Nilmanifold and Solvmanifold Techniques in Complex Geometry

Abstract: - Viviana del Barco (Université Paris-Saclay and UNR-CONICET): "Killing forms on nilpotent Lie groups"

Abstract. 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 $k$-forms are quite rare for $k\geq 2$. 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).

- Anna Fino (Università di Torino): "SKT metrics on nilmanifolds and solvmanifolds"

Abstact. An SKT (or pluriclosed) metric on a complex manifold is an Hermitian metric whose fundamental form is $\partial \overline \partial$-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.

- Hisashi Kasuya (Osaka University): "Results and problems on cohomology of solvmanifolds"

Abstract. 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.

- Sönke Rollenske (Philipps-Universität Marburg): "Dolbeault cohomology of complex nilmanifolds"

Abstract. By Nomizu's theorem, the de Rham cohomology of a compact nilmanifold $M=\Gamma \backslash G$ 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 $\Gamma$. 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.

complex variablesdifferential geometry

Audience: researchers in the topic

( chat )

Comments: The discussion is open at gitter.im/GTACOS-September2020/. The live discussion with the speakers for this series of talks will be held on September 15, see researchseminars.org/talk/tacos/5/

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Geometry and TACoS

Series comments: The idea of “Geometry & TACoS” is to have regular mathematical conferences on specific themes, related, in a broad sense, to Geometry and Topology of (Almost) Complex Structures. Each conference will consist of three blocks:

- A series of four lectures, by four speakers, which will be available on our YouTube channel at the beginning of each period.

- A discussion platform, available on Gitter, will be active for two weeks after the release of the videos. There, the participants will be able to interchange remarks, comments, and open questions concerning the topics of the four lectures, as well as interact with each other, the organizers and with the speakers (both in a public and a private way).

- The organizers will follow the development of the discussion and collect questions for the third part of the event, which is an informal coffee break (or TACoS break?) on Zoom, moderated by the organizers, during which the speakers may report on questions and comments that were asked in the chat rooms, and interact personally with the participants.

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