Seminars

 PUBLIC VIVA-VOCE NOTIFICATION 2.30 pm, http://meet.google.com/buj-vznb-cba Automorphism groups of Schubert varieties and rigidity of Bott-Samelson-Demazure-Hansen varieties Pinakinath Saha Chennai Mathematical Institute. 25-06-20 Abstract This thesis consists of three problems. Throughout this thesis we consider \$G\$ to be a simple algebraic group of adjoint type over the field \$\mathbb{C}\$ of complex numbers, \$B\$ to be a Borel subgroup of \$G\$ containing a maximal torus \$T\$ of \$G.\$ Let \$G/B\$ be the full flag variety consisting of all Borel subgroups of \$G.\$ For \$w\$ in \$W,\$ let \$X(w)\$ denote the Schubert variety in \$G/B\$ corresponding to \$w.\$ In the first problem, we show that given any parabolic subgroup \$P\$ of \$G\$ containing \$B\$ properly, there is an element \$w\$ in \$W\$ such that \$P=Aut^0(X(w)).\$ In the second problem, we consider \$G =PSO(2n+1,\mathbb{C})(n \ge 3)\$. Let \$w\$ be an element of Weyl group \$W\$ and \$\underline{i}\$ be a reduced expression of \$w.\$ Let \$Z(w, \underline{i})\$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of \$X(w)\$ corresponding to \$\underline{i}\$). Here we study the cohomology modules of the tangent bundle on \$Z(w_{0}, \underline{i}),\$ where \$w_{0}\$ is the longest element of the Weyl group \$W.\$ We describe all the reduced expressions of \$w_{0}\$ in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on \$Z(w_{0}, \underline{i})\$ vanish. In the third problem, we assume that the root system of \$G\$ is of type \$F_{4}.\$ In this problem, we study the cohomology modules of the tangent bundle on \$Z(w_{0}, \underline{i}),\$ where \$w_{0}\$ is the longest element of the Weyl group \$W.\$ We describe all the reduced expressions of \$w_{0}\$ in terms of a Coxeter element such that \$Z(w_{0}, \underline{i})\$ is rigid. Further, if \$G\$ is of type \$G_{2},\$ there is no reduced expression \$\underline{i}\$ of \$w_{0}\$ for which \$Z(w_{0}, \underline{i})\$ is rigid.