Public viva-voce Notification
Date: Wednesday, 23 April 2025
Time: 11.30 AM
Venue: Seminar Hall (Hybrid mode)
Torus quotients of flag varieties
Somnath Sudam Dake
Chennai Mathematical Institute.
23-04-25

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Abstract

The work done in this thesis is motivated by the important role that geometric invariant theory (GIT) techniques are expected to play in the geometric complexity theory (GCT) approach tolower bounds in algebraic complexity theory. The focus of this thesis is understanding quotients of important classical varieties under the natural torus action. Let n ≥ 1 be a integer and let T ⊂ SL(n, C) be the maximal torus of diagonal matrices. Let 1 ≤ r ≤ n − 1 be a integer and let Gr,n be the Grassmannian variety. There is a natural action of T on Gr,n by matrix multiplication. The thesis studies the GIT quotient of this action. We assume that (r, n) = 1. In our first result, we prove that the GIT quotient T\\(G3,7) ss T is projectively normal with respect to the descent of the line bundle L(7w3,7). We approach the proof of this theorem by transforming it into a computational problem through polyhedral theory and solving some of it using computer algorithms. In [KS09], Kannan and Sardar showed that there exists a unique minimal dimensional Schubert variety, Xwr,n , in Gr,n, admitting semistable points with respect to a suitable T-linearized line bundle. In our second result we explicitly construct a family of Richardson varieties inside Xwr,n such that the torus quotient of a Richardson variety in the family is a product of projective spaces.