Seminar on Hodge theory, local cohomology and the du Bois complex

Spring term 2025

The aim of this seminar is to study some recent developments in Hodge theory for singular subvarieties in smooth algebraic varieties. These are mainly due to Mihnea Popa and Mircea Mustaţă. Among the topics treated in these papers are: (new) characterization of local cohomological dimension (resp. defect) of singular subvarieties, local vanishing theorems for log resolutions, characterization of (higher) du Bois singularities in terms of comparison of Hodge and order filtration on the local cohomology sheaves (at least in the l.c.i. case) etc. These aspects, together with some older material on the du Bois complex and on the (more classical) notion of du Bois singularities will be discussed in the seminar.

More details, and references can be found in this seminar program.

Some of the references used in the program are gathered here.

The seminar will run remotely on Zoom (link will be send out separately). We meet on Fridays, 13:00-14:30. The precise schedule is below.

Schedule

  1. Introduction (Christian Sevenheck), 02.05.2025, (Notes, Video)
  2. Hyperresolutions, cubical resolutions and definition of the du Bois complex (Andreas Hohl), 16.05.2025 (Notes, Video)
  3. Properties of the du Bois complex and more examples (Paul Görlach), 23.05.2025 (Notes, Video)
  4. Definition of du Bois singularities, weak normality, relation to other singularity types (Yichen Qin) 06.06.2025 (Notes, Video)
  5. Reminders on (mixed) Hodge modules and on Hodge ideals (Carolina Tamborini), 13.06.2025 (Notes, Video)
  6. Hodge theory and local cohomology, Hodge module interpretation of the du Bois complex (Henry Dakin), 20.06.2025 (Notes, Video)
  7. Higher du Bois singularities (Christian Sevenheck), 27.06.2025 (Notes, Video)
  8. Local cohomological dimension resp. defect (Christian Lehn), 11.07.2025 (Notes, Video)
  9. The Hodge-du Bois diamond, Hodge theory of singular varieties, to be scheduled