Compact Hyper-Kähler category

Along with complex tori and Calabi-Yau manifolds, compact hyper-K\"ahler 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-K\"ahler 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-K\"ahler manifold has been constructed 15 years ago (O'Grady).

In this talk, I will introduce the notion of compact hyper-K\"ahler 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-K\"ahler categories in each even dimension. If time permits, I will discuss a specific modular example, whose Hochschild cohomology ring reveals some very interesting properties.