Stability conditions and exceptional collections

Barbara Bolognese

28-10-2021 - 14:15
Largo San Leonardo Murialdo,1


Bridgeland stability conditions have proved to be an extremely versatile tool to study the birational geometry of classical moduli spaces of sheaves, to solve counting problems, to help define new structure in mirror symmetry. After a gentle introduction, we will focus on the relation between stability conditions and exceptional collections.
A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. In general, proving the fullness of such a collection is a hard problem, often done on a case-by-case basis, with the aid of a deep understanding of the underlying geometry. Likewise, when an exceptional collection is not full, it is not straightforward to determine whether its residual category is the derived category of a variety. 
Taking after Bondal and Orlov, we examine two cases: the case of quadric hypersurfaces in P^{n+1} and the case of the index 2 Fano threefold Y (the generic intersection of two quadrics in P^5. In the first case, we prove that the classical result by Kapranov on the fullness of the standard exceptions is equivalent to the existence of a numerical stability condition on the residual category of the exceptional collection of the quadric. In the second case, we show how the same technique recovers the equivalence of the residual category of the exceptional collection {O_Y,O_Y(1)} with the derived category of a genus 2 curve. This is joint work with Domenico Fiorenza. Moreover, we will introduce ideas on methods to construct stability conditions via exceptional collections and hearts of finite length from a joint work in progress with Domenico Fiorenza and Alex Küronya.

org: Turchet Amos