Stability conditions for curves

This talk will be an introduction to the space of Bridgeland stability conditions on a triangulated category, focussing on the case of the derived category of coherent sheaves on a curve. These spaces of stability conditions have their roots in physics, and have a mirror theoretic interpretation as moduli of complex structures on the mirror variety.

For curves of genus g > 0, we will see that any stability condition comes from the classical notion of slope stability for torsion-free sheaves. On the projective line we can study the more complicated behaviour via a derived equivalence to the derived category of modules over the Kronecker quiver.