Wall crossing and equivalences of derived categories

The homological mirror symmetry program leads to the surprising prediction that many geometric invariants of a variety are preserved under birational modifications which preserve the canonical class. The most general formulation is the D-equivalence conjecture, which predicts that such modifications do not effect the derived category of coherent sheaves of a smooth projective variety. In particular, birationally equivalent Calabi Yau manifolds are predicted to have equivalent derived categories. I will discuss a proof of the D-equivalence conjecture for Calabi-Yau manifolds which are birationally equivalent to a moduli space of sheaves on some K3 surface. This is the first case in which the D-equivalence conjecture has been established for a complete birational equivalence class of Calabi-Yau manifolds in dimension >3. The proof combines techniques in the theory of derived categories of equivariant sheaves with new general methods for analyzing moduli problems (Theta-stratifications and good moduli spaces).