Work

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

Year: 2025

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2510.06922

Authors: Bröring, Louisa F., Pajwani, Jesse, Viergever, Anna M.

Institutions: University of Duisburg-Essen, University of Bristol, Heilbronn Institute for Mathematical Research, Leibniz University Hannover

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: UK Research and Innovation, University of Bath, Deutsche Forschungsgemeinschaft, Universität Duisburg-Essen

Awards: G3257303119, GRK 2553: Symmetrien und klassifizierende Räume: analytisch, arithmetisch und deriviert

OpenAlex ID: W4415097443

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