The $\mathbb{A}^1$-Euler characteristic of symmetric powers
Louisa F. Bröring, Jesse Pajwani, Anna M. Viergever
TL;DR
This survey investigates whether the quadratic $\\mathbb{A}^1$-Euler characteristic of symmetric powers can be computed from the characteristic of the variety itself via a dedicated power structure on the Grothendieck–Witt ring. It develops and compares two intertwined power-structure frameworks: one on the Grothendieck ring of varieties (via symmetric powers) and one on $\\mathrm{GW}(k)$ (via a proposed $a_*$), culminating in a precise conjecture that $\\chi_c(\\mathrm{Sym}^n X)=a_n(\\chi_c(X))$ for all $X$ and $n$. The paper gathers substantial evidence: structural results placing $K_0(\\mathrm{Sym}_k)$ inside tilde $K_0(Var_k)$, explicit computations for curves, and several cases where the conjecture holds (including dimension-zero varieties, many cellular types, and low $n$ for curves). It also outlines potential higher-level strategies to prove the conjecture via λ-structures and motivic spectra, and presents Göttsche-type formulas as a key payoff if the conjecture holds generally. Collectively, these results point toward a powerful computational framework for quadratically enriched enumerative geometry across fields, tied to power-structure compatibility with $\\chi_c$.
Abstract
The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an integer, which carries a lot of information, but is difficult to compute in practice. In this survey, we discuss a conjectural way for computing the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a variety in terms of the $\mathbb{A}^1$-Euler characteristic of the variety itself formulated using the theory of power structures. We discuss evidence towards the conjecture so far, techniques to approach it, and applications.