**Sukhendu Mehrotra**

CMI, Chennai

January 31, 2013

**Hyperholomorphic sheaves and deformations of K3 surfaces**:
I shall report on ongoing joint work with Eyal Markman on generalized
deformations of K3 surfaces. Any K3 surface $X$ (for example, a smooth
quartic surface in projective space) varies in a 20-dimensional family. On the other hand, the Hilbert scheme $M$ which parametrizes subsets (subchemes) of $n$ points on $X$ is known to deform in a 21-dimensional family. This means that the general deformation of M is {\em not} the Hilbert scheme of any K3. How to describe this extra parameter worth of deformations? Our work suggests that they arise from certain ``non-commutative'' deformations of $X$.