Nilmanifold and Solvmanifold Techniques in Complex Geometry

Killing forms on Riemannian manifolds are differential forms whose covariant derivative with respect to the Levi-Civita connection is totally skew-symmetric. They generalize to higher degrees the concept of Killing vector fields.

Examples of Riemannian manifolds with non-parallel Killing -forms are quite rare for . Nevertheless they appear, for instance, on nearly-K\”ahler manifolds, round spheres and Sasakian manifolds. The aim of this talk is to introduce recent results regarding the structure of 2-step nilpotent Lie groups endowed with left-invariant Riemannian metric and carrying non-trivial Killing forms. In the way, we will review aspects of the Riemannian geometry of nilpotent Lie groups endowed with left-invariant metrics and describe the methods to achieve the structure results. The talk is based on joint works with Andrei Moroianu (CNRS, France).

An SKT (or pluriclosed) metric on a complex manifold is an Hermitian metric whose fundamental form is dd^c-closed.

I will present some general results about SKT metrics on compact nilmanifolds and solvmanifolds,  considering also the link with symplectic geometry and generalized Kähler geometry.

One of the reasons why nilmanifolds and solvmanifolds provide many interesting examples for various geometries is that we can compute cohomology of them well. The contents of my video talk are as follows:

(1) I will give an overview of the study of de Rham and Dolbeault cohomology of nilmanifolds and solvmanifolds.

(2) I will explain details of techniques of computing cohomology of solvmanifolds I constructed.

(3) I will suggest an unsolved problem on Dolbeault cohomology of solvmanifolds with some observations on Oeljeklaus-Toma manifolds.

By Nomizu’s theorem, the de Rham cohomology of a compact nilmanifold can be represented by left-invariant differential forms, that is, it can be computed from the Lie-algebra and does not depend on the lattice.

If M is endowed with a left-invariant complex structure J, it is  natural to ask the same property for Dolbeault cohomology. I will sketch what is known and why, from a practical point of view, all relevant cases are already covered.

Starting from a key example, I will explain, why more recent approaches studying foliations instead of fibrations were neccessary to settle the case of real dimension six.