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 $\mathcal{X}$.

Comment: 41 pages; v2, v3: minor improvements, v4: revised according to the referee's report, to appear in EPIGA, v5: published version

• 14D06 - Fibrations, degenerations in algebraic geometry
• 14E15 - Global theory and resolution of singularities (algebro-geometric aspects)
• 14F08 - Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
• 14J45 - Fano varieties

• 1 zbMATH Open