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Compactly supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of cellular varieties

Jesse Pajwani, Herman Rohrbach, Anna M. Viergever

TL;DR

The paper investigates when the compactly supported $\\mathbb{A}^1$-Euler characteristic $\\chi_c^{\\mathrm{mot}}$ is compatible with the natural power structures on $\\mathrm{K}_0(\\mathrm{Var}_k)$ and $\\mathrm{GW}(k)$. It defines the class of symmetrisable varieties $\\mathrm{Sym}_k$, proving that $\\mathrm{K}_0$-étale linear varieties are symmetrisable and that this class includes cellular and many non-linear examples such as elliptic curves. The authors derive a general formula $\\chi_c^{\\mathrm{mot}}(X^{(n)}) = a_n(\\chi_c^{\\mathrm{mot}}(X))$ for all $n$ when $X$ lies in $\\mathrm{K}_0(\\mathrm{EtLin}_k)$, and they apply this to compute symmetric-power Euler characteristics for Grassmannians and certain del Pezzo surfaces, including a generating-series description. They further analyze base-change phenomena and provide a detailed treatment of Göttsche-type lemmas and odd-degree twists, yielding broad structural insights and concrete arithmetic refinements in motivic enumerative geometry.

Abstract

The compactly supported $\mathbb{A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an anologue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm{K}_0(\mathrm{Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm{GW}(k)$ of the base field $k$. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, Pajwani and Pál construct a power structure on $\mathrm{GW}(k)$ and show that the compactly supported $\mathbb{A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale $k$-algebras. In this paper, we define the class $\mathrm{Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb{A}^1$-Euler characteristic respects the power structures and study the algebraic properties of $\mathrm{K}_0(\mathrm{Sym}_k)$. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

Compactly supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of cellular varieties

TL;DR

The paper investigates when the compactly supported -Euler characteristic is compatible with the natural power structures on and . It defines the class of symmetrisable varieties , proving that -étale linear varieties are symmetrisable and that this class includes cellular and many non-linear examples such as elliptic curves. The authors derive a general formula for all when lies in , and they apply this to compute symmetric-power Euler characteristics for Grassmannians and certain del Pezzo surfaces, including a generating-series description. They further analyze base-change phenomena and provide a detailed treatment of Göttsche-type lemmas and odd-degree twists, yielding broad structural insights and concrete arithmetic refinements in motivic enumerative geometry.

Abstract

The compactly supported -Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an anologue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties taking values in the Grothendieck-Witt ring of the base field . The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, Pajwani and Pál construct a power structure on and show that the compactly supported -Euler characteristic respects these two power structures for -dimensional varieties, or equivalently étale -algebras. In this paper, we define the class of symmetrisable varieties to be those varieties for which the compactly supported -Euler characteristic respects the power structures and study the algebraic properties of . We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported -Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.
Paper Structure (13 sections, 37 theorems, 43 equations)

This paper contains 13 sections, 37 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Compactly supported A¹-Euler Characteristics and Symmetric Powers
  3. The compactly supported $\mathbb{A}^1$-Euler characteristic
  4. Power structures
  5. K₀-étale linear varieties
  6. Symmetrisable varieties
  7. Properties of symmetrisable varieties
  8. Göttsche's lemma for symmetric powers
  9. Odd Galois twists
  10. Computations of symmetric powers of symmetrisable varieties
  11. Grassmannians
  12. del Pezzo surfaces
  13. Curves of genus ≤1

Key Result

Theorem 1.1

Let $X$ be a $\mathrm{K}_0$-étale linear variety over field $k$ of characteristic $\neq 2$ (see Definition def:etale-linear), and for $n\in\mathbb{Z}_{\geq 0}$, write $X^{(n)} := \mathrm{Sym}^n(X)$. Then $\chi_c^{\mathrm{mot}}(X^{(n)}) = a_n(\chi_c^{\mathrm{mot}}(X))$ for every $n$, where $a_n$ deno

Theorems & Definitions (99)

  • Theorem 1.1: Theorem $\ref{['theorem:euler-char-symmetric-power-linear-variety']}$
  • Theorem 1.2: Corollary $\ref{['nonsmoothcurves']}$ and Lemma $\ref{['cor:positive-genus-not-etale-linear']}$
  • Theorem 1.3: Corollary \ref{['cor:generating-series-grassmannian']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 89 more