GIT quotient of minimal dimensional Schubert variety modulo a subtorus

Abstract: Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $ω<em>{r}$ denote the $r<sup>{th}$ fundamental weight. Let $\mathcal{L}(nω</em>{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $nω<em>{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w</em>{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(nω<em>{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w</em>{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r<sup>{th}$ stage of an iterated projective space bundle over $\mathbb{P}<sup>{q-1}$.

• The paper establishes the existence and uniqueness of the semistable minimal Schubert variety through explicit weight and combinatorial calculations.
• It constructs the GIT quotient as a projective bundle with a recursive Bott tower structure, employing diagonal transition functions for line bundle splitting.
• The approach integrates combinatorial indexing and Luna’s slice theorem to verify smoothness and elucidate the geometry of the quotient.

GIT Quotients of Minimal Dimensional Schubert Varieties Modulo Subtori in the Grassmannian

Introduction and Context

This work systematically investigates the structure of the Geometric Invariant Theory (GIT) quotient of a minimal dimensional Schubert variety in the Grassmannian $G_{r,n}$ under the action of a specific subtorus $T_{J_r}$ of a maximal torus $T \subset PSL(n,\mathbb{C})$. The setting leverages the known characterization of such minimal Schubert varieties $X(w_{r,n})$ as unique loci admitting semistable points with respect to a $T$-linearized very ample line bundle $\mathcal{L}(n\omega_r)$, under the congruence condition $n = rq + 1$ with $q \geq 2$.

Prior literature established one-dimensional torus quotients, and general properties of semistability and stability for torus actions on Schubert varieties and Grassmannians ("GIT quotient of Schubert varieties modulo one dimensional torus" [GKgit], Hausmann–Knutson, Seshadri, Kannan et al.). This paper generalizes to higher-rank subtori associated with the so-called "peaks" of the unique Weyl group element $w_{r,n}$, and thoroughly analyzes the quotient geometry, transition functions, and bundle structures.

Main Results

Existence and Uniqueness of the Semistable Schubert Variety

For $G = PSL(n,\mathbb{C})$, $T_{J_r}$0 maximal torus, and $T_{J_r}$1 the associated maximal parabolic, the authors review and then extend previous combinatorial criteria for determining the unique minimal-dimensional Schubert variety $T_{J_r}$2 in $T_{J_r}$3 that admits nonempty semistable locus for the $T_{J_r}$4-linearized bundle $T_{J_r}$5. The characterization of $T_{J_r}$6 and the reduced word for $T_{J_r}$7 are both explicit and foundational to the later bundle-theoretic results.

The authors use detailed analysis of Bruhat orderings, root combinatorics, and the Hilbert-Mumford criterion to show that $T_{J_r}$8 is also minimal among Schubert subvarieties for nonempty semistable locus under the action of $T_{J_r}$9, a proper subtorus of $T \subset PSL(n,\mathbb{C})$0 determined by "peaks" of $T \subset PSL(n,\mathbb{C})$1. They classify the semistable locus and prove, via explicit weight and combinatorial calculations, that for $T \subset PSL(n,\mathbb{C})$2 this locus always coincides with the stable locus—ensuring that the resulting quotient is a geometric quotient (i.e., orbits are separated and the action is free).

Structure of the GIT Quotient: Projective Bundle and Generalized Bott Tower

The chief structural result is that the GIT quotient $T \subset PSL(n,\mathbb{C})$3 is isomorphic to a projective bundle of the form

$T \subset PSL(n,\mathbb{C})$4

where $T \subset PSL(n,\mathbb{C})$5, with $T \subset PSL(n,\mathbb{C})$6 line bundles on $T \subset PSL(n,\mathbb{C})$7, and $T \subset PSL(n,\mathbb{C})$8 itself a GIT quotient of the analogous Schubert variety in $T \subset PSL(n,\mathbb{C})$9 by the corresponding subtorus. The transition functions are shown to be diagonal, ensuring $X(w_{r,n})$0 splits into a sum of line bundles.

By recursively applying this construction, the paper demonstrates that $X(w_{r,n})$1 is the $X(w_{r,n})$2-th stage of a generalized Bott tower (in the sense of [Bott tower]). Explicit open affine covers, local trivialisations, and cocycle conditions for the transition functions are presented, tracking the subtori actions and the combinatorics of the Schubert cell coordinates.

Explicit Fiber Type and Smoothness

The authors establish that the bundle is locally trivial with fiber type $X(w_{r,n})$3 over $X(w_{r,n})$4. The base case of the recursion ($X(w_{r,n})$5) is the projective space $X(w_{r,n})$6. Furthermore, using a careful analysis and Luna's slice theorem, the quotient variety is shown to be smooth.

Technical Contributions

• Combinatorial Indexing: The construction and partition of index sets $X(w_{r,n})$7 and subsets $X(w_{r,n})$8 generalizing the positions of free parameters in the Schubert cell matrices, together with recursion steps $X(w_{r,n})$9, provide a combinatorial machinery for tracking coordinates and equations in each iterated quotient.
• Weight Calculations and Invariants: The step-by-step construction of invariant monomials in the local ring and calculation of their weights under the subtorus action is carried out using analysis of root contributions. This underpins the verification of the Hilbert-Mumford criterion at each stage.
• Split Vector Bundle Structure: By constructing explicit trivializations and expressing transition functions as diagonal automorphisms (products of scalars arising from coordinate ring monomials), the work demonstrates that every projectivized bundle appearing in the Bott tower splits, further allowing identification of the cohomological type.
• Generalized Bott Tower Interpretation: The paper codifies the recursive structure of the quotient into the language of locally split vector bundles leading to projective bundles, verifying at every step the Bott tower axioms. This links the algebraic quotient geometry to the extensively-studied topological and algebro-geometric properties of generalized Bott manifolds.

Implications and Future Directions

This work provides a precise description of the quotient structure for a broad class of minimal Schubert varieties in Grassmannians under subtori actions, revealing a (split) Bott tower structure and quantifying the vector bundle splitting in terms of line bundles. The methods demonstrate how toric, Schubert, and GIT-theoretic constructions interact at the level of explicit coordinates, producing smooth geometric quotients directly tied to parabolic subgroup combinatorics. This provides a blueprint for understanding quotient presentations in other homogeneous varieties, potentially extending to other root systems or types ($T$0, $T$1, $T$2), as suggested by mention of Kannan and Pattanayak's classifying results.

The explicit identification of transition functions may facilitate further computations of cohomology, Chern classes, or invariants of the resulting Bott tower. The GIT perspective applied here may also shed light on moduli problems, descriptions of moduli of quiver varieties, and connections to polygon space compactifications (cf. Hausmann–Knutson), particularly for quotients modulo non-maximal tori.

Applications to rationality problems, orbit structure, and smooth resolution models for Schubert varieties are anticipated, and the techniques may further assist in classifying possible degenerations or extensions to quantum Schubert calculus.

Conclusion

This paper offers a rigorous, explicit, and constructive analysis of the GIT quotient of minimal Schubert varieties in the Grassmannian by suitably chosen subtori. By elucidating the split projective bundle and Bott tower structure in the quotient, and providing combinatorial recipes for constructing the associated line bundles and trivializations, the work advances the algebraic and geometric understanding of torus quotients in flag varieties and related moduli. The results clarify and extend earlier approaches to Schubert variety semistability and provide a solid foundation for further exploration of quotient structures and their applications in representation theory and algebraic geometry.