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

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.