Semiorthogonal decompositions in families

TL;DR

This paper reviews recent advances in the study of semiorthogonal decompositions of algebraic varieties, focusing on their behavior in families, including homological projective duality, residual categories, and singularity resolutions.

Contribution

It introduces new concepts such as residual categories, simultaneous resolutions, and absorption of singularities, expanding understanding of semiorthogonal decompositions in degenerating families.

Findings

Computed Serre dimensions of residual categories for complete intersections.

Established relations between residual categories and small quantum cohomology.

Developed constructions for simultaneous resolutions in nodal degenerations.

Abstract

We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behaviour in families. First, we overview new results concerning homological projective duality. Then we introduce residual categories, discuss their relation to small quantum cohomology, and compute Serre dimensions of residual categories of complete intersections. After that we define simultaneous resolutions of singularities and describe a construction that works in particular for nodal degenerations of even-dimensional varieties. Finally, we introduce the concept of absorption of singularities which works under appropriate assumptions for nodal degenerations of odd-dimensional varieties.