D-Branes, Pi-Stability and Theta-Stability

Abstract

We investigate some aspects of Pi-stability of D-branes on Calabi-Yau threefolds in cases where there is a point in moduli space where the grades nearly or completely align. We prove that an example of complete alignment is the case of a collapsed del Pezzo surface. It is shown that there is an open neighbourhood of such a point for which Pi-stability reduces to theta-stability of quiver representations. This should be contrasted to the case of the large radius limit where mu-stability of sheaves cannot be extended out over an open neighbourhood.