is triangular.

The innovative approach of this paper is to construct an explicit finite-dimensional cochain complex which allows to compute the de Rham cohomology \(H^*(G/\Gamma,E_\rho)\), even in the cases where the isomorphism above does not hold. The Doulbeault cohomology is also treated in this paper with a similar spirit.

Reviewer: Viviana del Barco (Rosario)

##### MSC:

17B56 | Cohomology of Lie (super)algebras |

22E25 | Nilpotent and solvable Lie groups |

32C35 | Analytic sheaves and cohomology groups |

58A12 | de Rham theory in global analysis |