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.
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.
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.