Abstract:
(joint works with Tarig Abdelgadir and Kazushi Ueda). It is known that the category of coherent sheaves on certain varieties, such as projective spaces and del Pezzo surfaces, are derived equivalent to the category of representations of certain quivers with relations. When we allow the variety to deform, even into the noncommutative directions, the relations of the quiver also deform correspondingly (with the quiver unchanged). This allows us to construct `approximate' compactified moduli space of noncommutative deformations of such varieties as the moduli space of relations of the corresponding quiver. In this talk I will explain the generality of this construction and what we know so far in the case of del Pezzo surfaces.