Ben Tighe (Oregon)

Tuesday, April 2, 2024 - 2:30pm

PDL C-38

Main talk (2:30pm)

**Title**: Rational v.s. k-du Bois Singularities**Abstract**: du Bois singularities are a broad class of singularities where one can study Hodge theory and deformation theory. Two fundamental results concerning these singularities are that log-canonical (Kollár-Kovács) and rational (Kovács, Saito) singularities are du Bois. While du Bois singularities are "very close" to being log-canonical, the du Bois property is much less restrictive than having rational singularities. In this talk, we will outline this gap by studying holomorphic and logarithmic extension for the "k-du Bois" property. Of particular interest will be the case when X is a singular symplectic variety, where the du Bois complex admits many useful symmetries.Pre-talk (1:45pm)

**Title:**The Holomorphic and Logarithmic Extension Properties

**Abstract**: Let X be a normal complex variety. An important property in the Hodge theory of singularities is holomorphic extension: every holomorphic p-form on the regular locus U "extends holomorphically" across the singularities. Holomorphic extension holds in all degrees when X has klt or rational singularities, but this can fail for broad classes of singularities. If X has log-canonical or du Bois singularities, then holomorphic forms on U instead extend meromorphically with logarithmic poles. We will investigate these properties with concrete examples and highlight how holomorphic/logarithmic extension can be detected by classic vanishing conditions corresponding to these singularities.