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Symmetric powers of null motivic Euler characteristic

Dori Bejleri, Stephen McKean

TL;DR

The paper develops a comprehensive framework for the motivic Euler characteristic $ ext{χ}^c$ over fields of characteristic not 2, proposing that symmetric powers preserve the kernel of $ ext{χ}^c$ and tying this to a power structure on $ ext{GW}(k)$ compatible with the one on $K_0( ext{Var}_k)$. It establishes foundational properties of $ ext{χ}^c$, computes its values on basic and bundle-type varieties, and shows its relation to étale and real Euler characteristics, while also detailing modifications of $K_0( ext{Var}_k)$ (inversions and localizations) and designing a power structure on $K_0( ext{Var}_k)$ (and on $K_0^{ ext{uh}}( ext{Var}_k)$) that makes these invariants compatible. Under a conjectural compatibility (a GW(k) power structure induced by $ ext{χ}^c$), the authors derive an enriched Göttsche formula for Hilbert schemes and compute local punctual Hilbert schemes, connecting to Yau–Zaslow-type enumerative results in enriched settings. They prove the conjecture in several cases (complex, real, and pythagorean fields) and show that, conditional on the conjecture, deep enrichment phenomena in $ ext{GW}(k)$-valued counts become available for Hilbert schemes and related enumerative geometries over broader base fields.

Abstract

Let k be a field of characteristic not 2. We conjecture that if X is a quasi-projective k-variety with trivial motivic Euler characteristic, then Sym$^n$X has trivial motivic Euler characteristic for all n. Conditional on this conjecture, we show that the Grothendieck--Witt ring admits a power structure that is compatible with the motivic Euler characteristic and the power structure on the Grothendieck ring of varieties. We then discuss how these conditional results would imply an enrichment of Göttsche's formula for the Euler characteristics of Hilbert schemes.

Symmetric powers of null motivic Euler characteristic

TL;DR

The paper develops a comprehensive framework for the motivic Euler characteristic over fields of characteristic not 2, proposing that symmetric powers preserve the kernel of and tying this to a power structure on compatible with the one on . It establishes foundational properties of , computes its values on basic and bundle-type varieties, and shows its relation to étale and real Euler characteristics, while also detailing modifications of (inversions and localizations) and designing a power structure on (and on ) that makes these invariants compatible. Under a conjectural compatibility (a GW(k) power structure induced by ), the authors derive an enriched Göttsche formula for Hilbert schemes and compute local punctual Hilbert schemes, connecting to Yau–Zaslow-type enumerative results in enriched settings. They prove the conjecture in several cases (complex, real, and pythagorean fields) and show that, conditional on the conjecture, deep enrichment phenomena in -valued counts become available for Hilbert schemes and related enumerative geometries over broader base fields.

Abstract

Let k be a field of characteristic not 2. We conjecture that if X is a quasi-projective k-variety with trivial motivic Euler characteristic, then SymX has trivial motivic Euler characteristic for all n. Conditional on this conjecture, we show that the Grothendieck--Witt ring admits a power structure that is compatible with the motivic Euler characteristic and the power structure on the Grothendieck ring of varieties. We then discuss how these conditional results would imply an enrichment of Göttsche's formula for the Euler characteristics of Hilbert schemes.
Paper Structure (26 sections, 51 theorems, 110 equations, 1 figure)

This paper contains 26 sections, 51 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline
  4. Related work
  5. Motivic Euler characteristic
  6. Background
  7. Extending the motivic Euler characteristic
  8. Properties and computations of the motivic Euler characteristic
  9. Basic properties
  10. Computations for various schemes
  11. Computations for bundles
  12. Recovering the étale Euler characteristic with compact support
  13. Modifications of $K_0(\mathrm{Var}_k)$
  14. Inverting universal homeomorphisms
  15. Localization at the Lefschetz motive
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $k$ be a field of characteristic not 2. Let be the power structure defined in Section sec:power structure K0. Then Conjecture conj:main is true if and only if there exists a power structure such that for all $A(t)\in 1+t\cdot K_0(\mathrm{Var}_k)[\![ t]\!]$ and $M\in K_0(\mathrm{Var}_k)$.

Figures (1)

  • Figure 1: $\mathbb{G}_m(\mathbb{R})$ with compactly supported Euler characteristic $-2$

Theorems & Definitions (135)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Levine--Raksit, Bachmann--Wickelgren
  • Theorem 2.5: Arcila-Maya--Bethea--Opie--Wickelgren--Zakharevich, Pajwani--Pál
  • Definition 2.6
  • Definition 2.7
  • ...and 125 more