Generating series of classes of exotic un-ordered configuration spaces
Sabir M. Gusein-Zade

TL;DR
This paper develops formulas for the generating series of classes of exotic un-ordered configuration spaces on complex varieties, extending previous work on ordered spaces and providing tools for computing various invariants.
Contribution
It introduces equations for the generating series of un-ordered exotic configuration spaces in the Grothendieck ring, generalizing Baryshnikov's ordered case to un-ordered varieties.
Findings
Derived equations for the classes in the Grothendieck ring
Provided formulas for additive invariants like Hodge–Deligne polynomial
Extended the theory to un-ordered configuration spaces on complex varieties
Abstract
A notion of exotic (ordered) configuration spaces of points on a space 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 being a complex quasiprojective variety, we give equations for the generating series of classes of these configuration spaces in the Grothendieck ring of complex quasiprojective varieties. The answer is formulated in terms of the (natural) power structure over the ring . This gives equations for the generating series of additive invariants of the configuration spaces such as the Hodge--Deligne polynomial and the Euler characteristic.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology
