Categorical absorptions of singularities and degenerations

TL;DR

This paper introduces a new operation called categorical absorption that simplifies the derived category of singular varieties by removing singularity-related subcategories, resulting in a smooth and proper category.

Contribution

It defines the concept of categorical absorption of singularities and constructs such an absorption for varieties with isolated ordinary double points, extending to families over smooth curves.

Findings

Categorical absorption removes singularity-related subcategories from derived categories.

Constructed categorical absorption for varieties with isolated ordinary double points.

Extended the smooth part of the derived category over smooth curve families.

Abstract

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$.