Exact functors on perverse coherent sheaves

TL;DR

This paper introduces an alternative definition of perverse coherent sheaves inspired by symplectic geometry, characterizing them via the concentration of derived sections over special subvarieties analogous to Lagrangians.

Contribution

It provides a new characterization of perverse coherent sheaves using microlocal methods and subvarieties analogous to Lagrangians, connecting symplectic geometry with algebraic geometry.

Findings

Perverse coherent sheaves are characterized by the concentration of $R ext{Gamma}_Z( ext{F})$ in degree 0.

Special subvarieties Z serve as analogs of Lagrangians in the symplectic setting.

The new definition aligns with existing notions of perverse sheaves in a coherent sheaf context.

Abstract

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $R\Gamma_Z(\mathcal{F})$ is concentrated in degree 0 for special subvarieties Z of X. These subvarieties Z are analogs of Lagrangians in the symplectic case.