arXiv subjects
Publications and source records attributed to Pinakinath Saha.
In this article, we compute both the algebraic and the analytic Brauer groups of a homogeneous space under the action of a connected, simply connected, semisimple complex algebraic group, where the stabilizer subgroup is closed and connected.
In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit description of the fan of a toric Schubert variety, leading to a combinatorial model for its cones. As an application, we obtain necessary and sufficient conditions for smoothness of toric Schubert varieties in terms of the Cartan integers associated to a reduced expression. Furthermore, we prove that for a Coxeter-type element $w \in W^P$, the interval $[e,w]_{W^P}$ is a supersolvable join-distributive lattice. Finally, we apply these results to the study of spherical and horospherical Schubert varieties, providing a combinatorial method for checking the smoothness via the associated toric Schubert varieties.
Let $G$ be a connected simply connected semisimple complex algebraic group and $P\, \subset\, G$ a parabolic subgroup. We give a necessary and sufficient condition for a line bundle -- on the blow-up of the generalized flag variety $G/P$ along a smooth Schubert variety -- to be ample (respectively, nef). Furthermore, it is shown that every such nef line bundle is actually globally generated. As a consequence, we are able to describe when such a blow-up is (weak) Fano.
Let $X$ be a normal projective variety equipped with an action of a semisimple algebraic group $G$, and assume that $X$ contains a unique closed orbit. Let $B$ be a Borel subgroup of $G$ and let $E$ be a $B$-equivariant vector bundle on $X$. In this article, we prove that $E$ is ample (respectively, nef) if and only if its restriction to the finite set of $B$-stable curves in $X$ is ample (respectively, nef). Moreover, we compute the nef cone of the blow-up of a nonsingular simple $G$-projective variety $X$ at a unique $B$-fixed point $x^-$, referred to as the sink of $X$. As an application, when $X$ is nonsingular, we calculate the Seshadri constants of any ample line bundle (not necessarily $G$-equivariant) at $x^-$. In addition, we compute the Seshadri constants of $B$-equivariant vector bundles at $x^{-}$.
Horospherical Schubert varieties are determined. It is shown that the stabilizer of an arbitrary point in a Schubert variety is a strongly solvable algebraic group. The connectedness of this stabilizer subgroup is discussed. Moreover, a new family of spherical varieties, called doubly spherical varieties, is introduced. It is shown that every nearly toric Schubert variety is doubly spherical.
In this article, we provide characterizations of toric Richardson varieties across all types through three distinct approaches: 1) poset theory, 2) root theory, and 3) geometry.
Let $G$ be a semi-simple simply connected algebraic group over the field $\mathbb{C}$ of complex numbers. Let $T$ be a maximal torus of $G,$ and let $W$ be the Weyl group of $G$ with respect to $T$. Let $Z(w,\, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple $\underline{i}$ associated to a reduced expression of an element $w \,\in\, W.$ We prove that for the tuple $\underline{i}$ associated to any reduced expression of a minuscule Weyl group element $w,$ the anti-canonical line bundle on $Z(w,\,\underline{i})$ is globally generated. As consequence, we prove that $Z(w,\,\underline{i})$ is weak Fano. Assume that $G$ is a simple algebraic group whose type is different from $A_2.$ Let $S\,=\,\{\alpha_{1},\,\cdots,\,\alpha_{n}\}$ be the set of simple roots. Let $w$ be such that support of $w$ is equal to $S.$ We prove that $Z(w,\,\underline{i})$ is Fano for the tuple $\underline{i}$ associated to any reduced expression of $w$ if and only if $w$ is a Coxeter element and $w^{-1}(\sum_{t=1}^{n}\alpha_{t})\,\in\, -S$.
Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety $E:=G\times^{B}F,$ which has a natural action of $G$; we call it $G$-induced variety or $(G,B)$-induced variety. In this article, we compute the connected component containing the identity automorphism of the group of all algebraic automorphisms of some particular $G$-induced varieties $E.$
Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.
Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w=(s_{i_{1}},s_{i_{2}},\ldots, s_{i_{r}})$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ denote the fibre bundle over $G/B$ with the fibre over $B/B$ is $Z_{w}.$ In this article, we give necessary and sufficient conditions for the varieties $Z_{w}$ and $\widetilde{Z_{w}}$ to be Fano (weak-Fano). We show that a line bundle on $Z_{w}$ is globally generated if and only if it is nef. We show that Picard group $\text{Pic}(\widetilde{Z_{w}})$ is free abelian and we construct a $\mathcal{O}(1)$-basis. We characterize the nef, globally generated, ample and very ample line bundles on $\widetilde{Z_{w}}$ in terms of the $\mathcal{O}(1)$-basis.
Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $\alpha$ is a co-minuscule root if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_{\alpha}$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$
Let $G$ be a simple simply-laced algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $\omega_\alpha$ is a minuscule fundamental weight if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_{\alpha}$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$
Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{\alpha_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_{4n}.$ Let $X$ be a Schubert variety in $G/P^{\alpha_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2\omega_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2\omega_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{\alpha_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence, we prove that the GIT quotient of $G/P^{\alpha_{4}}$ is projectively normal with respect to the descent of the $T_{G}$-linearized very ample line bundle $\mathcal{L}(2\omega_{4})$ and is isomorphic to the projective space $(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))$ as a polarized variety. Further, we prove that $R$ is generated by $R_1$ and $R_2$ for some Schubert varieties in $G/P^{\alpha_{4n}}$ (for $n \geq 2$). As a consequence, we prove that the GIT quotient of those Schubert varieties are projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(4\omega_{4n}).$ Moreover, for $G = Spin(2n,\mathbb{C})(n \ge 4)$ (respectively, $G=Sp(2n, \mathbb{C}) (n\ge 2)$) and a maximal torus $T_G$ of $G,$ we prove that the GIT quotient of $G/P^{\alpha_{1}}$ is projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(2\omega_{1})$ and is isomorphic to the projective space $(\mathbb{P}^{n-2},\mathcal{O}_{\mathbb{P}^{n-2}}(1))$ (respectively, $(\mathbb{P}^{n-1},\mathcal{O}_{\mathbb{P}^{n-1}}(1))$ as a polarized variety.
Let $G_{n,2n}$ be the Grassmannian parameterizing the $n$-dimensional subspaces of $\mathbb{C}^{2n}.$ The Picard group of $G_{n,2n}$ is generated by a unique ample line bundle $\mathcal{O}(1).$ Let $T$ be a maximal torus of $SL(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1).$ By \cite[Theorem 3.10, p.764]{Kum}, $2$ is the minimal integer $k$ such that $\mathcal{O}(k)$ descends to the GIT quotient. In this article, we prove that the GIT quotient of $G_{n,2n}$ ($n\ge 3$) by $T$ with respect to $\mathcal{O}(2)=\mathcal{O}(1)^{\otimes 2}$ is not projectively normal when polarized with the descent of $\mathcal{O}(2).$
Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular points. Further, we have computed the exact number of singular points of the GIT quotient of $G_{2,n}.$
Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ 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 this article, 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 (see Theorem \ref{theorem 8.1}).
Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ 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 this article, 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 (see Theorem 8.1). 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 (see Theorem 8.2).
Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. In this article we show that given any parabolic subgroup $P$ of $G$ containing $B$ properly, there is an element $w\in W$ such that $P$ is the connected component, containing the identity element of the group of all algebraic automorphisms of $X(w).$