{"id":"1d43f430-44f2-4e34-933c-00b472361ef7","arxiv_id":"2506.03026","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The local cohomological defect of an affine toric variety is characterized by Lefschetz cup-product maps on a projective toric variety of one dimension lower, which shows it is not a combinatorial invariant and allows examples in every dimension.","lead":"Toric varieties are geometric objects built from combinatorial fans, and this paper shows that a subtle singularity invariant, the local cohomological defect, is governed by cohomology of a simpler lower-dimensional space. The result proves the invariant is not determined by the fan's combinatorics, and it constructs explicit examples in every dimension.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 is not actually proved: the one-line reduction to Corollaries 3.5-3.6 omits the identification of the affine cone with the cone over (E, (-E)|_E); a sign or divisor error here would invalidate the main claim.","rationale":"The reader's verdict is CONDITIONAL, with the weakest assumption being the dependence on the authors' unpublished [KV25] and the asserted Proposition 3.7. I partially agree, but the more load-bearing issue is that Theorem 1.4 itself lacks a proof: the reduction to Corollaries 3.5-3.6 is a one-sentence assertion, and the identification it depends on is not supplied. This is an internal gap, not a disagreement with consensus. The paper contains substantial original material (the Ishida-complex description of the Chern class map, the pyramid theorem, shelling results) and the overall strategy is sound; I found no internal contradiction in the parts that are written out. In particular, the Euler-characteristic argument in Proposition 1.9 and the connecting-homomorphism analysis in Theorem 1.8 are coherent. However, because the main theorem's proof is omitted at the crucial step, the appropriate verdict is CONDITIONAL: the authors should either write out the identification or provide the computational verification described above. This does not change the reader's verdict but sharpens the condition.","tokens_in":20928,"tokens_out":29371,"duration_ms":309164,"concrete_test":"Perform the missing identification explicitly in the simplest nontrivial case: X=A^2, sigma=cone(e1,e2), rho=e1+e2. Compute the fan of E=P^1, the divisor (-E)|_E=O(1), and the cone sigma constructed from (E,O(1)) as in Sections 3.1-3.5; verify sigma is GL(2,Z)-isomorphic to the original cone. Then compute the LES of Theorem 1.4 for l=0,1 and compare it with the LES coming from Corollary 3.6; the two must agree termwise, including the sign of the connecting map. As a second check, implement Ish^l_sigma and the map c1^vee for the 4-dimensional cone sigma of Example 1.7 using a Macaulay2 script, and independently verify the resulting LES against the statement of Theorem 1.4.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central result, Theorem 1.4, is not proved in the text. Section 3.5 ends with 'rephrasing Corollaries 3.5 and 3.6 immediately gives a proof of Theorem 1.4', but the actual bridge is never written down. To get Theorem 1.4 from Corollary 3.6 one must: (i) choose coordinates so the interior ray rho is a coordinate ray; (ii) show that the star of rho in the subdivided fan, quotient by <rho>, is the fan of E; (iii) show that the support function of the Q-divisor (-E)|_E is the function psi used to build the cone sigma in Section 3.5; and (iv) verify that under this identification Ish^l_sigma is precisely the middle term of the short exact sequence of Corollary 3.6, with the connecting homomorphism equal to the dual of c1((-E)|_E). None of these steps appears. The identification is not a formality: it determines the sign and the divisor class (E versus -E), and a single sign error in the connecting homomorphism would change the long exact sequence and therefore the lcdef computations in Corollary 1.6 and the non-invariance example. The reader's verdict flags Proposition 3.7 as asserted; the same gap is more acute for Theorem 1.4, since the theorem is the paper's headline and its proof is a one-sentence reference.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an explicit fan-theoretic description of the cup product with the first Chern class of a line bundle on a proper toric variety, via the Grothendieck dual of the Atiyah class and Ishida complexes. The main result, Theorem 1.4, states that for an affine toric variety X associated to a cone sigma, inserting an interior ray rho yields a projective toric variety E such that the cohomology of the Ishida complex of sigma is governed by a long exact sequence involving the cohomology of Ishida complexes of E and the map dual to c_1((-E)|_E). From this the authors derive a criterion for the local cohomological defect in dimension four, prove that lcdef is not a combinatorial invariant, prove invariance under pyramids, and give several combinatorial criteria for lcdef in dimension four.","tokens_in":21206,"tokens_out":11367,"duration_ms":111883,"significance":"If the main theorem is correct, it is a substantial result: it gives a genuinely combinatorial description of the Lefschetz morphism on singular cohomology of toric varieties and connects it to the local cohomological defect of affine toric varieties. The paper also contains an attractive set of applications, including the non-invariance of lcdef and a pyramid-invariance theorem. The algebraic development in Sections 3.3 and 3.4 is detailed, and the exact sequences there are concrete and checkable. The main weakness is that the proof of the headline theorem is not actually written down: the bridge from the projective construction to the affine cone over (E, (-E)|_E) is asserted in one sentence, and Proposition 3.7 is likewise asserted rather than proved. Several load-bearing statements are imported from the authors' unpublished preprint [KV25], which further limits the self-containedness of the paper.","major_comments":[{"comment":"Proposition 3.7 is not proved. The entire proof is the sentence that Ish^{p+1}_sigma is the middle term in the short exact sequence of Corollary 3.6. That identification is not shown. The complex of Corollary 3.6 has terms indexed by both e_tau and b_tau, whereas the cone sigma, as stated in §3.5, has only the b_tau faces. One needs an explicit quasi-isomorphism, or at least a coherent argument identifying Ish^{p+1}_sigma with the middle term of that exact sequence. Without this identification, the equivalence between the vanishing of H^l(Ish^{p+1}_sigma) and the stated injectivity/surjectivity conditions on c_1(D) is unsupported.","section":"§3.5, Proposition 3.7"},{"comment":"The proof of Theorem 1.4 is reduced to a single sentence: \"rephrasing Corollaries 3.5 and 3.6 immediately gives a proof of Theorem 1.4.\" This is not a formality. One must prove that, after placing the interior ray rho on a coordinate axis, the star of rho in the subdivided fan modulo <rho> is the fan of E; that the original cone sigma is identified with the cone over (E, (-E)|_E); that the support function defining sigma is that of the restriction of -E to E; and that Ish^l_sigma is exactly the middle term of Corollary 3.6, with the connecting homomorphism equal to the dual of c_1((-E)|_E). The last point fixes the sign in the long exact sequence, and a sign error would change all lcdef computations in Corollary 1.6 and Example 1.7. These steps need to be written out.","section":"§3.5, proof of Theorem 1.4"},{"comment":"Several results that are load-bearing for the paper's applications are taken from the authors' previous preprint [KV25] without proof. These include the statement that singular cohomology of a proper toric variety is mixed of Hodge-Tate type, the surjectivity of the last differential of Ish^p_X for p >= n/2, and the equality lcdef = m for (m+3)-dimensional non-simplicial toric varieties with isolated non-simplicial locus. Since [KV25] is an unpublished preprint, the current manuscript's main applications depend on results that are not verifiable from the present text. The authors should either include precise proofs of these facts in an appendix or replace the dependence with a published reference.","section":"§1, Proposition 1.5, Corollary 1.6, and paragraph after Theorem 1.8"}],"minor_comments":[{"comment":"The claims that the two cones in Example 1.7 can be checked with Macaulay2, and that Example 4.4 has dim H^1 = dim H^2 = 1, would be much more useful if the actual Macaulay2 script or the output were included; as written, these computations are an assertion about a non-combinatorial invariant and should be reproducible.","section":"Examples 1.7 and 4.4"},{"comment":"The notation Ish^l_E in Theorem 1.4 and Corollary 1.6 is used for the cohomology of the Ishida complex of a projective toric variety E. Since the earlier definition of Ish^l_X is for the sheaf complex, it would help to explicitly state that H^i(E, Ish^l_E) denotes the hypercohomology of the sheaf complex, not the cohomology of the finite-dimensional complex Ish^l_P.","section":"§2.5 and Theorem 1.4"},{"comment":"The proof of Lemma 4.2 is dismissed as \"analogous\" to Lemma 4.1. Since this lemma is used in the proof of Theorem 1.8, a short proof or a precise reference to the analogous argument should be included.","section":"§4, Lemma 4.2"},{"comment":"The commutative diagram in §3.2 is hard to read because the arrows are not labeled and the vertical maps are not named. Labeling the maps would make the comparison with the exact sequence of complexes in §3.3 considerably clearer.","section":"§3.2, displayed diagram"}],"recommendation":"major_revision","confidential_remarks":"The central issue is that Theorem 1.4, the paper's headline result, is not actually proved in the text: the bridge from Corollaries 3.5 and 3.6 to the affine cone over (E, (-E)|_E) is missing, and Proposition 3.7 is asserted. In addition, because the paper relies heavily on the authors' previous preprint [KV25], the editor may wish to check whether that preprint has been accepted or otherwise made publicly verifiable. If the missing proof is supplied, the paper would be a strong contribution; in its current form, the main applications are not self-contained."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me be direct: this paper has real content. The fan-theoretic description of the Chern class map on singular cohomology (Cor 3.6) is new, and the consequence that lcdef of an affine toric variety can be read off from a Lefschetz-type map on a lower-dimensional projective toric variety is a nice bridge. The non-combinatoriality example, the pyramid invariance, and the shelling criteria are concrete and interesting. The combinatorics sections appear sound; I did not find internal contradictions.\n\nThe soft spot is exactly where the reader put their finger, and it is a little worse than they said. Theorem 1.4, the headline, is not proved. The text says that rephrasing Corollaries 3.5 and 3.6 immediately gives a proof, but the actual identification is never written down. You need to show that the cone over (E, (-E)|_E) is the cone ς used in §3.5, that the fan of E is the star of ρ quotiented by ρ, and that Ish^l_ς is the middle term of the short exact sequence of Cor 3.6 with the correct connecting homomorphism. None of those steps appear. Proposition 3.7 has the same problem: 'this follows from the fact that Ish^{p+1}_ς is the middle term' is an assertion, not a proof. This is load-bearing, because a sign or divisor-class error in the identification would change the LES, and the applications in Cor 1.6 and the non-invariance example rest on it.\n\nThat said, I don't think the gap is fatal. The identification is very likely correct; the authors are clearly capable of writing it out, and the independent example from CLS11 Exercise 12.3.11 supports the claimed lcdef computation. The reliance on [KV25] for Hodge-Tate type and surjectivity statements is a bit uncomfortable because that paper is unpublished, but it is their own prior work and the claims are plausible. The Macaulay2 check not shipping is minor.\n\nWho is this for? Specialists in toric geometry and local cohomology; someone looking for computational tools for lcdef. It deserves a serious referee. My recommendation: send to a good journal, but require the authors to expand the proof of Theorem 1.4 and Prop 3.7, explicitly writing the cone identification and the sign. If that verification is straightforward, the paper is a clear accept; if it reveals a twist, we need to know now.","headline":"A genuinely useful paper on toric local cohomological defect whose headline theorem is under-proved: the bridge from the fan computation to Theorem 1.4 needs to be written out.","tokens_in":21821,"tokens_out":3957,"would_cite":false,"duration_ms":39687,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14B15","14M25","32S50","52B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that the cup product with the first Chern class on the singular cohomology of a toric variety is fully described by fan combinatorics, and uses that description to prove that the local cohomological defect is not a…","keywords":["toric varieties","local cohomological dimension","local cohomological defect","Ishida complex","Lefschetz morphism","singular cohomology","Chern class map","Du Bois complex"],"falsifier":"Compute $H^2(\\mathrm{Ish}^3_\\sigma)$ for the two 14-ray cones in Example 1.7; if the two values turn out to be equal, the claim that the local cohomological defect is non-combinatorial is false. The same computation also tests the four-dimensional criterion, since lcdef $=1$ is equivalent to $\\dim H^2(E,\\mathbb{C}) \\geq 2$.","tokens_in":20676,"feed_emoji":"📐","tokens_out":6687,"duration_ms":61222,"temperature":0.7,"pith_summary":"This paper establishes a purely fan-theoretic description of the cup product with the first Chern class of a line bundle on the singular cohomology of a proper toric variety. It then uses that description to relate the local cohomological dimension of an affine toric variety to the Lefschetz morphism on a projective toric variety of one dimension lower. The payoff is concrete: the local cohomological defect, a measure of how far a Cohen–Macaulay variety is from being a local complete intersection, turns out not to be a combinatorial invariant of the cone. The paper also proves that the defect is unchanged under taking pyramids, which yields toric examples with every possible defect in every dimension.","feed_headline":"Local cohomological defect is not a combinatorial invariant","feed_subtitle":"Fan combinatorics plus a Lefschetz long exact sequence yields the counterexample and examples in every dimension.","key_machinery":"The Ishida complex $\\mathrm{Ish}^l_\\sigma$, a finite-dimensional complex of vector spaces built purely from the faces of a cone, is the central object; it is the degree-zero part of the Grothendieck dual of the Du Bois complex of reflexive differentials. The argument identifies the Grothendieck dual of the Chern class map with a connecting homomorphism in a short exact sequence of Ishida-type complexes, so the cup product becomes a purely combinatorial linear-algebra operation. Theorem 1.4 is the resulting long exact sequence linking the cone's Ishida cohomology to the Lefschetz morphism $c_1$ on the projective exceptional divisor.","core_discovery":"The central claim is Theorem 1.4: for an $n$-dimensional affine toric variety $X$ attached to a full-dimensional cone $\\sigma$, inserting an interior rational ray $\\rho$ gives a toric morphism $\\pi: \\widetilde{X} \\to X$ whose exceptional fibre $E$ is a projective toric variety of dimension $n-1$, and the cohomology of the Ishida complex $\\mathrm{Ish}^l_\\sigma$ sits in a long exact sequence with $H^*(E, \\mathrm{Ish}^{*}_E)$, whose maps are dual to the Chern class map of $(-E)|_E$. This makes the cohomological defect of the cone readable from the Lefschetz behaviour of a single projective toric variety. From this the authors derive a four-dimensional criterion (lcdef $= 1$ iff $\\dim H^2(E,\\mathbb{C}) \\geq 2$), and exhibit two cones with identical combinatorial data but different defects, proving non-invariance.","pith_inferences":["Because lcdef is computed from finite-dimensional vector spaces attached to a cone, the same long exact sequence could support an algorithm that decides the defect without resolving singularities; the computer algebra checks in the paper are a first step in that direction.","The fan-theoretic description of $c_1$ may extend to other natural classes on toric varieties, such as products of Chern classes or other cohomology operations, by iterating the same extension-of-complexes construction.","The non-invariance example suggests that any purported combinatorial formula for lcdef must use the actual positions of rays, not just the face poset; classifying the extra geometric data needed would be a natural next step."],"forward_implications":["The local cohomological defect of an affine toric variety can be computed from the maximal defect over faces of its cone, so the problem becomes algorithmic.","The defect is not a combinatorial invariant: Example 1.7 gives two cones with the same fan combinatorics but defects $1$ and $0$.","Taking a pyramid over a cone leaves the defect unchanged, so one can build $n$-dimensional affine toric varieties with any defect between $0$ and $n-3$.","In dimension four the defect is $1$ exactly when the exceptional projective threefold has $\\dim H^2(E,\\mathbb{C}) \\geq 2$; otherwise it is $0$.","The last differential of the Ishida complex vanishes in high degrees, recovering a prior vanishing result via a Hard Lefschetz-type injectivity statement."],"supporting_citations":[{"why":"Supplies the mixed Hodge–Tate statement for proper toric cohomology, the surjectivity of the last Ishida differential, and the lcdef $=m$ examples used to generate higher-dimensional examples.","marker":"[KV25]"},{"why":"Corollary 5.3 is restated as Theorem 1.1, giving the Ext-characterization of the local cohomological defect that motivates the cone definition.","marker":"[MP22]"},{"why":"Establishes the duality $R\\mathcal{H}om(\\Omega^p_X, \\omega_X) \\simeq \\mathrm{Ish}^{n-p}_X$, the fundamental identification on which the whole fan-theoretic translation rests.","marker":"[Ish87]"},{"why":"Provides the toric background, the acyclicity of structure sheaves used to pass from sheaf cohomology to Ishida complex cohomology, and Exercise 12.3.11 whose two threefolds drive the non-invariance example.","marker":"[CLS11]"},{"why":"Bounds the lcdef by $\\max\\{0,n-3\\}$ for Cohen–Macaulay varieties, so in dimension four only defects $0$ and $1$ are possible.","marker":"[DT16]"},{"why":"Introduces the local cohomological defect as the invariant studied throughout the paper.","marker":"[PS24]"},{"why":"Supplies the description of the first Chern class as a morphism of mixed Hodge modules, which becomes the Grothendieck-dual extension of Ishida complexes.","marker":"[RSW21]"},{"why":"Proves all cones are shellable, the fact used by Theorems 1.10 and 1.11 to compute $H^2(\\mathrm{Ish}^3_\\sigma)$ via shelling filtrations.","marker":"[BM71]"}],"fun_headline_variants":["Cohomological defect not fixed by fan combinatorics","Pyramid construction yields every cohomological defect","Lefschetz morphism exposes non-invariant defect","Toric cone defects elude combinatorial data","Local cohomology defect: counterexample via Lefschetz"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on two statements imported from the authors' preceding preprint, not reproved here: the cohomology of any proper toric variety is of a very restricted Hodge type, and the final differential of the Ishida complex is surjective in high degrees; the paper's injectivity, surjectivity, and non-invariance conclusions depend on them.","fun_headline_variants_meta":{"raw":{"variants":["Cohomological defect not fixed by fan combinatorics","Pyramid construction yields every cohomological defect","Lefschetz morphism exposes non-invariant defect","Toric cone defects elude combinatorial data","Local cohomology defect: counterexample via Lefschetz"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000608,"raw_usage":{"total_tokens":2795,"prompt_tokens":870,"completion_tokens":1925,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":1848}},"tokens_in":486,"tokens_out":1925,"duration_ms":13652,"temperature":1.0,"reasoning_tokens":1848,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:11:56.136054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $H^2(\\mathrm{Ish}^3_\\sigma)$ for the two 14-ray cones in Example 1.7; if the two values turn out to be equal, the claim that the local cohomological defect is non-combinatorial is false. The same computation also tests the four-dimensional criterion, since lcdef $=1$ is equivalent to $\\dim H^2(E,\\mathbb{C}) \\geq 2$.","supporting_citations":[],"review_version":1}