{"id":"5b2acb5b-5d3b-4486-ba3e-88ee82e2ad58","arxiv_id":"1908.05605","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Higher cohomology of the tangent bundle of the Bott-Samelson-Demazure-Hansen resolution of the longest Schubert variety in PSO(2n+1,C) vanishes if and only if the Coxeter element satisfies a2 != n-1.","lead":"This paper determines exactly which Bott-Samelson resolutions of Schubert varieties in the flag variety of the odd orthogonal group PSO(2n+1,C) have vanishing higher tangent cohomology, hence are rigid. The answer is a single inequality on the Coxeter element used to build the resolution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.2's Step 2 contains an indexing error in the i=t+1 subcase; as written, Lemma 3.3 — the vanishing H^1(w, α_j)=0 for j≠n−1 on which the whole proof reduces — is not established.","rationale":"The reader identifies Lemma 3.3 as the load-bearing structural input; my reading agrees. Lemma 3.3 is what lets the proof ignore every simple root except α_{n−1}; it is used in the induction within Lemma 3.3 itself, in Lemma 6.2, and in the final collapse of H^1(Z(w0,i)) to H^1(Z(w2,j2)) in Theorem 7.1. The proof of Lemma 3.3 depends on Lemma 3.2, and in Lemma 3.2 Step 2, subcase i=t+1, the written induction moves from the weight −(β_t+2α_n) to the conclusion H^0(v′, α_j)_{−(β_i+α_n)} with i=t+1; this does not follow from the induction hypothesis with index t, and the auxiliary pairing statement is off by a sign. If the intended step is to apply induction to the weight −(β_i+2α_n) contained in the same summand, the argument is likely repairable, but as printed it cannot be checked. I also note that the statement of Theorem 7.1 does not define a2 when k=1 (e.g., c=s_1...s_n), a case the proof never treats; this is secondary to the Lemma 3.3 gap. The proposed computational check on B_4/B_5 would test the truth of Lemma 3.3 independently of the faulty proof. Since the reader's conditional verdict already asks for clarification of the proof, I recommend no change to that verdict.","tokens_in":37202,"tokens_out":34857,"duration_ms":288111,"concrete_test":"Use the Demazure SES recursion (Lemmas 2.1–2.3) in Sage or LiE for root systems B_4 and B_5, enumerate all w in W, and compute H^1(w, α_j) for every j ≠ n−1; if any is nonzero, Lemma 3.3 is false. Independently re-derive Lemma 3.2 Step 2 with corrected indices to confirm that the induction can be repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.3 is the structural input that makes every subsequent H^1 computation reduce to α_{n−1}. Its proof depends on Lemma 3.2, and Lemma 3.2, Step 2, subcase 'Assume that i = t + 1' contains an indexing error. As written, from the summand V = C_{−(β_i+2α_n)} ⊕ C_{−(β_{i−1}+2α_n)} with i = t+1 one obtains H^0(v′, α_j)_{−(β_t+2α_n)} ≠ 0, then concludes by induction H^0(v′, α_j)_{−(β_i+α_n)} ≠ 0; the induction hypothesis with index t would give −(β_t+α_n), not −(β_i+α_n). The subsequent pairing statement ⟨−(β_t+α_n), α_t⟩ = 1 is also false (it equals −1). If the intended induction is instead applied to the weight −(β_i+2α_n) present in the same summand, the step may be repairable, but that is not what is written. Because Lemma 3.3 is used in its own induction, in Lemma 6.2, and throughout Section 7, the central theorem is not verifiable without correcting this argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":37437,"tokens_out":17167,"duration_ms":141910,"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":[{"comment":"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.","section":"Lemma 3.2, Step 2"}],"minor_comments":[{"comment":"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.","section":"Theorem 7.1, forward direction"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Lemmas 4.3 and 4.4"},{"comment":"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.","section":"Lemma 5.9"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a continuation of the authors' program and leans heavily on their earlier papers [4] and [5]. The transfer from type C_n to type B_n via the Weyl group isomorphism is legitimate, but the referee should be aware that the main theorem depends on the corrected version of Lemma 3.2. The indexing error identified in the report is locally repairable, so I do not see a reason to reject the paper, but the proof needs to be rewritten in that spot before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.","headline":"Clean main theorem and a real type-B extension, but Lemma 3.2 has an indexing error that currently blocks the key vanishing lemma.","tokens_in":745,"tokens_out":1877,"would_cite":true,"duration_ms":65733,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14F17","14D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Bott-Samelson-Demazure-Hansen variety","rigidity","tangent bundle cohomology","Coxeter element","type B_n","PSO(2n+1,C)","Schubert variety","vanishing theorem"],"falsifier":"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.","tokens_in":36972,"feed_emoji":"📐","tokens_out":12942,"duration_ms":117490,"temperature":0.7,"pith_summary":"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.","feed_headline":"One condition on a Coxeter element decides flag-resolution rigidity","feed_subtitle":"For PSO(2n+1,C), tangent-bundle cohomology vanishes exactly when a2≠n−1, so the variety is rigid.","key_machinery":"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 $\\beta_j=\\alpha_j+\\cdots+\\alpha_{n-1}$.","core_discovery":"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 $a_2\\ne n-1$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the reduced-expression lemmas from type $C_n$ that the paper transfers to type $B_n$, including the construction of $w_0$ from $n$ copies of a Coxeter element.","marker":"[4]"},{"why":"It proves the general vanishing $H^j=0$ for $j\\ge 2$, the parabolic structure of $H^0$, and the long exact sequence for tangent-bundle cohomology used throughout.","marker":"[5]"},{"why":"It gives the base vanishing $H^1(w,\\alpha_n)=0$ and the higher-cohomology vanishing for line bundles that the induction in Lemma 3.3 relies on.","marker":"[14]"},{"why":"It provides the statement about powers of Coxeter elements and fundamental weights that underlies the reduced-expression organization in Lemma 2.8.","marker":"[15]"},{"why":"It supplies the Demazure vanishing lemma from which the short exact sequences and weight-space computations are derived.","marker":"[7]"},{"why":"It introduces the BSDH desingularization and the fibration structure used to define the relative tangent bundles.","marker":"[6]"}],"fun_headline_variants":["Rigidity of BSDH variety: a2≠n−1 decides","Flag resolution rigid iff a2≠n−1 in PSO","Coxeter element decides rigidity: a2≠n−1","PSO(2n+1): rigid resolution iff a2≠n−1","Vanishing tangent cohomology: when a2≠n−1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Rigidity of BSDH variety: a2≠n−1 decides","Flag resolution rigid iff a2≠n−1 in PSO","Coxeter element decides rigidity: a2≠n−1","PSO(2n+1): rigid resolution iff a2≠n−1","Vanishing tangent cohomology: when a2≠n−1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001296,"raw_usage":{"total_tokens":5355,"prompt_tokens":1080,"completion_tokens":4275,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":696,"completion_tokens_details":{"reasoning_tokens":4177}},"tokens_in":696,"tokens_out":4275,"duration_ms":32723,"temperature":1.0,"reasoning_tokens":4177,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:32:19.375297+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"2, 435-468","cited_arxiv_id":null,"evidence_quote":"It supplies the reduced-expression lemmas from type $C_n$ that the paper transfers to type $B_n$, including the construction of $w_0$ from $n$ copies of a Coxeter element."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It proves the general vanishing $H^j=0$ for $j\\ge 2$, the parabolic structure of $H^0$, and the long exact sequence for tangent-bundle cohomology used throughout."},{"cited_title":"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","cited_arxiv_id":null,"evidence_quote":"It provides the statement about powers of Coxeter elements and fundamental weights that underlies the reduced-expression organization in Lemma 2.8."}],"review_version":1}