Geometry of Flag Varieties

Aug-Nov 2023

Instructor:

Manoj Kummini
Email: mkummini AT cmi DOT ac DOT in.
https://www.cmi.ac.in/~mkummini/index.html
Office: GL-05

Lectures :

• TBD.
• Details will be posted in the moodle course page (see below).

Moodle: All the information regarding this course will be posted on Moodle. You can directly access the course page at https://moodle.cmi.ac.in/course/view.php?id=813 . Otherwise, look for `Geometry of Flag Varieties' under `Mathematics / Aug-Nov 2023'.

References :

• Brion, Lectures on the geometry of flag varieties, Birkhauser. \texttt{http://arxiv.org/abs/math/0410240v1}
• Seshadri, Introduction to the theory of standard monomials, Hindustan Book Agency.
• Demazure, A very simple proof of Bott's theorem, Invent. Math., 1976.
• Mehta and Ramanathan, Frobenius Splitting and Cohomology Vanishing for Schubert Varieties Annals of Math. 1985.
• Brion and Kumar, Frobenius splitting methods in geometry and representation theory, Birkhauser.
• Billey and Lakshmibai, Singular loci of Schubert varieties, Birkhauser.

Brief description: This is an introductory course in commutative algebra.

• Grassmannians and its Schubert varieties; standard monomial theory. projective normality and arithmetic Cohen-Macaulayness.
• SL_n/Q and its Schubert varieties, Borel-Weyl-Bott theorem, Bott-Samelson desingularization.
• Frobenius splitting. Vanishing theorems.