Lieu : Séminaire en ligne

Résumé : Every vector bundle on a smooth curve has a canonical filtration, called the Harder-Narasimhan filtration, and the moduli of all vector bundles admits a stratification based on the properties of the Harder-Narasimhan filtration at each point. The theory of Theta-stratifications formulates this structure on a general algebraic stack. I will discuss how to characterize stratifications of this kind and the nice properties they enjoy, such as having well-behaved local cohomology. Even when studying classical moduli problems, such as the moduli of semistable coherent sheaves on a K3 surface, it will be necessary to use methods from derived algebraic geometry. We will explain how derived Theta-stratifications are part of a recent proof of a case of the D-equivalence conjecture : for any projective Calabi-Yau manifold X that is birationally equivalent to a moduli space of semistable coherent sheaves on a K3 surface, the derived category of coherent sheaves on X is equivalent to the derived category of this moduli space.