Characteristic classes of orbit stratifications, the axiomatic approach

Abstract

Consider a complex algebraic group $G$ acting on a smooth variety $M$ with finitely many orbits, and let $\Omega$ be an orbit. The following three invariants of $\Omega \subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\overline{\Omega },M]\in H_G^{\ast }(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(\Omega ,M)\in H_G^{\ast }(M)$, and (3) the equivariant motivic Chern class $mC(\Omega ,M)\in K_G(M)[y]$. The axioms for Chern-Schwartz-MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For $M$ a flag variety and $\Omega$ a Schubert cell---an orbit of the Borel group acting---this implies that CSM and MC classes coincide with the weight functions studied by Rimanyi-Tarasov-Varchenko. In this paper we review the general theory and illustrate it with examples.

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