Equivariant cohomology in algebraic geometry 1000-1M23EK
Torus action on vector space, weights, characters.
Basic information about actions of connected Lie groups on smooth manifolds, Lie algebra actions.
Slice theorem, equivariant CW-complexes.
Principal bundles, classifying spaces, Stiefel manifolds.
Borel equivariant cohomology, computations for homogeneous spaces (Grassmann manifolds, Flag varieties).
Differential interpretation of equivariant cohomology. Weil Algebra, connection, Mathai-Quillen twist, de Rham - Cartan theory.
Equivariant bundles and equivariant characteristic classes.
Equivariant formality in cohomology. Formality of projective varieties.
Borel localization theorem for torus action.
Atiyah-Bott-Beline-Vergne localization theorem, integration formula, Duistermaat-Heckman formula Hamiltonian actions.
GKM spaces, Chang-Skjelbred lemma.
Moment map.
Application of the localization theorem to computing Eukler characteristic of equivariant bundes.
Equivariant Schubert calculus.
Main fields of studies for MISMaP
mathematics
Type of course
Mode
Requirements
Prerequisites
Prerequisites (description)
Course coordinators
Learning outcomes
Student learns basic notions of equivariant cohomology theory.
Topological construction is understood and compared with the construction
based on differential geometry.
Student knows application to algebraic geometry, in particular for
homogeneous spaces.
Student achieves knowledge of the discipline allowing to start independent
research.
Assessment criteria
1/3 solving problems during classes
1/3 essay
1/3 oral exam
Bibliography
D. Anderson, W. Fulton: Equivariant Cohomology in Algebraic Geometry
V. Guillemin, S. Sternberg: Supersymmetry and Equivariant de Rham Theory, Springer 1999
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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