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arXiv · 2510.06922

The $\mathbb{A}^1$-Euler characteristic of symmetric powers

Louisa F. Bröring·Jesse Pajwani·Anna M. Viergever
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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.

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Louisa F. Bröring, Jesse Pajwani, Anna M. Viergever. 2025-10-08. The $\mathbb{A}^1$-Euler characteristic of symmetric powers. https://arxiv.org/abs/2510.06922

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