# Giorgio Scattareggia: An obstruction theory for the moduli spaces of coherent systems

**Time: **
Wed 2019-03-06 13.15 - 15.00

**Location: **
Room 306, House 6, Kräftriket, Department of Mathematics, Stockholm University

**Participating: **
Giorgio Scattareggia (Bergen)

Abstract: Informally, a perfect obstruction theory for a moduli space *M* is a perfect complex in the derived category of *M* which encodes all the information about the infinitesimal properties of *M*. In particular, if a moduli space *M* admits a perfect obstruction theory of rank *r*, then it has expected dimension *r*, i.e., at every point of *M* the dimension of the tangent space minus the dimension of the obstruction space is equal to *r*. On the other hand, whenever a moduli space has expected dimension *r*, one suspects that it has an obstruction theory of rank *r*.

A coherent system on a curve *C* is a pair *(E, V)*, where *E* is a finite rank vector bundle on *C* and *V* is a linear subspace of the space of global sections of *E*. In other words, coherent systems are the generalization of linear series for higher rank vector bundles. One can define a condition of (semi-)stability for coherent systems and use it to construct their moduli spaces as coarse GIT quotients; the stability condition depends on the choice of a real parameter and it differs from the stability condition of the bundle *E*. The deformation theory of the moduli spaces of stable coherent systems is known; in particular, such moduli spaces have an expected dimension which depends on the rank and the degree of the bundles *E* and on the dimension of the vector spaces *V*.

In this talk we briefly recall the definition of obstruction theory and we sketch the construction of a perfect obstruction theory for the moduli spaces of stable coherent systems.