Oberseminar WiSe 2009/10 Zusammenfassung des Vortrags von E. Markman (Amherst, USA)
The Beauville-Bogomolov class as a characteristic class
Let M be a compact hyper-Kahler manifold. The second cohomology of M admits an integral symmetric bilinear form, called the Beauville-Bogomolov pairing. This pairing can be interpreted as a Hodge class BB(M) in the fourth cohomology of M. Let M be deformation equivalent to the Hilbert scheme of n points on a K3 surface, n>3. Then c2(M) and BB(M) are linearly independent. We prove that BB(M) is a characteristic class. The proof involves a construction of a universal object E in the derived category of M ×M.
We end with brief comments about joint work in progress with Sukhendu Mehrotra. Consider the special case where M above is a moduli space of stable sheaves on a K3 surface S. We reconstruct the derived category of the surface S as a natural triangulated category C(M,E), associated to the above mentioned universal object E on M ×M. The deformability of the pair (M,E), in directions in which M is no longer a moduli space of sheaves on a K3 surface, may then be interpreted as non-commutative ``deformations'' of the derived category of S.
We expect the construction to generalize to the case of an abelian surface S and generalized Kummer variety M.