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Motivic Coh and Quot zeta functions of singular curves

Yifeng Huang, Ruofan Jiang

Abstract

We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced singular curves. We prove that Quot zeta functions are motivically rational, using a novel parametrization and the geometry of affine Grassmannians, and that they satisfy an arbitrary-rank reflection principle, via harmonic analysis. We show that the a normalized high-rank limit of Quot zeta functions converges to the Coh zeta function. As a first application, we compute explicit formulas for these zeta functions for all $y^2 = x^n$ singularities, revealing a surprising and previously unknown connection to Rogers--Ramanujan type $q$-series. Further applications to affine Springer fibers and commuting varieties are also discussed.

Motivic Coh and Quot zeta functions of singular curves

Abstract

We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced singular curves. We prove that Quot zeta functions are motivically rational, using a novel parametrization and the geometry of affine Grassmannians, and that they satisfy an arbitrary-rank reflection principle, via harmonic analysis. We show that the a normalized high-rank limit of Quot zeta functions converges to the Coh zeta function. As a first application, we compute explicit formulas for these zeta functions for all singularities, revealing a surprising and previously unknown connection to Rogers--Ramanujan type -series. Further applications to affine Springer fibers and commuting varieties are also discussed.
Paper Structure (49 sections, 63 theorems, 167 equations)

This paper contains 49 sections, 63 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Statements of results
  3. Relation to $q$-series
  4. Relation to other works
  5. Outline of methods and plan of the paper
  6. Quot schemes and commuting varieties
  7. Classical varieties
  8. Formal varieties
  9. Motivic Quot-to-Coh and Proof of Theorem \ref{['thm:A']}
  10. Framing data
  11. An effective bound
  12. Side remarks: examples
  13. Torsion-free bundles over reduced curves, via lattices
  14. The local setup
  15. Some basic structures
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field, and $X$ be any variety over $k$. The motivic Coh zeta function is the rank $d\to\infty$ limit of the rank $d$ Quot zeta function with a suitable rescaling: where the convergence is coefficient-wise and in the sense of the dimension filtration on ${K_0(\mathrm{Stck}_{k})}$behrenddhillon2007.

Theorems & Definitions (154)

  • Theorem 1.1: $\subseteq$ \ref{['thm:A-effective']}
  • Theorem 1.2: $\subseteq$ \ref{['thm:rationality-motivic']}
  • Corollary 1.3: $\subseteq$ \ref{['thm:rationality-intro', 'thm:local-to-global']}
  • Theorem 1.4: $=$ \ref{['thm:cusp_simplification', 'thm:node-full']}, and Remark \ref{['rmk:motivic-torus']}
  • Theorem 1.5: $\subseteq$ \ref{['thm:reflection-local']}
  • Theorem 1.6: $\subseteq$ Theorems \ref{['thm:cusp_simplification']}, \ref{['thm:node-full']}, and shane2024multiple
  • Definition 2.1
  • Lemma 2.2: Artinian reduction
  • proof
  • Lemma 2.3
  • ...and 144 more