Characteristic classes and stable envelopes for homogeneous spaces

Abstract

Recent development in representation theory and enumerative geometry led A. Okounkov and his coauthors to the definition of stable envelopes, which can be interpreted as characteristic classes associated to a torus acting on a symplectic algebraic variety. It turns out that if the symplectic variety is the cotangent bundle of a smooth variety, then the stable envelopes can be expressed by the following invariants of singular varieties: 
– Chern–Schwartz–MacPherson classes in cohomology, 
– motivic Chern classes in K-theory, 
– or elliptic classes of Borisov–Libgober in elliptic cohomology, 
depending which cohomology theory we consider. The stable envelopes have an additional parameter, the slope, which is a Q-divisor. For a distinguished slope the above classes applied to the Białynicki-Birula cells (a.k.a. atracting sets) differ from the stable envelopes only by a normalization factor. In general one has to introduce a twist resembling the construction of multiplier ideals. This leads to a definition of twisted characteristic classes. We will discuss the K-theoretic classes in detail and we will show how our construction originates from the relative Borisov–Libgober elliptic classes. We will concentrate on application to Schubert varieties.

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