The \mathbbA¹-Euler characteristic of symmetric powers

2025/10/08 by Bröring, Louisa F., Pajwani, Jesse, Viergever, Anna M.
#14F42 #14G27 #14N10 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.06922

Abstract

The \mathbbA¹-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 \mathbbA¹-Euler characteristic of the symmetric powers of a variety in terms of the \mathbbA¹-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.

Citations

• Symmetric powers of null motivic Euler characteristic
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Related

• Compactly supported \mathbbA¹-Euler characteristics of symmetric powers of cellular varieties
• Quadratic Euler Characteristic of Symmetric Powers of Curves
• Symmetric powers of null motivic Euler characteristic
• Compactly supported \mathbbA¹-Euler characteristic and the Hochschild complex
• Unusual formulae for the Euler characteristic