Speaker: Cristian Martinez

Abstract: Stability conditions have become one of the modern tools to study geometric properties of moduli spaces of sheaves.

 

Given a chern character v and an ample class H on a smooth projective complex surface, there is a distinguished open set of stability conditions so that the only semistable objects of type v are coherent sheaves that are Gieseker semistable with respect to H. Moving away from this chamber to its boundary corresponds to a contraction of the Gieseker moduli. This, for instance, accounts for all smooth MMPs on surfaces. One of the key ingredients in the construction of stability conditions on surfaces is the existence of an inequality on the Chern classes of a semistable sheaf (the Bogomolov-Gieseker inequality). On some threefolds a generalized inequality is satisfied by a class of “semistable” complexes, allowing for the construction of stability conditions. In this talk I will explore some of these ideas and show a class of stable complexes violating the generalized Bogomolov-Gieseker inequality on blow-ups of smooth threefolds.