Derived Theta-stratifications and the D-equivalence conjecture

Abstract: The D-equivalence conjecture predicts that birationally equivalent projective Calabi-Yau manifolds have equivalent derived categories of coherent sheaves. It is motivated by homological mirror symmetry, and has inspired a lot of recent work on connections between birational geometry and derived categories. In dimension 3, the conjecture is settled, but little is known in higher dimensions. I will discuss a proof of this conjecture for the class of Calab-Yau manifolds that are birationally equivalent to some moduli space of stable sheaves on a K3 surface. This is the only class for which the conjecture is known in dimension >3. The proof uses a more general structure theory for the derived category of an algebraic stack equipped with a Theta-stratification, which we apply in this case to the Harder-Narasimhan stratification of the moduli of sheaves.

algebraic geometry

Audience: researchers in the topic

UBC Vancouver Algebraic Geometry Seminar

Series comments: Recordings and slides are available on the seminar webpage:

wiki.math.ubc.ca/mathbook/alggeom-seminar/Main_Page