Perverse schobers and birational geometry

TL;DR

This paper explores how perverse schobers, as categorical analogs of perverse sheaves, naturally arise in the Homological Minimal Model Program, providing new insights into birational transformations and their effects on derived categories.

Contribution

It introduces the concept of schober-type diagrams for flops of relative dimension 1 and analyzes their cohomological properties, connecting them to classical geometric structures.

Findings

Schober structures appear in the Homological Minimal Model Program.

Categorical analogs of cohomology are determined for flop diagrams.

The case of g=sl(3) relates to classical spaces of complete triangles.

Abstract

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a "web of flops" provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g=sl(3) we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.