Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$

Pinakinath Saha; S.Senthamarai Kannan

REVIEW 1 major objections 4 minor 15 references

Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For PSO(2n+1,C), a BSDH resolution of the full flag variety is rigid exactly when the second block of its Coxeter element does not start at n−1.

desk verdict Clean main theorem and a real type-B extension, but Lemma 3.2 has an indexing error that currently blocks the key vanishing lemma. read the letter →

arxiv 1908.05605 v1 pith:LOYQ32VG submitted 2019-08-13 math.AG math.COmath.RT

S.Senthamarai Kannan , Pinakinath Saha This is my paper · ORCID

classification math.AGmath.COmath.RT MSC 14M1514F1714D15

keywords Bott-Samelson-Demazure-HansenvarietyrigiditytangentbundlecohomologyCoxeterelementtypeB_nPSO(2n+1C)Schubertvanishingtheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines exactly when a Bott-Samelson-Demazure-Hansen (BSDH) resolution of the full flag variety is rigid in type B_n, that is, for G=PSO(2n+1,C) with n≥3. For the longest Weyl group element w0 written using n copies of a Coxeter element c, it proves that all higher cohomology of the tangent bundle vanishes—so the variety is rigid—exactly when the block decomposition of c satisfies $a_2 \ne n-1$. This matters because in non-simply-laced types rigidity is not automatic and depends on the reduced expression chosen; the paper turns that dependence into a single combinatorial condition. The proof isolates $H^{1}$ as the only obstruction and reduces the computation to the single exceptional root $\alpha_{n-1}$; all higher cohomology was already known to vanish.

What carries the argument

The central machinery is the Demazure short exact sequence (SES) of B-modules, which relates the cohomology of the tangent bundle of a BSDH variety to that of the variety obtained by deleting its last simple reflection, with error term $H^1(w,\alpha_i)$. The paper combines this with an inductive weight-space analysis over the Weyl group. The pivotal new input is Lemma 3.3: $H^1(w,\alpha_j)=0$ for every $w\in W$ and every $j\ne n-1$; this reduces the entire problem to weights of $\alpha_{n-1}$. Around this, the proof uses a type-C-to-B transfer of reduced-expression lemmas (Lemmas 2.6–2.8) and the previously known facts that $H^j=0$ for $j\ge 2$ and $H^0(Z(w_0,\underline{i}),T)$ is a parabolic subalgebra. The surviving $H^1$ weights turn out to be one-dimensional spaces indexed by $-(\beta_j+\alpha_n)$, where $\beta_j=\alpha_j+\cdots+\alpha_{n-1}$.

What would settle it

Take $n=3$ in $PSO(7,\mathbb{C})$, let $c=s_3s_2s_1$, and form $\underline{i}=(3,2,1,3,2,1,3,2,1)$; the theorem predicts $H^1(Z(w_0,\underline{i}),T)\ne 0$ because $a_2=2=n-1$. A direct computation of $H^1(s_3s_2,\alpha_2)$ via the Demazure short exact sequence should show that the weight $-(\beta_1+\alpha_3)$ with $\beta_1=\alpha_1+\alpha_2$ is present; if that weight space is zero, the main theorem is false.

Watch

Extended reading notes

Core claim

The central claim is Theorem 7.1: let $G=PSO(2n+1,\mathbb{C})$ with $n\ge 3$, let $c$ be a Coxeter element, and let $\underline{i}=(i_1,\ldots,i_n)$ be a reduced expression of $w_0$ obtained from $n$ reduced expressions of $c$ as in Lemma 2.8. Then $H^j(Z(w_0,\underline{i}),T(w_0,\underline{i}))=0$ for all $j\ge 1$ if and only if $c=\prod_{j=1}^k [a_j,a_{j-1}-1]$ with $a_0=n+1$ and $a_2\ne n-1$. Equivalently, the BSDH variety is rigid exactly under this condition. The 'only if' direction shows that when $a_2=n-1$, the shorter expression $s_ns_{n-1}$ already carries a nonzero $H^1$ class, and Lemma 6.1 forces it to survive surjectively in $H^1$ of the full $Z(w_0,\underline{i})$. The 'if' direction is a chain of isomorphisms that identifies $H^1(Z(w_0,\underline{i}),T)$ with $H^1$ of successively smaller BSDH varieties, whose weight spaces are computed explicitly and vanish precisely when $a_2\ne n-1$.

Load-bearing premise

Everything rests on Lemma 3.3, the claim that $H^1(w,\alpha_j)=0$ for every Weyl group element $w$ and every simple root $\alpha_j$ other than $\alpha_{n-1}$; if that induction has a hidden counterexample, the isomorphism chain that proves the 'if' direction collapses.

Editorial extensions

If this is right

Whenever $a_2\ne n-1$, the BSDH variety $Z(w_0,\underline{i})$ is rigid and, by Corollary 7.2, admits no deformations.
Whenever $a_2=n-1$, the BSDH variety is not rigid: a nonzero class in $H^1(s_ns_{n-1},(n,n-1))$ surjects onto $H^1(Z(w_0,\underline{i}),T)$, so rigidity fails.
Because $H^j=0$ for $j\ge 2$ was already known, the theorem settles the full cohomology of the tangent bundle for these varieties, not just the first degree.
The rank-2 case PSO(5,C) escapes the statement: the same construction has nonzero $H^1$, so the hypothesis $n\ge 3$ is essential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test whether the same dichotomy holds for arbitrary reduced expressions of $w_0$ in type $B_n$, not only those built by repeating one Coxeter element; the paper's weight analysis suggests $H^1$ would still be governed by $\alpha_{n-1}$.
The explicit description of $H^1(w_r,\alpha_{n-1})$ as a direct sum of one-dimensional weight spaces at $-(\beta_j+\alpha_n)$ makes it possible to write down the nonzero deformation class when $a_2=n-1$, rather than only proving its existence.
A parallel computation for the exceptional group $F_4$ would show whether controlling a single long root near the short root is a general feature of non-simply-laced Lie types or something specific to the $B_n/C_n$ families.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem gives a sharp criterion for vanishing of higher cohomology of the tangent bundle, hence rigidity, for the special reduced expressions of w0 in type B_n built from a Coxeter element. That is a genuine extension of the type C_n case, not a routine translation: the presence of n-1 long simple roots forces a genuinely new vanishing statement (Lemma 3.3) and a lot of new weight combinatorics. The reduction of the whole H^1 computation to the single root alpha_{n-1} is elegant, and the weight calculations in Sections 4 and 5 are detailed and mostly plausible. The use of the Weyl group isomorphism to transfer lemmas from type C_n is legitimate, since the Weyl groups are isomorphic. The soft spot is real and load-bearing. Lemma 3.3, on which every later H^1 vanishing depends, is proved via Lemma 3.2, and Lemma 3.2, Step 2, subcase i = t+1, has a concrete indexing error. From the summand V = C_{-(beta_i+2alpha_n)} + C_{-(beta_{i-1}+2alpha_n)} with i = t+1, the text picks the weight -(beta_t+2alpha_n), invokes induction, and concludes -(beta_i+alpha_n). The induction hypothesis with index t would give -(beta_t+alpha_n), not -(beta_i+alpha_n). The subsequent pairing <-(beta_t+alpha_n), alpha_t> = 1 is also off; it is -1 for that weight, and 0 for the weight actually needed. The step looks repairable--apply induction to the -(beta_i+2alpha_n) weight and use the zero pairing--but as written the proof of Lemma 3.3 does not go through, and the chain of isomorphisms in Section 7 depends on it. There are also many 'proceeding recursively' passages and typos that make verification harder, but those are secondary. One small thing: the f argument in Theorem 7.1 is terse but correct. Since H^0(s_n, alpha_n)_{alpha_n+alpha_{n-1}} = 0, exactness forces the map from H^1(u, alpha_{n-1}) to H^1(Z(u,j), T) to be injective on that weight. Who is this for? Someone working on deformation theory of Schubert or BSDH varieties in non-simply-laced types. The result is probably true, and the paper deserves a serious referee--the gap is specific and fixable, not a sign of a wrong main idea. I would send it to review rather than desk reject, but the referee must insist on a corrected Lemma 3.2.

Referee Report

1 major / 4 minor

Summary. The paper studies the cohomology of the tangent bundle on Bott-Samelson-Demazure-Hansen varieties Z(w_0, i) for G = PSO(2n+1, C) with n ≥ 3, where i is a reduced expression of the longest Weyl group element obtained by concatenating n reduced expressions of a Coxeter element c. The main theorem (Theorem 7.1) asserts that H^j(Z(w_0,i), T) = 0 for all j ≥ 1 if and only if, in the decomposition c = ∏_{j=1}^k [a_j, a_{j-1}-1] with a_0 = n+1, one has a_2 ≠ n-1. The proof reduces the computation to the single simple root α_{n-1} via Lemma 3.3, computes H^0 and H^1 weight spaces for intermediate subexpressions, and propagates vanishing through the long exact sequence from [5]. The paper ends with a remark showing the statement fails for n = 2.

Significance. If the main theorem holds, it gives a complete rigidity criterion for a natural family of BSDH varieties in type B_n, in analogy with the type C_n result of [4], and it sharpens the general vanishing results of [5]. The paper is a substantial computational contribution: the structural reduction that H^1(w, α_j) vanishes for all j ≠ n-1 is a genuinely new input for the non-simply-laced case. The authors also deserve credit for stating the criterion in a precise combinatorial form, for explicitly excluding n = 2 in Remark 7.3, and for organizing the computations around the long exact sequence of the relative tangent bundle. However, the proof is not yet in fully verifiable form because a key induction step in the proof of Lemma 3.2 contains an indexing error.

major comments (1)

[Lemma 3.2, Step 2] In Lemma 3.2, Step 2, subcase 'Assume that i = t + 1' of Case 3, the argument as written is not valid. From the summand V = C_{−(β_i+2α_n)} ⊕ C_{−(β_{i−1}+2α_n)} with i = t+1, the proof asserts H^0(v′, α_j)_{−(β_t+2α_n)} ≠ 0 and then, by the induction hypothesis, H^0(v′, α_j)_{−(β_i+α_n)} ≠ 0. The induction hypothesis applied to the weight −(β_t+2α_n) would give H^0(v′, α_j)_{−(β_t+α_n)} ≠ 0, not the claimed weight −(β_i+α_n); moreover, ⟨−(β_t+α_n), α_t⟩ = −1, not 1, so the subsequent pairing statement is false. The step is repairable: one should apply the induction hypothesis to the weight −(β_i+2α_n), which is also present in the same summand, to obtain H^0(v′, α_j)_{−(β_i+α_n)} ≠ 0, and then ⟨−(β_i+α_n), α_t⟩ = 1. Since Lemma 3.2 is the key input for Lemma 3.3, and Lemma 3.3 is used in Lemma 6.2 and throughout Section 7, this correction is necessary for the proof of the main theorem as written.

minor comments (4)

[Theorem 7.1, forward direction] In the displayed LES in the proof of (⇒), the term H^0(s_n, α_n) is used as an abbreviation for H^0(Z(s_n, (n)), T(s_n, (n))); please state this identification explicitly. Also, the sentence 'Hence f is non zero homomorphism' should be replaced by the precise statement that the weight α_n+α_{n−1} is not in the image of the incoming map, so it survives in H^1(Z(u,j),T); a nonzero map f alone would not imply the desired nonvanishing.
[Throughout] There are many typographical errors that should be corrected in a careful proofreading pass: 'surjectie' and 'Thereore' in Lemma 6.3; a nonexistent reference to '(6.2.4)' in Lemma 5.5; in Lemma 4.1(2) the weight C_{−(β_{n−1}+α_n)} should be C_{−(β_{n−1}+2α_n)}; and in Lemma 6.5 the expression H^0(Z(u_1,i_1), T(u′_1,i_1)) mixes the tuples i_1 and i′_1.
[Lemmas 4.3 and 4.4] The phrases 'it is easy to see' and 'applying SES repeatedly' are used for several substantial computations; for a computation-heavy paper, it would help the reader if the base cases and the recursive pattern for these repeated SES applications were spelled out, at least for the first two steps of each recursion.
[Lemma 5.9] The proof of Lemma 5.9 is very compressed, especially the derivation that T_{l+1}(α_{n−2}) = −α_{l−1} after the recursion; please expand this argument so that the induction is transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rigidity criterion is derived from independent lemmas, not from the vanishing it predicts.

full rationale

The derivation chain is not circular in the sense defined here. Theorem 7.1 is proved by contrapositive for the a2 = n-1 case, exhibiting a nonvanishing H^1 via Lemma 6.1, and by a chain of B-module isomorphisms for a2 != n-1, ending at Lemma 5.2(2) and Lemma 3.3. No parameter is fitted to the target cohomology, and no reduced-expression condition is defined in terms of the vanishing of H^1. The paper does cite prior works with overlapping authorship, especially [4], [5], and [14]; these provide published tools such as H^j = 0 for j >= 2, the parabolic structure of H^0, and the Weyl-group transfer from type C_n, but none of them assumes the target iff statement. Under the stated standard, this is ordinary reliance on prior theorems rather than circularity. A possible indexing slip in Lemma 3.2, Step 2, subcase i = t+1, where the induction hypothesis appears to be applied to a mismatched weight, would be a correctness gap rather than a circularity, so it is not reflected in this score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical or combinatorial entities. Its central claim rests on standard cohomology machinery for homogeneous vector bundles, on the Weyl group transfer from type C_n to B_n, and on prior published results of the same research group. These are all stated assumptions rather than free parameters.

assumptions (3)

standard math Standard facts on cohomology of homogeneous vector bundles: Demazure vanishing (Lemma 2.1), the short exact sequence SES in Section 2, and the structure of indecomposable B-modules over a minimal parabolic (Lemma 2.4, from [1]).
Used throughout Sections 3 to 7 to reduce cohomology of the tangent bundle to weight-space computations.
domain assumption The Weyl group of B_n is isomorphic to that of C_n via s_i maps to s_i, so the reduced-expression lemmas of [4] for type C_n transfer to B_n (Lemmas 2.6, 2.7, 2.8).
This is standard but glosses over the difference in root lengths; the paper relies on it to lift several lemmas from [4] without reproof.
domain assumption H^j(w, alpha_n) = 0 for all w in W (used via [14, Corollary 5.6]), and the results of [5] that H^j(Z(w,i),T) = 0 for j>=2 and that H^0(Z(w0,i),T) is a parabolic subalgebra.
These external results are taken as given; they come from papers of the same research group but are published and not re-derived in this paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$." pith.science (2026). https://pith.science/paper/LOYQ32VG

@misc{pith2026190805605,
  author       = {Pith},
  title        = {Pith review of: Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbbC)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOYQ32VG}},
  note         = {Machine review of arXiv:1908.05605}
}

abstract

Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i})$ vanish (see Theorem \ref{theorem 8.1}).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 4 canonical work pages

[5]

B.Narasimha Chary, S.Senthamarai Kannan, A.J.Parameswaran, Automorphism group of a Bott- Samelson-Demazure-Hansen variety, Transformation Groups 20 (2015), no.3, 665-698

work page 2015
[4]

2, 435-468

B.Narasimha Chary, S.Senthamarai Kannan, Rigidity of Bott-Sam elson-Demazure-Hansen variety for P Sp(2n,C), Journal of Lie Theory 27 (2017), no. 2, 435-468

work page 2017
[1]

Balaji, S

V. Balaji, S. Senthamarai Kannan, K.V. Subrahmanyam, Cohomo logy of line bundles on Schubert varieties-I, Transformation Groups 9 (2004), no.2, 105-131

work page 2004
[2]

Bott and H

R. Bott and H. Samelson, Applications of the theory of Morse to s ymmetric spaces, Amer. J. Math. 80 (1958), 964-1029

1958
[3]

Brion, S

M. Brion, S. Kumar, Frobenius Splitting Methods in Geometry and R epresentation theory, Progress in Mathematics, Vol. 231, Birkh¨ auser, Boston, Inc., Boston, MA, 2005

2005
[6]

Demazure, Desingularisation des varieties de schubert gener alisees

M. Demazure, Desingularisation des varieties de schubert gener alisees. Ann. Sci. Ecole Norm. Sup 7(1974), 53-88

1974
[7]

Demazure, A very simple proof of Bott’s theorem, Invent

M. Demazure, A very simple proof of Bott’s theorem, Invent. Ma th. 33 (1976), 271-272

1976
[8]

Hansen, On cycles on ﬂag manifolds, Math

H.C. Hansen, On cycles on ﬂag manifolds, Math. Scand. 33 (1973), 269-274

1973

Show all 15 references

[9]

Humphreys, Introduction to Lie algebras and Representat ion theory, Springer-Verlag, Berlin Heidelberg, New York, 1972

J.E. Humphreys, Introduction to Lie algebras and Representat ion theory, Springer-Verlag, Berlin Heidelberg, New York, 1972

1972
[10]

Humphreys, Linear Algebraic Groups, Springer-Verlag, Be rlin Heidelberg, New York, 1975

J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, Be rlin Heidelberg, New York, 1975

1975
[11]

Humphreys, Conjugacy classes in semisimple algebraic group s, Math

J.E. Humphreys, Conjugacy classes in semisimple algebraic group s, Math. Surveys Monographs, vol.43, Amer. Math. Soc., 1995

1995
[12]

Huybrechts, ”Complex Geometry: An Introduction”, Sprin ger-Verlag, Berlin Heidelberg, New York, 2005

D. Huybrechts, ”Complex Geometry: An Introduction”, Sprin ger-Verlag, Berlin Heidelberg, New York, 2005

2005
[13]

Jantzen, Representations of Algebraic Groups, (Second Edition ), Mathematical Surveys and Monographs, Vol.107, 2003

J.C. Jantzen, Representations of Algebraic Groups, (Second Edition ), Mathematical Surveys and Monographs, Vol.107, 2003

2003
[14]

Senthamarai Kannan, On the automorphism group of a smoo th Schubert variety

S. Senthamarai Kannan, On the automorphism group of a smoo th Schubert variety. Algebr. Repres- ent. Theory 19 (2016), no.4, 761-782

2016
[15]

Yang and A.Zelevinsky, Cluster algebras of ﬁnite type via Cox eter elements and principal minors, Transformation Groups 13 (2008), no.3-4, 855-895

S.W. Yang and A.Zelevinsky, Cluster algebras of ﬁnite type via Cox eter elements and principal minors, Transformation Groups 13 (2008), no.3-4, 855-895. RIGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN V ARIETY FOR P SO(2n + 1,C) 35 Chennai Mathematical Institute, Plot H1, SIPCOT IT ...

2008

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

B.Narasimha Chary, S.Senthamarai Kannan, A.J.Parameswaran, Automorphism group of a Bott- Samelson-Demazure-Hansen variety, Transformation Groups 20 (2015), no.3, 665-698

work page 2015

[2] [4]

2, 435-468

B.Narasimha Chary, S.Senthamarai Kannan, Rigidity of Bott-Sam elson-Demazure-Hansen variety for P Sp(2n,C), Journal of Lie Theory 27 (2017), no. 2, 435-468

work page 2017

[3] [1]

Balaji, S

V. Balaji, S. Senthamarai Kannan, K.V. Subrahmanyam, Cohomo logy of line bundles on Schubert varieties-I, Transformation Groups 9 (2004), no.2, 105-131

work page 2004

[4] [2]

Bott and H

R. Bott and H. Samelson, Applications of the theory of Morse to s ymmetric spaces, Amer. J. Math. 80 (1958), 964-1029

1958

[5] [3]

Brion, S

M. Brion, S. Kumar, Frobenius Splitting Methods in Geometry and R epresentation theory, Progress in Mathematics, Vol. 231, Birkh¨ auser, Boston, Inc., Boston, MA, 2005

2005

[6] [6]

Demazure, Desingularisation des varieties de schubert gener alisees

M. Demazure, Desingularisation des varieties de schubert gener alisees. Ann. Sci. Ecole Norm. Sup 7(1974), 53-88

1974

[7] [7]

Demazure, A very simple proof of Bott’s theorem, Invent

M. Demazure, A very simple proof of Bott’s theorem, Invent. Ma th. 33 (1976), 271-272

1976

[8] [8]

Hansen, On cycles on ﬂag manifolds, Math

H.C. Hansen, On cycles on ﬂag manifolds, Math. Scand. 33 (1973), 269-274

1973