On some symplectic quotients of Schubert varieties
| dc.creator | Mare, Augustin-Liviu | |
| dc.date | 2006-06-19 | |
| dc.date | 2009-05-28 | |
| dc.date.accessioned | 2026-07-07T13:18:41Z | |
| dc.date.available | 2026-07-07T13:18:41Z | |
| dc.description | Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a projective space, which is obtained by identifying $G/P$ with a coadjoint orbit of the compact Lie group $K$, where $G=K^{\mathbb C}$. The maximal torus $T$ of $K$ acts linearly on the projective space and it leaves $X$ invariant: let $Ψ: X \to {\rm Lie}(T)^*$ be the restriction of the moment map relative to the Fubini-Study symplectic form. By a theorem of Atiyah, $Ψ(X)$ is a convex polytope in ${\rm Lie}(T)^*$. In this paper we show that all pre-images $Ψ^{-1}(μ)$, $μ\in Ψ(X)$, are connected subspaces of $X$. We then consider a one-dimensional subtorus $S\subset T$, and the map $f: X\to {\mathbb R}$, which is the restriction of the $S$ moment map to $X$. We study quotients of the form $f^{-1}(r)/S$, where $r\in {\mathbb R}$. We show that under certain assumptions concerning $X$, $S$, and $r$, these symplectic quotients are (new) examples of spaces for which the Kirwan surjectivity theorem and Tolman and Weitsman's presentation of the kernel of the Kirwan map hold true (combined with a theorem of Goresky, Kottwitz, and MacPherson, these results lead to an explicit description of the cohomology ring of the quotient). The singular Schubert variety in the Grassmannian $G_2({\mathbb C}^4)$ of 2 planes in ${\mathbb C}^4$ is discussed in detail. | |
| dc.description | This is a substantially revised version of the paper initially called "On the cohomology of symplectic quotients of Schubert varieties by certain circle actions". It now has 20 pages and 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0606474 | |
| dc.identifier | http://arxiv.org/abs/math/0606474 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/231504 | |
| dc.subject | Symplectic Geometry | |
| dc.subject | Differential Geometry | |
| dc.subject | 53D20, 14L24 | |
| dc.title | On some symplectic quotients of Schubert varieties | |
| dc.type | text |