Perverse sheaves of categories and some applications

TL;DR

This paper explores perverse sheaves of categories, revealing their connections to algebraic geometry, derived categories, and Calabi--Yau varieties, and demonstrating their utility in understanding autoequivalences and degenerations.

Contribution

It introduces the concept of perverse sheaves of categories and demonstrates their applications in encoding derived categories, semiorthogonal decompositions, and representing degenerate Calabi--Yau varieties.

Findings

Perverse sheaves of categories encode derived categories of coherent sheaves on $ P^1$ bundles.

They relate to semiorthogonal decompositions and autoequivalences as spherical twists.

They can represent certain degenerate Calabi--Yau varieties.

Abstract

We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on $\mathbb{P}^1$ bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi--Yau varieties.