Generating series of classes of exotic un-ordered configuration spaces

2022/03/21 by S. M. Gusein‐Zade, Gusein-Zade, Sabir M.
Mathematics · #18F30 #32S35 #55R80 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2203.10798

openalex publication_date 2022/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A notion of exotic (ordered) configuration spaces of points on a space X was suggested by Yu.~Baryshnikov. He gave equations for the (exponential) generating series of the Euler characteristics of these spaces. Here we consider un-ordered analogues of these spaces. For X being a complex quasiprojective variety, we give equations for the generating series of classes of these configuration spaces in the Grothendieck ring K₀(\rmVar_ℂ) of complex quasiprojective varieties. The answer is formulated in terms of the (natural) power structure over the ring K₀(\rmVar_ℂ). This gives equations for the generating series of additive invariants of the configuration spaces such as the Hodge--Deligne polynomial and the Euler characteristic.

Related

• On the power structure over the Grothendieck ring of varieties and its applications
• Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points
• Motivic Chern classes of configuration spaces
• Higher order generalized Euler characteristics and generating series
• Configuration spaces of orbits and their S_n-equivariant E-polynomials