# 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.

## Eventi passati

Please have a look on the page of past seminars