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.