On the length of perverse sheaves and D-modules

TL;DR

This paper demonstrates that the length function for perverse sheaves and D-modules on smooth complex algebraic varieties is an absolute Q-constructible function, leading to Zariski constructibility results for loci of local systems under derived functors.

Contribution

It establishes the Q-constructibility of the length function for perverse sheaves and D-modules, providing a new geometric perspective on their behavior under derived functors.

Findings

Length function is absolute Q-constructible.

Loci of local systems with prescribed length are Zariski constructible.

Describes the structure of these loci via algebraic subtori and set operations.

Abstract

We prove that the length function for perverse sheaves and algebraic regular holonomic D-modules on a smooth complex algebraic variety Y is an absolute Q-constructible function. One consequence is: for "any" fixed natural (derived) functor F between constructible complexes or perverse sheaves on two smooth varieties X and Y, the loci of rank one local systems L on X whose image F(L) has prescribed length are Zariski constructible subsets defined over Q, obtained from finitely many torsion-translated complex affine algebraic subtori of the moduli of rank one local systems via a finite sequence of taking union, intersection, and complement.