Compact hyper-K\"ahler categories

TL;DR

This paper introduces the concept of hyper-K"ahler categories, exploring their construction, deformation theory, and specific examples like non-commutative Hilbert schemes and categorical resolutions related to K3 surfaces.

Contribution

It defines hyper-K"ahler categories and investigates their construction methods, deformation properties, and detailed examples, advancing categorical approaches in algebraic geometry.

Findings

Hyper-K"ahler categories can be constructed via specific techniques.

Deformation theory of these categories is developed.

Examples include non-commutative Hilbert schemes and categorical resolutions.

Abstract

We define and study the notion of hyper-K\"ahler category. On the theoretical side, we focus on construction techniques and deformation theory of such categories. We also study in details some examples : non-commutative Hilbert schemes of points on a K3 surface and a categorical resolution of a relative compactified Prymian constructed by Markushevich and Tikhomirov.