Prerequisites: Basic knowledge of algebraic geometry, e.g. at
the level of Harstshorne's book, ch. I-III, or equivalent; basic
knowledge oc characteristic classes, e.g. at the level of
Milnor-Stasheff book, or equivalent.
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)
- Luis Sola Conde: Geometry of rational
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-
Bott-Samelson 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,
GKM-spaces and the moment map (main examples toric varieties,
homogeneous spaces and their subvarieties). Equivariant cohomology
theories: Borel theory and K-theory, localization theorems, formula
for cohomology/K-theory of GKM-spaces. Equivariant cohomology and
K-theory 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 (CSM-classes) and
K-theory (motivic Chern classes). Equivariant characteristic classes
of toric varieties and Schubert varieties in G/B. Action of the Hecke
algebra on characteristic classes.
- Chiss, Ginzburg, Representation Theory and Complex
Geometry, mainly Ch. 5, 6.
- Fulton, Harris, Representation theory, Ch. 7,8,9,14
- Humphreys, Introduction to Lie algebras and representation
Rational homogeneous varieties
Program of the school: Two lectures each morning, 90 min each,
followed by two 60 min excercise sessions in the afternoon.
Organizers: Aleksandra Borowka (Institute of Mathematics, Polish
Academy of Sciences) and Jakub Koncki, Eleonora Romano, Jaroslaw
Wisniewski (Institute of Mathematics, the University of Warsaw).
The school is supported by:
Institute of Mathematics,
Faculty of Mathematics, Informatics and Mechanics, the University of
Warsaw and by research
projects Algebraic geometry:
varieties and structure (2013/08/A/ST1/00804) and
Complex contact manifolds and geometry of secants (2017/26/E/ST1/00231)
The school will take place in a Warsaw University
pension in Lukecin,
on Western part of Polish Baltic sea shore, see
The accommodation (full board, shared double room) will cost about 120
zloty (PLN) a day (1 Euro is approx. 4.3 PLN, but the exchange rate is
not fixed). A fee of 50 PLN will apply.
Graduate students and young researchers with inadequate support from
their home institutions are encouraged to apply for accommodation cost
waiver (we may ask you to provide additional letter of
recommendation). The organizers will not pay for participants'
Travel to Lukecin:
The closest airport is in Szczecin (Poland), you may consider
also traveling through Berlin (Germany) and then taking a train to
Szczecin (it takes about 2hrs),
see the German railway
page, or a bus,
from Berlin airports to Szczecin and Koszalin
The organizers will provide a bus from Szczecin Glowny train
station to Lukecin on the day of arrival, Sunday evening.
All inquiries should be directed
to Eleonora Romano,
elrom AT mimuw.edu.pl.
September Schools Homepage