On the GIT quotient of Grassmannians by one dimensional torus

TL;DR

The paper analyzes the Geometric Invariant Theory quotients of Grassmannians $G(r,n)$ by a one-parameter subgroup of $SL(n,\mathbb C)$ with respect to the Plücker line bundle. It derives an explicit global structure for the quotient $X$ as a homogeneous fiber bundle $X\cong H\times_{P_{s,r}}\mathbb P(M_{s-p,r-p})$ in key cases, and shows $X$ is $H$-spherical with a rich Levi-subgroup orbit structure. The work further computes geometric and representation-theoretic properties: the Picard group, Fano property, automorphism group, cohomology of line bundles, and the $H$-module decomposition of global sections, and proves projective normality of the descended line bundle. These results illuminate how subtorus actions interpolate between torus and Levi quotients, providing explicit descriptions and moduli-theoretic interpretations in terms of parabolic inductions and determinantal varieties.

Abstract

We consider the action of the one-parameter subgroup of the special linear group corresponding to a simple root on Grassmannians and describe the structure of the associated Geometric Invariant Theory (GIT) quotients with respect to Plücker line bundle. Using the combinatorics of Weyl group elements, we explicitly describe the semistable loci and identify cases where the resulting quotient admits the structure of a parabolic induction of a projective space. We further analyze the orbit structure under the Levi subgroup, compute the Picard group, connected component of the automorphism group and examine key geometric features such as Fano properties, cohomology of line bundles, and projective normality with respect to the descended linearization.

Theorems & Definitions (52)

• Theorem 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 3.1
• Theorem 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Lemma 3.5