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2.00 pm, Lecture Hall 2 PUBLIC VIVA-VOCE NOTIFICATION Torus quotients and Automorphism group of a Bott-Samelson variety B. Narasimha Chary Chennai Mathematical Institute. 27-04-16 Abstract This thesis consists of two problems: Problem 1 deals with the study of the homogeneous coordinate ring of Torus quotient of the homogeneous space. More precisely, let $G$ be a simple adjoint algebraic group over the field of complex numbers $\mathbb C$. We fix a maximal torus $T$ of $G$. Let $B$ be a Borel subgroup of $G$ containing $T$. For any dominant character $\chi$ of $T$, let $\mathcal L_{\chi}$ be the corresponding $T$-linearized line bundle on the flag variety $G/B$. Let $T\backslash \backslash(G/B)^{ss}_{T}(\mathcal L_{\chi}) $ be the GIT quotient of $G/B$ by $T$ with respect to the line bundle $\mathcal L_{\chi}$. We are interested in the following question: When the homogeneous coordinate ring of $T\backslash \backslash(G/B)^{ss}_{T}(\mathcal L_{\chi})$ is isomorphic to a polynomial ring; equivalently, when $T\backslash \backslash(G/B)^{ss}_T(\mathcal L_{\chi})$ is isomorphic to a weighted projective space. We prove that it is a polynomial ring if $\chi$ satisfies a combinatorial property in terms of a ``Coxeter element" of the Weyl group $W$ of $G$. Problem 2: We use the same notations as above. Let $X(w)$ be the Schubert variety in $G/B$ corresponding to $w\in W$. Let $Z(w,\underline i)$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline i$ of $w \in W$. We compute the connected component $Aut^0(Z(w, \underline i))$ of the automorphism group of $Z(w,\underline i)$ containing the identity automorphism. In particular, we prove that the Bott-Samelson-Demazure-Hansen varieties corresponding to the different reduced expressions of $w$ need not be isomorphic. We also prove some vanishing results for the cohomology of the tangent bundle of $Z(w, \underline i)$. As an application, we see that the varieties $Z(w, \underline i)$ are rigid when $G$ is simply-laced and their deformations are unobstructed in general.
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