42nd Autumn School in Algebraic Geometry
Homogeneous Spaces, Characteristic Classes
Lukecin, Poland, September 9  September 13, 2019
Teachers: Luis Sola Conde (Universita di Trento), Andrzej Weber (University of Warsaw)
Lecture series:
 Luis Sola Conde: Geometry of rational
homogeneous spaces.
During these lectures we will look at
rational homogeneous spaces from the point of view of projective
geometry. We will start by going through their classification and the
description of their Mori cones and contractions. We will then study
how their cohomology rings can be written in terms of certain
features of their groups of automorphisms (roots, Weyl groups),
introducing Schubert cycles and their standard desingularizations
BottSamelson varieties. Along the different lectures we will
describe projectively many examples, of classical (Grassmannians,
flags, quadrics,...) and exceptional type.
 Andrzej Weber: Characteristic classes of singular
varieties in equivariant theories.
Spaces with torus action,
GKMspaces and the moment map (main examples toric varieties,
homogeneous spaces and their subvarieties). Equivariant cohomology
theories: Borel theory and Ktheory, localization theorems, formula
for cohomology/Ktheory of GKMspaces. Equivariant cohomology and
Ktheory of homogeneous spaces, Demazure operations, Schubert
varieties and their fundamental classes, inductive definition of
Grothendieck polynomials. Equivariant characteristic classes of
singular varieties in equivariant cohomology (CSMclasses) and
Ktheory (motivic Chern classes). Equivariant characteristic classes
of toric varieties and Schubert varieties in G/B. Action of the Hecke
algebra on characteristic classes.

