Sergio Console

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

Curvature invariants, Killing vector fields, connections and cohomogeneity

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

Level sets of scalar Weyl invariants and cohomogeneity

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

On index number and topology of flag manifolds

\mathfrak{g}^\mathbb{C}

Second Stiefel‐Whitney class and spin structures on flat manifolds of diagonal type

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.

Submanifolds and Holonomy

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.