{"id":"4ae9c9af-5f41-4718-8ab0-1adf51d03ae4","arxiv_id":"2607.12672","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For complex varieties with klt or rational singularities, negative K-theory is shown to be governed by mixed Hodge weights and higher singularity types, with new proof in dimension three and partial results in dimension four.","lead":"This paper shows that the negative K-groups of singular complex varieties—algebraic invariants that detect singularities—are controlled by the weights of their mixed Hodge structures and by higher singularity types. It proves explicit formulas for threefolds and fourfolds with klt or rational singularities and lays out conjectures for all dimensions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 12.1(i) for n≥6 rests on the MMP ordering in Theorem 9.9, which the paper's own footnote says is not unconditional for dlt pairs; a failure there breaks the Chow surjectivity α_Q and the rational identification.","rationale":"The reader's verdict is CONDITIONAL, and the stated concern about [Bur] is real for the integral claims (Propositions 6.1, 7.3, Theorems 11.2 and 12.1(ii)), but it does not threaten the rational statement of Theorem 12.1(i). The rational statement depends instead on the Chow/MMP results in §9, especially Theorem 9.9. That theorem is the most delicate part of the paper, and its proof contains an explicit caveat about dlt MMP only being unconditionally known in dimension ≤5. Since the central theorem is claimed for all n≥4, this is a genuine load-bearing concern for the main rational result, though not a demonstrated counterexample. The paper gives substantial independent support elsewhere: machine-checkable computations are absent, but the main rational arguments are built on established results (CHSW08, PP24, Sh25) and the Kummer/cone examples make the integral caveats concrete. I would keep the verdict CONDITIONAL, with the focus shifted from [Bur] to the MMP ordering for n≥6.","tokens_in":54352,"tokens_out":42780,"duration_ms":384810,"concrete_test":"Construct (or identify) a Q-factorial klt variety of dimension n≥6 with an isolated singularity whose log resolution has at least two non-uniruled threefold strata, run the (Y,G)-MMP of Theorem 9.9 explicitly, and check at every flip/divisorial contraction that (Y_i,Γ_i) remains dlt and that the ordering conditions (1) and (2) of Theorem 9.9 hold. A more targeted check: verify dlt-preservation for a single flip of the type used in the proof; if any flip creates an lc center that is not a stratum of Γ_i, Proposition 5.6 cannot be applied and the proof of Theorem 9.9 breaks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For the rational central claim Theorem 12.1(i), the cited unpublished [Bur] is not the main load-bearing input: the rational klt proof uses Corollary 1.7, Corollary 7.6, and Theorem I, not Theorem 2.10. The least secure step is Theorem 9.9, specifically the MMP ordering. The proof runs an MMP for a dlt pair (Y,Γ) via the klt perturbation G=Γ−εg*D, and the footnote explicitly concedes that running such an MMP for a dlt pair is not known without Special Termination and is unconditional only in dimension ≤5. For n≥6, the proof needs each transform (Y_i,Γ_i) to remain dlt so that Proposition 5.6 can be used to identify lc centers and produce the pairing/ordering T_q⊂F_q. If a flip produces a pair that is not dlt, the equality T_i=F_i∩E_i and the triangular ordering can fail. Then the surjectivity of α_Q in Theorem 9.6 fails, so H^{n+2}_cdh(X,Q(2)) need not vanish, and the final isomorphism K_{−n+2}(X)_Q ≃ gr^W_2 H^n(X,Q) collapses for n≥6. This is not an internal inconsistency, but it is a clearly flagged conditional step in the proof of the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a program relating negative K-theory of complex varieties to mixed Hodge theory and to the theory of higher singularities (Du Bois, D_m, pre-m-rational). The main results are: Theorem H computes weight-zero and weight-one cdh cohomology; Theorem D/F give vanishing of the two bottom K-groups under rational singularities; Theorem 12.1 gives, for a projective variety X of dimension n≥3 with isolated klt-type singularities, an isomorphism K_{-n+2}(X)_Q ≃ KH_{-n+2}(X)_Q ≃ H^{n-2}(Pic X•)_Q ≃ gr^W_2 H^n(X,Q), together with integral/LCI refinements in special cases. The proofs use cdh-descent, higher Chow groups, restriction maps on codimension-two cycles on log resolutions, and an MMP-based ordering of exceptional strata. The appendix derives the conjectural picture from a generalized Bloch–Beilinson conjecture and collects examples, including Kummer varieties and Cayley-threefold examples showing integral obstructions.","tokens_in":54715,"tokens_out":7582,"duration_ms":82749,"significance":"If the main theorem is correct, it gives a striking and non-obvious formula: for isolated klt-type singularities, the next-to-bottom negative K-group is not a mysterious torsion invariant but is the weight-two Hodge piece of middle cohomology, computed from Picard groups of strata in a log resolution. The weight-zero and weight-one sections are clean and likely standard, and the examples (Kummer varieties, Cayley surfaces, cones, toric varieties, quotient singularities) are valuable in calibrating the conjectures. The paper is also honest in separating conjectures from theorems and in flagging conditional input. However, a central part of the claimed theorem for n≥4 is conditional on an MMP statement whose proof the paper itself admits is not unconditional, so the paper is not yet in a state where the headline theorem can be accepted as proved in full.","major_comments":[{"comment":"This is the load-bearing conditional step. The proof of Theorem 9.9 runs an MMP for the dlt pair (Y,Γ) via the klt perturbation G=Γ−εg*D; the footnote explicitly concedes that running such an MMP for a dlt pair is not known without Special Termination and is unconditional only in dimension ≤5. The ordering property (i)(2)/(ii)(2) of Theorem 9.9 is then used in Theorem 9.6 to prove the surjectivity of α_Q, hence Lemma 8.5(ii), Theorem I(ii), and ultimately Theorem 12.1(i) for n≥4 (in particular for all n≥6). If that MMP step fails, the triangular ordering T_q⊂F_r only if q≤r can fail, the diagram chase proving α_Q surjective collapses, and the vanishing of H^{n+2}_cdh(X,Q(2)) is not established. The theorem as stated is therefore not proved unconditionally for n≥4; the manuscript should either supply the missing MMP input, restrict the statement to the range where the MMP is known, or sta","section":"§9, proof of Theorem 9.9, footnote after eq. (9.10)"},{"comment":"Theorem 2.10 is cited to the first author's unpublished and forthcoming work [Bur], and it is not proved in this paper. It underpins the integral statements in Proposition 6.1, the final claim of Proposition 7.3, Theorem 11.2, and the integral versions of the main consequences. If [Bur] is not available, these integral klt-type conclusions are conditional. The rational central claim can probably avoid Theorem 2.10, but the paper should state clearly in the main theorems which parts depend on [Bur], or else include a proof of Theorem 2.10 in this paper.","section":"§2, Theorem 2.10 and its use"}],"minor_comments":[{"comment":"The proof says 'Thanks to Corollary 3.8 we have K_{-n}(X)≃KH_{-n}(X)', but the cited statement appears to be Theorem 3.8 (Weibel's conjecture and K_{-n}-regularity), not Corollary 3.8. Please check the cross-reference.","section":"§10, proof of Theorem 10.1"},{"comment":"In the definition of log pullback, the second displayed formula should read f_*Δ_Y=Δ, not f^*Δ_Y=Δ. As written it is not the usual pushforward formula for log pullback.","section":"§5, eq. (5.1)"},{"comment":"The text says 'Since such a blow-up always exists, there will be no loss of generality in assuming it' regarding factorisation through a plt blow-up. This is plausible, but it should be justified or made precise, since Theorem 9.6 and Theorem 9.9 explicitly assume this factorization.","section":"§8, after Example 8.1"},{"comment":"The phrase 'it is straightforward to see that it survives to the E∞-page' for H^n_cdh(X,Z(1)) can be made precise by pointing at the relevant differentials and the vanishing of H^{n-2}_cdh(X,Z(0)) and H^{n+2}_cdh(X,Z(2)). A one-sentence explanation would improve readability.","section":"§12, proof of Theorem 12.1"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is ambitious and contains a substantial body of correct-looking material, but the main theorem for n≥4 rests on the MMP ordering in Theorem 9.9, which the manuscript itself flags as not unconditional. This is not a matter of presentation; it is a load-bearing gap that should be fixed before publication, either by completing the MMP input or by explicitly restricting/conditionalizing the theorem. The heavy reliance on unpublished work [Bur] for the integral statements should also be made visible in the main theorem statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Top line: this is a substantive preprint that gives the first systematic Hodge-theoretic picture of negative K-theory for klt/rational varieties, and the central rational formulas for threefolds and fourfolds look right. The main thing to know before citing it as settled is that the proof of the general n≥6 statement rests on an MMP step the authors themselves flag as not yet unconditional, and the integral statements lean on an unpublished theorem of the first author. Neither point is concealed; both need scrutiny.\n\nWhat is actually new: the conjectural hierarchy B/C/E/G; the complete weight-zero and weight-one cdh cohomology computations; the quotient-singularity trace map (Proposition 15.1); the Chow/MMP surjectivity theorem (9.9); and the applications to quotient, cone, toric, secant, and Kummer examples. The unconditional rational results for K_{-n}, K_{-n+1}, and K_{-n+2} for isolated klt singularities in low dimension are a real advance. The paper is also honest: conjectures are cleanly separated from theorems, and where Bloch's conjecture on 0-cycles is used for rational threefolds, that is stated explicitly.\n\nSoft spots in proportion. First, Theorem 2.10 is cited to the first author's forthcoming work and is not proved here. It underpins the integral weight purity used in Propositions 6.1 and 7.3 and in the integral statements of Theorems 11.2 and 12.1. The rational isomorphism in Theorem 12.1(i) does not actually need Theorem 2.10 — Corollary 1.7 and Corollary 7.6 carry it — so this is a problem for the integral claims, not for the main rational result. Second, and more serious for n≥6: the proof of Theorem 9.9 runs an MMP for a dlt pair through a klt perturbation, and the footnote concedes that doing this for dlt pairs is not known unconditionally without Special Termination, hence only known in dimension ≤5. For n≥6, failure of that MMP step would break the Chow surjectivity in Theorem 9.6, the vanishing of H^{n+2}_cdh(X,Q(2)), and finally the isomorphism in Theorem 12.1(i). The authors flag this, but it is still load-bearing. Third, the exact sequence expressing K_{-1} via local analytic class groups is deferred to [BPS]; that is a self-containedness gap, not an evident mathematical error.\n\nOn the citation pattern: the heavy use of [PP24], [PP25], [Sh25], and [Bur] is not circularity in the usual sense. The published ones are external and checkable; the real burden is simply that [Bur] is forthcoming and Theorem 2.10 is load-bearing for integral statements.\n\nBottom line: for people working in negative K-theory, singularity theory, or the MMP, this is one of the more interesting preprints around. It deserves a serious referee. I would send it to review, with the request that the referee check Theorem 9.9 and the dependence on Theorem 2.10 carefully, and that the authors state the unconditional range of Theorem 12.1(i) precisely rather than leaving the reader to triangulate.","headline":"Substantive new framework for negative K-theory via Hodge theory and MMP, but the n≥6 case and integral statements rest on flagged conditional/unpublished inputs.","tokens_in":55213,"tokens_out":2623,"would_cite":true,"duration_ms":28902,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19E20","19E15","14C30","14J17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for projective varieties with isolated klt-type singularities, the rational negative K-group K_{-n+2} is not an exotic torsion invariant but is canonically isomorphic to the weight-two Hodge piece of middle cohomology,","keywords":["negative K-theory","mixed Hodge theory","cdh cohomology","higher rational singularities","klt singularities","Chow groups","homotopy K-theory","minimal model program"],"falsifier":"Take a projective variety X of dimension n ≥ 3 with isolated klt-type singularities and compute the explicit Picard-strata group H^{n-2}(Pic X•)_Q from a log resolution; if it is not isomorphic to gr^W_2 H^n(X,Q), or if K_{-n+2}(X)_Q is nonzero while gr^W_2 H^n(X,Q) = 0, Theorem 12.1 fails. The paper's Kummer-threefold example already shows the integral version fails through torsion, so the rational isomorphism is the precise claim to test.","tokens_in":54224,"feed_emoji":"🧮","tokens_out":8413,"duration_ms":86596,"temperature":0.7,"pith_summary":"The paper aims to explain negative K-theory of singular complex varieties through mixed Hodge theory: it conjectures, and proves in important cases, that the vanishing and precise descriptions of the groups K_{-i} are governed by the weight filtration on cohomology and by higher singularity notions. Its central theorem states that for a projective variety of dimension n at least 3 with isolated klt-type singularities, the rational group K_{-n+2} is isomorphic to the weight-two piece of middle cohomology, computed explicitly from Picard groups of the strata of a log resolution. This turns a mysterious invariant of singularities into a Hodge-theoretic quantity, and explains when it vanishes and when it is merely torsion. A sympathetic reader should care because the paper replaces a case-by-case study of singular K-theory with a systematic mechanism, and it names the precise conjectural conditions under which the mechanism should hold unconditionally.","feed_headline":"Negative K-theory equals a Hodge piece on singular spaces","feed_subtitle":"For isolated klt-type varieties, the next-to-bottom rational K-group is the weight-two Hodge part of middle cohomology.","key_machinery":"The engine is cdh-motivic cohomology H^k_cdh(X,Q(j)), which appears on the E_2-page of the Atiyah–Hirzebruch type spectral sequence converging to KH_{-i}(X)_Q. The paper computes the weight-zero and weight-one cdh cohomology in terms of the integral weight filtration and of Pic(X•), the complex of Picard groups attached to a cubical hyperresolution of X; its cohomology H^{n-2}(Pic(X•))_Q is the concrete Picard-strata formula from the main theorem. The decisive step is proving that the borderline weight-two group H^{n+2}_cdh(X,Q(2)) vanishes for isolated klt-type singularities, via surjectivity of restriction maps CH^2(Y)_Q → ⊕_j CH^2(S_j)_Q on a log resolution, or between threefold and surfa","core_discovery":"The central theorem is Theorem 12.1: for a projective variety X of dimension n ≥ 3 with isolated singularities of klt type, there are natural isomorphisms K_{-n+2}(X)_Q ≃ KH_{-n+2}(X)_Q ≃ H^{n-2}(Pic(X•))_Q ≃ gr^W_2 H^n(X,Q). In words, the next-to-bottom negative K-group is rationally controlled by the weight-two piece of middle cohomology, and can be computed as a quotient and kernel of restriction maps among Picard groups of the surface and curve strata in a log resolution. The paper also formulates systematic conjectures B, C, E, and G: weight bounds on mixed Hodge structure imply vanishing of rational homotopy K-theory; pre-m-rational singularities imply the same for ordinary K-theory; a","pith_inferences":["Editorial extension: the formula K_{-n+2}(X)_Q ≃ gr^W_2 H^n(X,Q) suggests reading the group as a higher-dimensional Q-factoriality defect, a global-versus-local measure of Weil divisors modulo Cartier divisors; a natural next step is to write an exact sequence expressing it through class groups of the singular germs, as the paper explicitly does for threefolds.","Editorial extension: the same machinery points to the next hard case, namely the weight-two cdh groups H^{n+4}_cdh(X,Q(3)) (and higher weights), whose vanishing under klt-type hypotheses would give analogous control of K_{-n+3}; the paper stops at weight two, and its Chow-restriction technique suggests an MMP-based ordering for fivefold strata as the needed input.","Editorial extension: the clean rational theorem makes the integral failure exhibited by the Kummer threefold more informative: it suggests that integral refinements require control of integral weight filtrations, so a concrete test is to search for klt-type examples beyond the Kummer one where integral weights misbehave.","Editorial extension: because quotient singularities have all negative rational K-groups zero but sometimes nonzero integral K_{-1}, the Kummer-type torsion can be read as a new invariant of rational homology manifolds that is invisible in cohomology and is governed by the integral weight filtration."],"forward_implications":["For a projective threefold of klt type, K_{-3} = 0 and K_{-2} is the cokernel of the restriction map from Picard groups of surface strata to Picard groups of curve strata; when the singularities are isolated, K_{-1}(X)_Q ≃ gr^W_2 H^3(X,Q).","For any n ≥ 3 with isolated klt-type singularities, K_{-n+2}(X)_Q vanishes exactly when the middle cohomology has no weight-two part, and it vanishes automatically for pre-1-rational (or D_1) singularities.","Rational singularities imply K_{-n}(X)_Q = K_{-n+1}(X)_Q = 0; quotient-singular and toric varieties have all negative rational K-groups equal to zero, so their nonzero negative K-theory is torsion.","The full conjectural picture, including the cdh-cohomology vanishing and the formulas beyond vanishing, would follow from a generalized Bloch–Beilinson filtration on higher Chow groups, so the proved cases give concrete evidence for that filtration.","The methods extend by localization to quasi-projective varieties with isolated singularities and to local rings of such singularities, so the same formulas describe K-groups of germs."],"fun_headline_variants":["Negative K-group is weight-two Hodge on singular spaces","On klt singularities, negative K-theory is Hodge weight 2","Singular spaces: negative K-theory equals Hodge piece","Hodge weight two controls negative K-theory on singularities","Negative K-theory: Hodge weight 2 on singular spaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is an unpublished integral weight-purity theorem for klt-type varieties — H^i_0(X,Z) = H^i_1(X,Z) = 0 for i > 0 — cited to the first author's forthcoming work in Section 2 and not proved here; all klt-type vanishings and descriptions over Z in the paper depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Negative K-group is weight-two Hodge on singular spaces","On klt singularities, negative K-theory is Hodge weight 2","Singular spaces: negative K-theory equals Hodge piece","Hodge weight two controls negative K-theory on singularities","Negative K-theory: Hodge weight 2 on singular spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002019,"raw_usage":{"total_tokens":7627,"prompt_tokens":582,"completion_tokens":7045,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":326,"completion_tokens_details":{"reasoning_tokens":6955}},"tokens_in":326,"tokens_out":7045,"duration_ms":43788,"temperature":1.0,"reasoning_tokens":6955,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T06:22:16.285355+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a projective variety X of dimension n ≥ 3 with isolated klt-type singularities and compute the explicit Picard-strata group H^{n-2}(Pic X•)_Q from a log resolution; if it is not isomorphic to gr^W_2 H^n(X,Q), or if K_{-n+2}(X)_Q is nonzero while gr^W_2 H^n(X,Q) = 0, Theorem 12.1 fails. The paper's Kummer-threefold example already shows the integral version fails through torsion, so the rational isomorphism is the precise claim to test.","supporting_citations":[],"review_version":2}