Basic Geometry of Flag Manifolds and Schubert Varieties


Title: Basic Geometry of Flag Manifolds and Schubert Varieties
Summary:

This course aims to be an introduction to the study of geometry of
flag manifolds and Schubert varieties. Only prerequisites are a
working knowledge of linear algebra and basic group theory. No prior
knowledge of geometry is required. One of the objectives of this
course is to teach algebraic geometry using the very concrete examples
provided by flag manifolds and Schubert varieties.

Outline/Topics:

* Preliminaries from Linear Algebraic Groups

** Root Systems

** Coxeter Groups

** Bruhat Decomposition

** Parabolic subgroups

** Flag Manifolds G/B, G/P

* Algebraic Geometry of Flag Manifolds with concrete examples

** Projective line

** Projective spaces

** Grassmannians

** Low dimensional flag manifolds

* Schubert Varieties

** Optional: Schubert Calculus and Schur Polymomials

* Line Bundles

* Borel-Weil-Bott Theorem

* Line bundles on Schubert varieties

* Schubert Polynomials

References:

* Hiller, Geometry of Coxeter Groups

* Brion, Lectures on the Geometry of Flag Varieties

http://arxiv.org/abs/math/0410240

* J. Bernstein, I.M. Gelfand, S.I. Gelfand
Schubert cells and cohomologies of spaces G/P.
Uspekhi Mat. Nauk 38, No.3, 3-26(1973).

http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BGG-CoxeterF-Usp.pdf

* Fulton, Young Tableaux

* Vogan, Geometry of Flag Manifolds and Representation Theory

* Aker, Algebraic Geometry Fall Workshop 2009
Reklamlar

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