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Mathematics Seminar Date: Thursday, 03 October 2024 Time: 11:50 AM - 12:50 PM Venue: Lecture Hall 1 GIT quotient of Hessenberg variety modulo one dimensional torus Arkadev Ghosh Chennai Mathematical Institute. 03-10-24 Abstract Let G=PSL(n,C),T be a maximal torus of G and B be a Borel subgroup of G containing T. Let $S={\alpha_{1},...,\alpha_{n-1}}$ be the set of simple roots of G relative to (B,T). Let W=N_{G}(T)/T be the Weyl group of G relative to T. Let {\lambda_{1},...,\lambda_{n-1}} be one parameter subgroups of T dual to {\alpha_{1},...,\alpha_{n-1}}. Let X(w_{s,r}) be the minimal dimensional Schubert variety admitting semistable points for the \lambda_{s}(G_{m})-linearized ample line bundle \mathcal{L}(n\omega_{r}) on G_{r,n}. Let L_{S\setminus\{\alpha_{s}\}} be the Levi subgroup of the maximal parabolic P_{S\setminus\{\alpha_{s}\}} corresponding to \alpha_{s}, and let $B_{L_{S\setminus\{\alpha_{s}\}}}=B\cap L_{S\setminus\{\alpha_{s}\}}$. Then for any irreducible component \mathcal{X}_{v} of the Hessenberg variety with $v\in (W^{S\setminus\{\alpha_{s}\}})^{-1}$ with there are exactly two simple roots \alpha_{r}, \alpha_{t} that are made negative by v, and $v^{S\setminus\{\alpha_{r}\}}=w_{s,r}$, n does not divide rs, there is an ample line bundle \mathcal{L}(\chi) on G/B such that the GIT quotient of \mathcal{X}_{v} modulo \lambda_{s}(G_m) is isomorphic to $L_{S\setminus\{\alpha_{s}\}}\times ^{B_{L_{S\setminus\{\alpha_{s}\}}}}\mathbb{P}^{1}$. Further, if there are exactly three simple roots made negative by v, say \alpha_{r},\alpha_{t_{1}}, and \alpha_{t_{2}}, then we prove that there is an ample line bundle \mathcal{L}(\chi) on G/B such that the GIT quotient of \mathcal{X}_{v} modulo \lambda_{s}(G_m) is isomorphic to $L_{S\setminus\{\alpha_{s}\}}\times ^{B_{L_{S\setminus\{\alpha_{s}\}}}}(\mathbb{P}^{1}\times\mathbb{P}^{1})$.
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