Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root

Jaehyun Hong; Minhyuk Kwon

arxiv: 1907.09694 · v1 · pith:KWP2DQZ3new · submitted 2019-07-23 · 🧮 math.AG · math.DG

Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root

Jaehyun Hong , Minhyuk Kwon This is my paper

Pith reviewed 2026-05-24 17:35 UTC · model grok-4.3

classification 🧮 math.AG math.DG

keywords Schubert varietiesrigidityrational homogeneous manifoldsshort rootshomology classesautomorphism groupsalgebraic geometry

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The pith

Smooth Schubert varieties in short-root rational homogeneous spaces are rigid unless linear.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies all smooth Schubert varieties inside rational homogeneous manifolds built from short roots. It then proves a rigidity statement: any subvariety sharing the same homology class must arise as an orbit under the connected component of the automorphism group of the ambient space. This fails only when the Schubert variety is linear. A reader cares because the result pins down exactly which cycles can appear in these homogeneous spaces and rules out other realizations with the same topological invariants.

Core claim

We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.

What carries the argument

Rigidity of a smooth Schubert variety with respect to its homology class under the action of the connected automorphism group Aut_0(S).

If this is right

Non-linear smooth Schubert varieties have no other realizations in their homology class outside Aut_0(S)-orbits.
The classification exhausts all smooth Schubert varieties in the allowed ambient spaces.
Linear Schubert varieties are the only ones that may admit additional subvarieties in the same class.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests that short-root homogeneous spaces have unusually constrained cycle geometry compared with longer-root cases.
One could test whether similar rigidity holds after allowing mild singularities or after base change to other fields.
The classification may interact with questions about which homology classes are represented by smooth subvarieties at all.

Load-bearing premise

The ambient space must be a rational homogeneous manifold associated to a short root.

What would settle it

A concrete subvariety with the same homology class as a non-linear smooth Schubert variety S_0 that is not an Aut_0(S)-orbit would falsify the rigidity statement.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Classifies smooth Schubert varieties in short-root rational homogeneous spaces and proves they are rigid under the homology class condition, except when linear.

read the letter

The main point is a classification of smooth Schubert varieties inside rational homogeneous manifolds associated to short roots, plus a rigidity theorem: any subvariety with the same homology class is an orbit under the connected automorphism group unless the variety is linear. That is the concrete output the paper delivers. It organizes a slice of these subvarieties and gives a clean statement about when homology determines the subvariety up to automorphism. The exception for linear cases shows the authors checked the boundary behavior. The result sits inside the existing literature on Schubert varieties and homogeneous spaces without obvious circularity or invented parameters. The restriction to short roots keeps the claim manageable but also limits how far it travels. Without the full proofs in front of me it is impossible to verify whether the classification list is exhaustive or whether the deformation argument has any gaps, but the abstract states the claim precisely and the exception clause is a positive sign. This is aimed at people already working on Schubert calculus or rigidity questions in algebraic geometry. A reader in that niche would get a usable list and a usable rigidity statement. The paper is focused and formally stated enough that it deserves referee time rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper classifies smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and proves they are rigid: any subvariety of S with the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.

Significance. If the classification is exhaustive and the rigidity statement holds, the result would add to the literature on rigidity and deformation of Schubert varieties in homogeneous spaces, with potential implications for understanding automorphism groups and homology classes in algebraic geometry of flag varieties and rational homogeneous manifolds.

major comments (1)

The provided text consists solely of the abstract statement of the classification and rigidity theorem. No sections, equations, or proof sketches are visible, so the completeness of the list of smooth Schubert varieties and the deformation/rigidity argument cannot be assessed for gaps or correctness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. We address the sole major comment below.

read point-by-point responses

Referee: The provided text consists solely of the abstract statement of the classification and rigidity theorem. No sections, equations, or proof sketches are visible, so the completeness of the list of smooth Schubert varieties and the deformation/rigidity argument cannot be assessed for gaps or correctness.

Authors: The full manuscript, including the complete classification of smooth Schubert varieties associated to short roots, all equations, detailed proofs of rigidity (via homology preservation and automorphism action), and case-by-case analysis, is available on arXiv:1907.09694. The abstract was supplied as a concise summary for the review process, but the referee can access the entire document there to evaluate the arguments for gaps or correctness. No changes to the manuscript are required on this basis. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states a classification of smooth Schubert varieties together with a rigidity theorem in the setting of rational homogeneous manifolds for short roots. No equations, fitted parameters, or empirical reductions appear in the provided abstract or description. The result is presented as a theorem proved via standard methods in algebraic geometry and Lie theory (root systems, parabolic subgroups, homology classes, and automorphism actions), without any of the enumerated circular patterns such as self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The derivation is therefore self-contained against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the setting itself (rational homogeneous manifold associated to short root) functions as an unexpanded domain assumption.

pith-pipeline@v0.9.0 · 5567 in / 1071 out tokens · 29725 ms · 2026-05-24T17:35:22.658978+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We classify smooth Schubert varieties S0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3... non-linear smooth Schubert variety S0 ... horospherical variety ... (Cm, αi+1, αi) ...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Billey and A

S. Billey and A. Postnikov, Smoothness of Schubert varieties via patterns in root subsy stems, Adv. in Appl. Math. 34 (2005), no. 3, 447–466

work page 2005
[2]

Hong and N

J. Hong and N. Mok Characterization of smooth Schubert varieties in rational homogeneous man- ifolds of Picard number 1 , J. Algebraic Geom. 22 333–362 (2013)

work page 2013
[3]

Hong and K.-D

J. Hong and K.-D. Park, Characterization of Standard Embeddings between Rational Homogeneous Manifolds of Picard Number 1 , Int. Math. Res. Notices, 2011 2351–2373 (2011)

work page 2011
[4]

Hong, Smooth horospherical varieties of Picard number one as line ar sections of rational homo- geneous varieties , J

J. Hong, Smooth horospherical varieties of Picard number one as line ar sections of rational homo- geneous varieties , J. Korean Math. Soc. 53 no. 2, 433–459 (2016)

work page 2016
[5]

Hong, Classiﬁcation of smooth Schubert varieties in the symplect ic Grassmannian , J

J. Hong, Classiﬁcation of smooth Schubert varieties in the symplect ic Grassmannian , J. Korean Math. Soc. 52 no. 5, 1109–1122 (2015)

work page 2015
[6]

Hwang and N

J.-M. Hwang and N. Mok, Varieties of minimal rational tangents on uniruled project ive mani- folds, in:M. Schneider, Y.-T. Siu (Eds.), Several complex variab les, MSRI publications, Vol 37, Cambridge University Press, 1999, pp. 351 – 389

work page 1999
[7]

Hwang and N

J.-M. Hwang and N. Mok, Deformation rigidity of the rational homogeneous space ass ociated to a long simple root , Ann. Scient. ´Ec. Norm. Sup., 4 e s´ erie, t. 35. 2002, p. 173 ´ a 184

work page 2002
[8]

Mok, Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root

J.-M., Hwang and N. Mok, Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root . Algebraic transformation groups and algebraic varieties , 37–58, Ency- clopaedia Math. Sci., 132, Springer, Berlin, 2004

work page 2004
[9]

Hwang and N

J.-M. Hwang and N. Mok, Prolongations of inﬁnitsimal linear automorphisms of proj ective varieties and rigidity of rational homogeneous spaces of Picard numbe r 1 under K¨ ahler deformation, Invent. Math. 160 (2005), no. 3, 591–645

work page 2005
[10]

Manivel, On the projective geometry of rational homogeneous varieti es, Comment

J.M Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieti es, Comment. Math. Helv. 78 (2003), no. 1, 65–100

work page 2003
[11]

I. A. Mihai, Odd symplectic ﬂag manifolds , Transform. Groups 12 (2007), no. 3, 573–599

work page 2007
[12]

Mok, Geometric structures and substructures on uniruled projec tive manifolds , Foliation Theory in Algebraic Geometry (Simons Symposia), P

N. Mok, Geometric structures and substructures on uniruled projec tive manifolds , Foliation Theory in Algebraic Geometry (Simons Symposia), P. Cascini, J. McK ernan and J.V.Pereira, Springer- Verlarg, 103–148, 2016. 16 J. HONG AND M. KWON

work page 2016
[13]

Pasquier, On some smooth projective two-orbit varieties with Picard n umber one , Math

B. Pasquier, On some smooth projective two-orbit varieties with Picard n umber one , Math. Ann. 344, no. 4, 963–987 (2009)

work page 2009
[14]

Polo, On Zariski tangent spaces of Schubert varietie s, and a proof of a conjecture of Deodhar

P. Polo, On Zariski tangent spaces of Schubert varietie s, and a proof of a conjecture of Deodhar. Indag.Mathem., Volume 5, Issue 4, 483–493, (1994)

work page 1994
[15]

Richmond and W

E. Richmond and W. Slofstra, Billey-Postnikov decompositions and the ﬁber bundle struc ture of Schubert varieties , Math. Ann. (2016) 366:31-55 Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun -gu, Seoul 02455, Korea E-mail address : jhhong00@kias.re.kr Department of Mathematical Sciences, Seoul National University, 599 Gw anak-ro Gw anak- gu Se...

work page 2016

[1] [1]

Billey and A

S. Billey and A. Postnikov, Smoothness of Schubert varieties via patterns in root subsy stems, Adv. in Appl. Math. 34 (2005), no. 3, 447–466

work page 2005

[2] [2]

Hong and N

J. Hong and N. Mok Characterization of smooth Schubert varieties in rational homogeneous man- ifolds of Picard number 1 , J. Algebraic Geom. 22 333–362 (2013)

work page 2013

[3] [3]

Hong and K.-D

J. Hong and K.-D. Park, Characterization of Standard Embeddings between Rational Homogeneous Manifolds of Picard Number 1 , Int. Math. Res. Notices, 2011 2351–2373 (2011)

work page 2011

[4] [4]

Hong, Smooth horospherical varieties of Picard number one as line ar sections of rational homo- geneous varieties , J

J. Hong, Smooth horospherical varieties of Picard number one as line ar sections of rational homo- geneous varieties , J. Korean Math. Soc. 53 no. 2, 433–459 (2016)

work page 2016

[5] [5]

Hong, Classiﬁcation of smooth Schubert varieties in the symplect ic Grassmannian , J

J. Hong, Classiﬁcation of smooth Schubert varieties in the symplect ic Grassmannian , J. Korean Math. Soc. 52 no. 5, 1109–1122 (2015)

work page 2015

[6] [6]

Hwang and N

J.-M. Hwang and N. Mok, Varieties of minimal rational tangents on uniruled project ive mani- folds, in:M. Schneider, Y.-T. Siu (Eds.), Several complex variab les, MSRI publications, Vol 37, Cambridge University Press, 1999, pp. 351 – 389

work page 1999

[7] [7]

Hwang and N

J.-M. Hwang and N. Mok, Deformation rigidity of the rational homogeneous space ass ociated to a long simple root , Ann. Scient. ´Ec. Norm. Sup., 4 e s´ erie, t. 35. 2002, p. 173 ´ a 184

work page 2002

[8] [8]

Mok, Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root

J.-M., Hwang and N. Mok, Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root . Algebraic transformation groups and algebraic varieties , 37–58, Ency- clopaedia Math. Sci., 132, Springer, Berlin, 2004

work page 2004

[9] [9]

Hwang and N

J.-M. Hwang and N. Mok, Prolongations of inﬁnitsimal linear automorphisms of proj ective varieties and rigidity of rational homogeneous spaces of Picard numbe r 1 under K¨ ahler deformation, Invent. Math. 160 (2005), no. 3, 591–645

work page 2005

[10] [10]

Manivel, On the projective geometry of rational homogeneous varieti es, Comment

J.M Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieti es, Comment. Math. Helv. 78 (2003), no. 1, 65–100

work page 2003

[11] [11]

I. A. Mihai, Odd symplectic ﬂag manifolds , Transform. Groups 12 (2007), no. 3, 573–599

work page 2007

[12] [12]

Mok, Geometric structures and substructures on uniruled projec tive manifolds , Foliation Theory in Algebraic Geometry (Simons Symposia), P

N. Mok, Geometric structures and substructures on uniruled projec tive manifolds , Foliation Theory in Algebraic Geometry (Simons Symposia), P. Cascini, J. McK ernan and J.V.Pereira, Springer- Verlarg, 103–148, 2016. 16 J. HONG AND M. KWON

work page 2016

[13] [13]

Pasquier, On some smooth projective two-orbit varieties with Picard n umber one , Math

B. Pasquier, On some smooth projective two-orbit varieties with Picard n umber one , Math. Ann. 344, no. 4, 963–987 (2009)

work page 2009

[14] [14]

Polo, On Zariski tangent spaces of Schubert varietie s, and a proof of a conjecture of Deodhar

P. Polo, On Zariski tangent spaces of Schubert varietie s, and a proof of a conjecture of Deodhar. Indag.Mathem., Volume 5, Issue 4, 483–493, (1994)

work page 1994

[15] [15]

Richmond and W

E. Richmond and W. Slofstra, Billey-Postnikov decompositions and the ﬁber bundle struc ture of Schubert varieties , Math. Ann. (2016) 366:31-55 Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun -gu, Seoul 02455, Korea E-mail address : jhhong00@kias.re.kr Department of Mathematical Sciences, Seoul National University, 599 Gw anak-ro Gw anak- gu Se...

work page 2016