{"id":"19db4480-b09c-427b-adf5-1afeecef0a90","arxiv_id":"2412.19793","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A vector bundle on a smooth projective toric variety that has the same full cohomology as an increasingly ordered direct sum of line bundles must itself split as that direct sum.","lead":"This paper proves a Horrocks-style splitting criterion for vector bundles on any smooth projective toric variety, under the condition that the bundle has the same cohomology as a direct sum of line bundles whose consecutive differences are ample. The result extends previous criteria for products of projective spaces and Picard rank 2 toric varieties to the general toric case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1(III) relies on an unproved identification between the diagonal and point stratifications; this bound is load-bearing for Proposition 3.3 and hence for Theorem 1.","rationale":"The paper's proof is structurally sound: the Beilinson-type spectral sequence, the twisting by O(-D_n), the matching of E_1 terms from the cohomology hypothesis, and the induction in Proposition 2.4 all check out. The decisive input from the toric resolution is concentrated in Proposition 3.3, and the only non-routine claim there is Lemma 3.1(III). The proof of (III) hinges on an asserted identification of Bondal strata on the anti-diagonal kernel with strata in the point resolution. This identification is not derived in the text, and the cited example in [HHL24] does not obviously contain the label correspondence. Since the inequality p ≤ dim P_{-E} is exactly what converts Batyrev–Borisov vanishing into the required H^q vanishing for q < p, this is the single most load-bearing concern. I agree with the reader's assessment. I do not see grounds to escalate: the identification is plausible and likely true, but it needs a rigorous derivation or a precise external statement. A concrete check on a Picard-rank-2 toric surface such as F_1 could expose a counterexample or lend confidence. Therefore the verdict should remain CONDITIONAL, i.e., UNCHANGED relative to the reader's verdict.","tokens_in":6729,"tokens_out":26952,"duration_ms":267828,"concrete_test":"Compute the HHL diagonal resolution for a smooth projective toric surface of Picard rank 2, e.g., the Hirzebruch surface F_1, using [HHL24, §3]. For each homological degree p, list the line bundles O(E')⊠O(E) that occur in K^p and check the second-factor condition p ≤ dim P_{-E}. Also derive the induced stratification of the anti-diagonal subtorus L_R from the character-pullback description of Bondal labels in [FH22, §5] and compare it with the Bondal stratification of X used for the point resolution, verifying that the second-factor labels match. If any p-dimensional stratum has dim P_{-E} < p, Lemma 3.1(III) is false and Theorem 1 is unsupported; if the match holds, the asserted correspondence is at least consistent on a nontrivial example and the gap reduces to a missing general proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1 is obtained by applying the recipe in Proposition 2.4 to the HHL resolution of the diagonal K with A = Ample(X). The only point where specific features of K enter is Proposition 3.3, whose argument rests on Lemma 3.1(III): every summand O(E')⊠O(E) of K^p satisfies p ≤ dim P_{-E}. The proof of (III) in Lemma 3.1 is a short paragraph asserting that the Bondal stratification on the anti-diagonal kernel L_R agrees with the Bondal stratification on M_{X,R}/M_X used for the resolution of a point, and that a p-dimensional stratum labeled O(E')⊠O(E) in the diagonal resolution receives the label O(E) in the point resolution. This correspondence is stated but not derived; the cited [HHL24, Exa. 3.13] only covers the kernel inheriting the stratification, not the label identification. Proposition 3.3 needs the inequality p ≤ dim P_{-E} to convert Batyrev–Borisov vanishing into H^q(O(E-D)) = 0 for q < p. If the correspondence fails, the cohomological support of K in the ample cone is unproved, and the induction in Proposition 2.4 cannot go through. The author's own Remark 3.2 notes a subtle difference between the HHL and FH22 stratifications, indicating this is not a routine bookkeeping point. The theorem may be true, but this step is not yet established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Horrocks-type splitting criterion for vector bundles on smooth projective toric varieties: if a vector bundle E has the same cohomology as a sum E' of line bundles whose successive difference divisors are ample, then E is isomorphic to E' (Theorem 1). The proof is organized around a general Fourier-Mukai recipe in Section 2, which reduces the splitting statement to the existence of a diagonal resolution that is `cohomologically supported' in the ample cone. The main technical input, Proposition 3.3, claims that the Hanlon-Hicks-Lazarev resolution of the diagonal on a toric variety has this support property, using the dimension bound in Lemma 3.1(III).","tokens_in":6981,"tokens_out":30695,"duration_ms":311741,"significance":"If the proof is completed, the result would be a substantial generalization of Horrocks' criterion to arbitrary smooth projective toric varieties, under an ordered-ampleness hypothesis analogous to the Eisenbud-Erman-Schreyer criterion for products of projective spaces. The general recipe in Proposition 2.4 is a useful organizational framework, and the intended use of the Hanlon-Hicks-Lazarev diagonal resolution is natural and promising. The paper is concise and clearly written, but the main theorem currently depends on a toric stratification assertion that is not fully established.","major_comments":[{"comment":"The proof of the bound p ≤ dim P_{-E} is not complete. The paragraph in Lemma 3.1(III) asserts that the Bondal stratification on the kernel L_R agrees with the Bondal stratification on M_{X,R}/M_X used for the resolution of a point, and that a stratum labeled O(E')⊠O(E) in the diagonal resolution receives the label O(E) in the point resolution; this label projection is the step that makes [FH22, Lem. 5.6] applicable. The cited [HHL24, Exa. 3.13] only establishes that the kernel inherits a stratification, not the equality of labels, and Remark 3.2 explicitly notes a subtle difference between the HHL and FH22 stratifications. Since Proposition 3.3 needs p ≤ dim P_{-E} to convert Batyrev-Borisov vanishing into H^q(O(E-D))=0 for q<p, this is load-bearing for Theorem 1. Please supply a detailed derivation or a precise reference for the label correspondence, including the claim in Remark 3.2 that the union of HHL strata with a given label is the unique FH22 stratum.","section":"§3, Lemma 3.1(III)"},{"comment":"In the proof of Proposition 2.4, the sentence 'E0,0_1(E) = E0,0_1(E') = O^{r_n}_X' is asserted without justification. The term E0,0_1(E') is a sum over the K^0 summands G⊠O(E) of G ⊗ H^0(X, E'⊗O(E)); it is not automatic that only the O_X⊠O_X summand contributes. This equality is used to extract the first summand in the induction, so the proof needs an explicit argument. For instance, the author should justify that, after twisting, all other possible contributions to H^0 vanish, or add a hypothesis on K^0 (such as K^0 having O_X⊠O_X as its only contributing summand) to Proposition 2.4.","section":"§2, Proposition 2.4"}],"minor_comments":[{"comment":"The divisor D+(1-ǫ)~E-E is a Q-divisor; the appeal to Batyrev-Borisov vanishing should state the version for round-ups of Q-divisors, since [CLS11, Thm. 9.3.5(b)] is usually stated for Cartier divisors.","section":"§3, Proposition 3.3"},{"comment":"The diagram in (2.1) refers to 'dotted diagonals with k > 0' without defining k; please define k (for example, k = p - q or the total degree) so that Definition 2.1 is unambiguous.","section":"§2, diagram (2.1)"},{"comment":"The reference [Tho00] appears in the bibliography but is not cited in the body of the text.","section":"References"},{"comment":"There are minor typographical issues in the rendered text, such as 'V arieties' in the title and 'vari eties' in the abstract.","section":"Title/Abstract"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is attractive and the overall strategy is credible, but the proof of Lemma 3.1(III) is too terse for a load-bearing step. I would be willing to look at a revision that expands the toric stratification argument and clarifies the identification of the term E0,0_1(E') in Proposition 2.4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe short version: this is a genuine new splitting criterion for vector bundles on arbitrary smooth projective toric varieties, and the proof strategy is attractive. But the key technical step—Lemma 3.1(III)—has a load-bearing identification that is asserted, not shown. If that step holds, Theorem 1 follows cleanly; I think it probably does, but the paper needs to close that gap before I'd call it fully proved.\n\nWhat is actually new: Theorem 1 is the first Horrocks-type criterion that works for all smooth projective toric varieties, not just P^n, products of P^n, or Picard rank 2. The hypothesis that consecutive divisors are ample is strong, but it's a natural analogue of the EES15 condition. The recipe in Section 2 is a nice abstraction; Proposition 2.4 cleanly recovers Horrocks, Ottaviani, and EES15, and it isolates exactly what one needs from the resolution of the diagonal. The application to toric varieties uses Batyrev–Borisov vanishing in a clever way to show the HHL resolution is cohomologically supported in the ample cone. The exposition is clear and the paper is short.\n\nThe soft spot is Proposition 3.3, and specifically Lemma 3.1(III). The proof of (III) asserts that the Bondal stratification of the diagonal kernel L_R agrees with the stratification used for the resolution of a point, and that a stratum labeled O(E') ⊠ O(E) in the diagonal resolution receives the label O(E) in the point resolution. This is the step that converts the dimension bound into a cohomology vanishing, so everything rests on it. The citation to [HHL24, Exa. 3.13] only covers the kernel inheriting the stratification; it doesn't establish the label identification. And the author's own Remark 3.2 says the HHL and FH22 stratifications differ in a subtle way, which makes the asserted correspondence less routine. I don't think the statement is false—it's quite plausible—but the proof as written is incomplete.\n\nThere's a minor point too: the strong ampleness hypothesis is not discussed in terms of examples or necessity, but that's not a flaw; it's just a limitation.\n\nWho should read this: people working on splitting criteria, toric geometry, or derived categories of toric varieties. It deserves a serious referee. The recommendation: send it to peer review, and ask the author to expand the proof of Lemma 3.1(III) or give a precise reference for the label correspondence. If that gets fixed, this is a nice paper.","headline":"A genuinely new and plausible splitting criterion for vector bundles on smooth projective toric varieties, but the proof's key technical lemma has a load-bearing identification that is asserted rather than demonstrated, so the paper needs a fix before it is fully convincing.","tokens_in":7557,"tokens_out":4625,"would_cite":true,"duration_ms":41504,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","14F06","14F08"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that on a smooth projective toric variety, a vector bundle whose cohomology matches an ordered sum of line bundles in every twist must itself be that sum.","keywords":["vector bundles","toric varieties","splitting criteria","cohomology vanishing","resolution of the diagonal","line bundles","integral transform","spectral sequence"],"falsifier":"Construct the diagonal resolution explicitly for a small smooth projective toric surface, such as $\\mathbb{P}^1 \\times \\mathbb{P}^1$, and search for a summand $\\mathcal{O}(E') \\boxtimes \\mathcal{O}(E)$ in the second or higher term with $\\dim P_{-E} < p$; such a summand would violate Lemma 3.1(III). Alternatively, produce a vector bundle $E$ and an ordered line-bundle sum $E'$ with equal cohomology in all twists but $E \\not\\cong E'$, refuting Theorem 1 directly.","tokens_in":6470,"feed_emoji":"📐","tokens_out":11015,"duration_ms":88238,"temperature":0.7,"pith_summary":"This paper proves a splitting criterion for vector bundles on arbitrary smooth projective toric varieties. If a vector bundle $E$ has the same cohomology $H^q(X, E \\otimes L)$ as a fixed direct sum of line bundles $E' = \\bigoplus_i \\mathcal{O}(D_i)^{r_i}$ for every degree $q \\geq 0$ and every twist $L \\in \\operatorname{Pic} X$, and the divisors $D_{i+1} - D_i$ are ample for consecutive summands, then $E$ is isomorphic to $E'$. In other words, the full cohomology table of a bundle, once it matches an increasingly ordered line-bundle sum, completely determines the bundle. This generalizes the classical splitting criterion for projective spaces to all smooth projective toric varieties. The proof builds a spectral sequence from a resolution of the diagonal by line bundles and uses it to peel off one line-bundle summand at a time.","feed_headline":"Cohomology equality forces toric bundles to split","feed_subtitle":"A splitting criterion: match the cohomology of an ordered line-bundle sum in every twist and the bundle is that sum.","key_machinery":"The load-bearing object is the resolution of the diagonal by line bundles constructed in [HHL24], used as the kernel of an integral transform. Its terms are box products $G \\boxtimes L$ of a locally free sheaf with a line bundle from a canonical collection on $X$. The key estimate, Lemma 3.1(III), says that any summand $\\mathcal{O}(E') \\boxtimes \\mathcal{O}(E)$ appearing in the $p$-th term has $p \\leq \\dim P_{-E}$, where $P_{-E}$ is the polytope of global sections of the line bundle $\\mathcal{O}(-E)$. Combined with the standard toric vanishing theorem, this makes the resolution cohomologically supported in the ample cone: for every summand of its $p$-th term, $H^q(X, \\mathcal{O}(E-D))=0$ for $q<p$ and every ample divisor $D$. The spectral sequence of the integral transform then has vanishing higher diagonals, so the zeroth term $\\mathcal{O}^{r_n}$ is a direct summand of $E$, and induction removes the remaining summands.","core_discovery":"The central claim is Theorem 1: on a smooth projective toric variety $X$, if $E$ is a vector bundle and $E'=\\bigoplus_{i=1}^n \\mathcal{O}(D_i)^{r_i}$ is a direct sum of line bundles such that $D_{i+1}-D_i$ is ample for every $0<i<n$, then equality of all cohomology groups $H^q(X,E\\otimes L)=H^q(X,E'\\otimes L)$ for $q\\ge 0$ and all $L\\in\\operatorname{Pic} X$ forces $E\\cong E'$. The proof first establishes a general recipe: whenever a variety has a locally free resolution of the diagonal that is cohomologically supported in a cone $A$ and the line bundles are ordered by consecutive differences in $A$, cohomology equality implies splitting. For toric varieties the needed support statement is proved for $A=\\operatorname{Ample}(X)$: every line-bundle summand in the $p$-th term of the diagonal resolution satisfies $p\\le \\dim P_{-E}$, the dimension of the section polytope of $-E$, and the toric vanishing theorem then kills all higher spectral-sequence terms. The theorem is obtained by induction, peeling the highest line-bundle summand off $E$ and repeating.","pith_inferences":["Implicit in the proof: the ordering hypothesis is used only to locate the top summand inside a cone; if a toric variety admitted a diagonal resolution supported in a larger cone, the same argument would give a splitting criterion with weaker hypotheses.","Testable extension: on a smooth projective toric surface of Picard rank two, one could search computationally for a vector bundle whose cohomology matches a non-ordered line-bundle sum; finding one would show the ample-ordering hypothesis is necessary.","The dimension bound $p \\leq \\dim P_{-E}$ can be checked directly for small examples by writing out the diagonal resolution and comparing the stratification labels; any violation would pinpoint where the support argument fails.","If the theorem is correct, multigraded cohomology tables on toric varieties are rigid enough to certify splitting, which suggests a practical route to automatic splitting certificates in computer algebra systems."],"forward_implications":["A vector bundle on any smooth projective toric variety that has the same cohomology as an ordered line-bundle sum in every twist is actually isomorphic to that sum, so cohomology tables determine such bundles completely.","The classical splitting criterion for projective space appears as the special case in which the Picard group is cyclic and every line-bundle sum can be ordered by degree; the theorem recovers it without an inductive restriction argument.","The theorem gives a new obstruction to indecomposability: a bundle that matches an ordered line-bundle sum cohomologically must be decomposable.","The recipe in Section 2 turns any resolution of the diagonal supported in a cone $A$ into a splitting criterion for line bundles ordered inside $A$, so new resolutions immediately yield new criteria on other varieties."],"supporting_citations":[{"why":"Constructs the resolution of the diagonal by line bundles whose summands and vanishing behavior are the main input.","marker":"[HHL24]"},{"why":"Provides the Picard-rank-two case and the monad/spectral-sequence recipe adapted in Section 2.","marker":"[BS24]"},{"why":"Supplies the product-of-projective-spaces criterion and the spectral-sequence lemmas used in the general recipe.","marker":"[EES15]"},{"why":"Establishes the equality between the dimension of the label stratum and the section polytope used in Lemma 3.1(III).","marker":"[FH22]"},{"why":"Introduces the stratification of the real torus by divisors that labels the terms of the resolution.","marker":"[Bon06]"},{"why":"Supplies the toric vanishing theorem that turns the dimension bound into cohomology vanishing.","marker":"[CLS11]"}],"fun_headline_variants":["Cohomology equality splits toric bundles","All-twist cohomology match implies toric sum","Toric splitting from identical cohomology","When cohomology agrees, toric bundles split","Horrocks-type theorem for toric varieties"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that in the diagonal resolution from [HHL24], every line-bundle summand in the $p$-th term comes from a $p$-dimensional stratum whose label matches the corresponding stratum in the resolution of a point, which yields the bound $p \\leq \\dim P_{-E}$; if this correspondence fails, the resolution need not be cohomologically supported in the ample cone and the induction cannot start.","fun_headline_variants_meta":{"raw":{"variants":["Cohomology equality splits toric bundles","All-twist cohomology match implies toric sum","Toric splitting from identical cohomology","When cohomology agrees, toric bundles split","Horrocks-type theorem for toric varieties"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000298,"raw_usage":{"total_tokens":1668,"prompt_tokens":833,"completion_tokens":835,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":762}},"tokens_in":449,"tokens_out":835,"duration_ms":8302,"temperature":1.0,"reasoning_tokens":762,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:50:26.201512+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct the diagonal resolution explicitly for a small smooth projective toric surface, such as $\\mathbb{P}^1 \\times \\mathbb{P}^1$, and search for a summand $\\mathcal{O}(E') \\boxtimes \\mathcal{O}(E)$ in the second or higher term with $\\dim P_{-E} < p$; such a summand would violate Lemma 3.1(III). Alternatively, produce a vector bundle $E$ and an ordered line-bundle sum $E'$ with equal cohomology in all twists but $E \\not\\cong E'$, refuting Theorem 1 directly.","supporting_citations":[],"review_version":1}