|Location:||MSRI: Simons Auditorium|
The classical slope (degree/rank) of a vector bundle or torsion-free sheaf
gives a "codimension one" notion of stability that can be used to prove the first case
of Kodaira vanishing (Hom(L,O_X) = 0 for an ample line bundle L on a projective variety X).
A "codimension two" family of slopes in the derived category of a smooth projective
variety X can be used to similarly prove the next case of Kodaira vanishing (Hom(L,O) = 0).
I will talk about this, the Reider's theorem analogue when X = S is a surface, and then
irresponsibly speculate about what might be going on in general.