Harvard-MIT Algebraic Geometry Seminar: Filtrations on Local Cohomology and Injectivity Theorems

Local cohomology modules of a subvariety carry several natural filtrations that reflect its singularities. The Hodge filtration comes from Hodge theory, while the Ext filtration is defined algebraically using powers of the defining ideal. In this talk, I will discuss a proof that the Hodge filtration is contained in the Ext filtration in every cohomological degree, answering a question of Mustata and Popa. This generalizes the containment of the Hodge filtration in the pole order filtration for hypersurfaces, established in cohomology by Deligne–Dimca and in general by Saito. The main observation is a connection between the Ext filtration and Hartshorne’s algebraic de Rham complex equipped with its infinitesimal filtration. If time permits, I will discuss how the same approach proves the higher injectivity conjecture of Popa–Shen–Vo and several variants. This is joint work with Qianyu Chen and Bradley Dirks.