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Motivic classes of stacks in finite characteristic and applications to stacks of Higgs bundles

Ruoxi Li

TL;DR

The paper extends motivic class theory to stacks in finite characteristic by introducing a surjective universally injective relation, develops a robust plethystic and λ-ring framework, and defines a counting measure compatible with Higgs-bundle moduli. It proves explicit motivic formulas for stacks of semistable Higgs bundles over a smooth projective curve, generalizing characteristic-zero results (Fedorov, Soibelman, Mozgovoy–Schiffmann) to arbitrary characteristic. A key innovation is Mellit’s simplification in the universal λ-ring quotient, yielding streamlined expressions for motivic classes and finite-field counts via the motivic zeta-function, its plethystic exponential/ logarithm, and motivic Hall algebra techniques. The work connects motivic volumes with étale realizations and counting over finite fields, enabling direct comparisons with known volume formulas and Donaldson–Thomas type invariants, while providing a coherent framework applicable to both Higgs bundles and broader stack-theoretic motivic questions.

Abstract

We define a ring of motivic classes of stacks suitable for symmetric powers in finite characteristic. Let $X$ be a smooth projective curve over a field of arbitrary characteristic. We calculate the motivic classes of the moduli stacks of semistable Higgs bundles on $X$. This recovers results of Fedorov, A. Soibelman and Y. Soibelman in characteristic zero, as well as those of Mozgovoy and Schiffmann for finite fields. We also obtain a simpler formula for the motivic classes of the stacks of Higgs bundles in the universal $λ$-ring quotient using Mellit's results.

Motivic classes of stacks in finite characteristic and applications to stacks of Higgs bundles

TL;DR

The paper extends motivic class theory to stacks in finite characteristic by introducing a surjective universally injective relation, develops a robust plethystic and λ-ring framework, and defines a counting measure compatible with Higgs-bundle moduli. It proves explicit motivic formulas for stacks of semistable Higgs bundles over a smooth projective curve, generalizing characteristic-zero results (Fedorov, Soibelman, Mozgovoy–Schiffmann) to arbitrary characteristic. A key innovation is Mellit’s simplification in the universal λ-ring quotient, yielding streamlined expressions for motivic classes and finite-field counts via the motivic zeta-function, its plethystic exponential/ logarithm, and motivic Hall algebra techniques. The work connects motivic volumes with étale realizations and counting over finite fields, enabling direct comparisons with known volume formulas and Donaldson–Thomas type invariants, while providing a coherent framework applicable to both Higgs bundles and broader stack-theoretic motivic questions.

Abstract

We define a ring of motivic classes of stacks suitable for symmetric powers in finite characteristic. Let be a smooth projective curve over a field of arbitrary characteristic. We calculate the motivic classes of the moduli stacks of semistable Higgs bundles on . This recovers results of Fedorov, A. Soibelman and Y. Soibelman in characteristic zero, as well as those of Mozgovoy and Schiffmann for finite fields. We also obtain a simpler formula for the motivic classes of the stacks of Higgs bundles in the universal -ring quotient using Mellit's results.

Paper Structure

This paper contains 34 sections, 32 theorems, 156 equations.

Table of Contents

  1. Introduction and main results
  2. Motivic classes of stacks
  3. Motivic zeta-function and plethystic exponents
  4. Counting measure
  5. Moduli stacks of Higgs bundles
  6. Explicit formulas for motivic classes
  7. Mellit's simplification
  8. Organization of the article
  9. Acknowledgements
  10. Motivic classes in finite characteristic
  11. Definitions of the rings of motivic classes of schemes and stacks
  12. Dimensional completion
  13. Motivic zeta-function and pre-$\lambda$-ring structure on $\overline{\mathop{\mathrm{Mot}}\nolimits}(k)$
  14. Stack motivic functions
  15. Bilinear form
  16. ...and 19 more sections

Key Result

Theorem 1.3

Assume that $M$ is a scheme, $\mathcal{A}_0=\mathop{\mathrm{Spec}}\nolimits k$ and $\mathcal{A}_1,\mathcal{A}_2,\ldots,\mathcal{A}_n,\ldots$ are stacks. Put $A(z)=[\mathcal{A}_0]+[\mathcal{A}_1]z+[\mathcal{A}_2]z^2+\cdots+[\mathcal{A}_n]z^n+\ldots$. Then we have where $\Delta$ is the "large diagonal" in $M^{\sum_i k_i}$ which consists of $\left(\sum_i k_i\right)$-tuples of points of $M$ with at l

Theorems & Definitions (81)

  • Definition 1.2: Definition \ref{['motstacks']}
  • Theorem 1.3: Theorem \ref{['power']}
  • Theorem 1.4: Theorem \ref{['countingstack']}
  • Theorem 1.5: Theorem \ref{['higgsformula']}
  • Theorem 1.6: Theorem \ref{['degenerate1']}
  • Definition 2.1
  • Definition 2.3
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 71 more