Roland Abuaf (IHES Paris) will speak in the geometry seminar on 8th December, 2015. His title is Hyperkähler categories and his abstract is below. Roland’s talk will be at 2pm, in the Salle de Profs on the 9th floor of building NO. The seminar that week will be a double bill, with Ljudmila Kamenova also speaking, on Kobayashi’s pseudometric of compact complex manifolds at 3.15pm in the same room.
Along with complex tori and Calabi-Yau manifolds, compact hyper-Käahler manifolds are building blocks for the decomposition of Ricci-flat manifolds. All complex tori are “classified”, while we know how to construct a large number of different deformation classes of CY manifolds in each complex dimension (bigger than 3). The situation for compact hyper-Kähler manifolds is quite different. We have absolutely no clue on how to classify them and we only know 2 infinite series and 2 sporadic examples of such manifolds. The last new example of compact hyper-Kähler manifold has been constructed more than 15 years ago (O’Grady).
In this talk, I will introduce the notion of compact hyper-Kähler categories, study some of their invariance properties and some construction techniques. In particular, I will show that one can construct a large number of different deformation classes of compact hyper-Kähler categories in each even dimension. If time permits, I will discuss a specific modular example, whose Hochschild cohomology ring reveals some very interesting properties.