Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups

Abstract

It is conjectured that the Dolbeault cohomology of a complex nilmanifold X is computed by left-invariant forms. We prove this under the assumption that X is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six.Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

On the structure of double complexes

“…In many (and conjecturally all) cases [16], the inclusion A inv X ⊆ A X is an E 1 -isomorphism. In particular, this holds in complex dimensions up to 3 [18].…”

Vertical–horizontal decomposition of Laplacians and cohomologies of manifolds with trivial tangent bundles

“…Sönke Rollenske is grateful to the other authors for the invitation to join the project at a relatively late stage. He is also grateful to A. Fino and J. Ruppenthal for many discussions about the Dolbeault cohomology of nilmanifolds that culminated in the paper [13]. We are also pleased to thank the anonymous referee for valuable remarks and suggestions for a better presentation of our results.…”

“…In complex case, the J -compatible ascending series may not give a torus fibration resolution (see Example 3.6 in [23] or [13]). But our main theorem implies the following result.…”

“…The above result applies in a number of important cases. A different way to generalize the weak version of Cordero-Fernández-Gray-Ugarte's theorem was considered in [13] and like in loc. cit.…”

Locally conformally Kähler solvmanifolds: a survey

“…Also, the first example of a compact symplectic manifold that does not admit any Kähler metric happens to be a nilmanifold, the well known Kodaira-Thurston manifold, which is a primary Kodaira surface ( [58,90]). It is well known that the de Rham cohomology of nilmanifolds and completely solvable solvmanifolds can be computed in terms of invariant forms (according to Nomizu [70] and Hattori [52], respectively), and it is conjectured that the Dolbeault cohomology of nilmanifolds can also be computed using invariant forms (this conjecture has been proved for several particular cases, see [32,30,83,39]). In [18] it was shown that any nilmanifold with an invariant complex structure has holomorphically trivial canonical bundle, while it is known that this does not happen generally for solvmanifolds (see [38]).…”