Anna Fino

Dolbeault Cohomology of compact nilmanifolds

AbstractLetM=G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space

HyperKähler torsion structures invariant by nilpotent Lie groups

\mathcal{C}\left( \mathfrak{g} \right)

Ricci solitons and geometry of non-reductive homogeneous 4-spaces

of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification

On generalized Gauduchon metrics

\mathfrak{g}^\mathbb{C}

Hypercomplex eight-dimensional nilpotent Lie groups

of the Lie algebra ofG. to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.

The Calabi–Yau equation on 4-manifolds over 2-tori

Let M =Γ \G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.