[Paper Review] On the length of perverse sheaves and D-modules

Nero Budur, Pietro Gatti+2 more|arXiv (Cornell University)|Sep 4, 2017

Algebraic Geometry and Number Theory16 references5 citations

TL;DR

This paper establishes that the length function for perverse sheaves and regular holonomic D-modules on smooth complex algebraic varieties is an absolute Q-constructible function. The key result shows that for any fixed derived functor between constructible complexes or perverse sheaves on smooth varieties, the loci of rank one local systems whose image has prescribed length are Zariski constructible subsets defined over Q, formed from finitely many torsion-translated algebraic subtori via finite unions, intersections, and complements.

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.

Motivation & Objective

• To establish that the length function on perverse sheaves and regular holonomic D-modules over smooth complex algebraic varieties is an absolute Q-constructible function.
• To show that for any fixed derived functor between constructible complexes or perverse sheaves on smooth varieties, the loci of rank one local systems with prescribed image length are Zariski constructible over Q.
• To provide a geometric and arithmetic description of length jump loci in terms of torsion-translated subtori in the moduli space of local systems.
• To extend the understanding of length invariants in derived categories using the Riemann-Hilbert correspondence and perverse cohomology functors.
• To demonstrate that such loci are not arbitrary but have a precise algebraic-geometric structure defined over Q, even in higher-rank local system settings.

Proposed method

• Utilizes the Riemann-Hilbert correspondence to relate D-modules to perverse sheaves, preserving length under equivalence.
• Applies perverse cohomology functors to define length on the derived category via alternating sums of perverse cohomology sheaf lengths.
• Employs the theory of absolute Q-constructible sets, defined as finite combinations (union, intersection, complement) of torsion-translated complex affine algebraic subtori over Q.
• Uses induction on dimension and support of cones in distinguished triangles to show Q-constructibility of length functions.
• Analyzes the structure of intermediate extensions and direct images (e.g., $Rj_*$, $Rf_*$) of local systems via short exact sequences involving skyscraper sheaves.
• Applies the decomposition theorem in the proper case to interpret length as the number of semi-simple direct summands in the image.

Experimental results

Research questions

• RQ1 Is the length function for perverse sheaves and D-modules on smooth complex varieties an absolute Q-constructible function?
• RQ2 Can the loci of rank one local systems whose image under a fixed derived functor has prescribed length be described as Zariski constructible subsets defined over Q?
• RQ3 What is the geometric and arithmetic structure of length jump loci in the moduli space of local systems?
• RQ4 How does the length of direct images of intersection complexes behave under algebraic maps between smooth varieties?
• RQ5 To what extent do length loci in higher-rank local system moduli spaces inherit Q-constructible structure?

Key findings

• The length function on perverse sheaves and regular holonomic D-modules on smooth complex algebraic varieties is an absolute Q-constructible function.
• For any algebraic map $f: X o Y$ and smooth subvariety $S o X$, the loci $igackslash \{ L o ext{M}_B(S,1) \mid \ell({}^pR^if_*(IC_{\overline{S}}(L))) = k \big\}$ are absolute Q-constructible subsets.
• In the proper case, the length of $^pR^if_*(IC_{\overline{S}}(L))$ counts the number of semi-simple direct summands, and such loci are Q-constructible.
• For rank 2 local systems on $\mathbb{C} \setminus \{a,b\}$, the loci of prescribed length for $Rj_*(L[1])$ are unions of linear and quadratic subvarieties in $\mathbb{C}^3$, including points like $(2,2,2)$ and lines such as $\{x=2, y=z\}$.
• The length jump loci for $Rj_*(L[1])$ are shown to be linear on the simple locus, with $\ell(Rj_*(L[1])) \geq 6$ only at the point $(2,2,2)$.
• The function $m \mapsto \ell(Rj_!(q(L)[n]) \otimes^L_R R/m)$ is Q-constructible, established via induction on the dimension of the support of the cone in a distinguished triangle.