BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Viviana del Barco\, Anna Fino\, Hisashi Kasuya\, Sönke Rollenske DTSTART:20200901T054500Z DTEND:20200901T064500Z DTSTAMP:20260423T024024Z UID:tacos/4 DESCRIPTION:Title: Ni lmanifold and Solvmanifold Techniques in Complex Geometry\nby Viviana del Barco\, Anna Fino\, Hisashi Kasuya\, Sönke Rollenske as part of Geome try and TACoS\n\n\nAbstract\n- Viviana del Barco (Université Paris-Saclay and UNR-CONICET): "Killing forms on nilpotent Lie groups"\n\nAbstract. Ki lling forms on Riemannian manifolds are differential forms whose covariant derivative with respect to the Levi-Civita connection is totally skew-sym metric. They generalize to higher degrees the concept of Killing vector fi elds.\nExamples of Riemannian manifolds with non-parallel Killing $k$-form s 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 o f 2-step nilpotent Lie groups endowed with left-invariant Riemannian metr ic and carrying non-trivial Killing forms. In the way\, we will review asp ects of the Riemannian geometry of nilpotent Lie groups endowed with left- invariant metrics and describe the methods to achieve the structure result s. The talk is based on joint works with Andrei Moroianu (CNRS\, France).\ n\n\n- Anna Fino (Università di Torino): "SKT metrics on nilmanifolds and solvmanifolds"\n\nAbstact. An SKT (or pluriclosed) metric on a complex m anifold is an Hermitian metric whose fundamental form is $\\partial \\over line \\partial$-closed.\nI will present some general results about SKT met rics on compact nilmanifolds and solvmanifolds\,  considering also the li nk with symplectic geometry and generalized Kähler geometry.\n\n\n- Hisas hi Kasuya (Osaka University): "Results and problems on cohomology of solvm anifolds"\n\nAbstract. One of the reasons why nilmanifolds and solvmanif olds provide many interesting examples for various geometries is that we c an compute cohomology of them well. The contents of my video talk are as f ollows:\n(1) I will give an overview of the study of de Rham and Dolbeault cohomology of nilmanifolds and solvmanifolds.\n(2) I will explain detail s of techniques of computing cohomology of solvmanifolds I constructed. \n(3) I will suggest an unsolved problem on Dolbeault cohomology of solvm anifolds with some observations on Oeljeklaus-Toma manifolds.\n\n\n- Sön ke Rollenske (Philipps-Universität Marburg): "Dolbeault cohomology of com plex nilmanifolds"\n\nAbstract. By Nomizu's theorem\, the de Rham cohomol ogy of a compact nilmanifold $M=\\Gamma \\backslash G$ can be represented by left-invariant  differential forms\, that is\, it can be computed fro m the Lie-algebra and does not depend on the lattice $\\Gamma$.\nIf M is e ndowed with a left-invariant complex structure J\, it is  natural to ask the same property for Dolbeault cohomology. I will sketch what is known a nd why\, from a practical point of view\, all relevant cases are already c overed.\nStarting from a key example\, I will explain\, why more recent ap proaches studying foliations instead of fibrations were neccessary to sett le the case of real dimension six.\n\nThe discussion is open at https://gi tter.im/GTACOS-September2020/. The live discussion with the speakers for t his series of talks will be held on September 15\, see https://researchsem inars.org/talk/tacos/5/\n LOCATION:https://researchseminars.org/talk/tacos/4/ END:VEVENT END:VCALENDAR