Techniques of computations of Dolbeault cohomology of solvmanifolds.

*(English)*Zbl 1261.22009The computation of the Dolbeault cohomology of nilmanifolds and solvmanifolds endowed with an invariant complex structure by means of Lie algebra cohomology is still an open problem. For nilmanifolds (i.e., compact quotients \(G / \Gamma\) of a nilpotent Lie group \(G\) by a lattice \(\Gamma\)) endowed with an invariant complex structure, it is an open conjecture which holds for wide classes of nilmanifolds and no counterexample is known (for more details, see, e.g., [S. Rollenske, W. Ebeling (ed.) et al., Complex and differential geometry. Berlin: Springer. Springer Proceedings in Mathematics 8, 369–392 (2011; Zbl 1266.53066)]).

The answer to the analog question in the case of the de Rham cohomology is positive for nilmanifolds by a result of K. Nomizu [Ann. Math. (2) 59, 531–538 (1954; Zbl 0058.02202)], but it turns out to be more involved for solvmanifolds (i.e., compact quotients \(G / \Gamma\) of solvable Lie groups \(G\) by lattices \(\Gamma\)). Indeed for solvmanifolds it is not true in general that the de Rham cohomology is given by the one of the corresponding Lie algebra.

The author deals with the computation of the Dolbeault cohomology of a certain class of solvmanifolds, namely, he considers solvmanifolds \(G/\Gamma\), where \(G={\mathbb C}^{n}\ltimes _{\phi}N\) such that

(1) \(N\) is a simply connected nilpotent Lie group with a left-invariant complex structure \(J\);

(2) for any \(t\in {\mathbb C}^{n}\), \(\phi(t)\) is a holomorphic automorphism of \((N,J)\);

(3) \(\phi\) induces a semi-simple action on the Lie algebra of \(N\).

He assumes that \(G\) has a lattice \(\Gamma=\Gamma^{\prime}\ltimes_{\phi}\Gamma^{\prime\prime}\) such that \(\Gamma^{\prime}\) and \(\Gamma^{\prime\prime}\) are lattices of \({\mathbb C}^{n}\) and \(N\), respectively, and that the inclusion \(\bigwedge^{p,q}{\mathfrak n}^{\ast}\subset A^{p,q}(N/\Gamma^{\prime\prime})\) induces an isomorphism \(H^{p,q}_{\bar\partial}(\bigwedge^{p,q}{\mathfrak n}^{\ast})\cong H^{p,q}_{\bar\partial }(N/\Gamma^{\prime\prime})\) for the nilmanifold \(N/\Gamma^{\prime\prime}\), i.e., one can apply the above mentioned results on the Dolbeault cohomology of nilmanifolds. Then, he proves that one can compute the Dolbeault cohomology \(H^{p,q}_{\bar\partial}(G/\Gamma)\) of \(G/\Gamma\) by using finite dimensional cochain complexes.

As an application, he considers some examples, observing that, for some solvmanifold, the Dolbeault cohomology depends on the choice of lattices \(\Gamma\). Moreover, he gives examples of non-Kähler complex solvmanifolds satisfying the Hodge symmetry.

The answer to the analog question in the case of the de Rham cohomology is positive for nilmanifolds by a result of K. Nomizu [Ann. Math. (2) 59, 531–538 (1954; Zbl 0058.02202)], but it turns out to be more involved for solvmanifolds (i.e., compact quotients \(G / \Gamma\) of solvable Lie groups \(G\) by lattices \(\Gamma\)). Indeed for solvmanifolds it is not true in general that the de Rham cohomology is given by the one of the corresponding Lie algebra.

The author deals with the computation of the Dolbeault cohomology of a certain class of solvmanifolds, namely, he considers solvmanifolds \(G/\Gamma\), where \(G={\mathbb C}^{n}\ltimes _{\phi}N\) such that

(1) \(N\) is a simply connected nilpotent Lie group with a left-invariant complex structure \(J\);

(2) for any \(t\in {\mathbb C}^{n}\), \(\phi(t)\) is a holomorphic automorphism of \((N,J)\);

(3) \(\phi\) induces a semi-simple action on the Lie algebra of \(N\).

He assumes that \(G\) has a lattice \(\Gamma=\Gamma^{\prime}\ltimes_{\phi}\Gamma^{\prime\prime}\) such that \(\Gamma^{\prime}\) and \(\Gamma^{\prime\prime}\) are lattices of \({\mathbb C}^{n}\) and \(N\), respectively, and that the inclusion \(\bigwedge^{p,q}{\mathfrak n}^{\ast}\subset A^{p,q}(N/\Gamma^{\prime\prime})\) induces an isomorphism \(H^{p,q}_{\bar\partial}(\bigwedge^{p,q}{\mathfrak n}^{\ast})\cong H^{p,q}_{\bar\partial }(N/\Gamma^{\prime\prime})\) for the nilmanifold \(N/\Gamma^{\prime\prime}\), i.e., one can apply the above mentioned results on the Dolbeault cohomology of nilmanifolds. Then, he proves that one can compute the Dolbeault cohomology \(H^{p,q}_{\bar\partial}(G/\Gamma)\) of \(G/\Gamma\) by using finite dimensional cochain complexes.

As an application, he considers some examples, observing that, for some solvmanifold, the Dolbeault cohomology depends on the choice of lattices \(\Gamma\). Moreover, he gives examples of non-Kähler complex solvmanifolds satisfying the Hodge symmetry.

Reviewer: Sergio Console (Torino)

##### MSC:

22E25 | Nilpotent and solvable Lie groups |

53C30 | Differential geometry of homogeneous manifolds |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

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