{"id":"6f94983c-ff26-4c20-b7b6-35f501923b28","arxiv_id":"2505.04303","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Bott-Chern complexity of a non-projective Calabi-Yau Kähler pair is at least 3, and this bound is attained by singular non-projective K3 surfaces with Picard rank zero.","lead":"Mathematicians define a new invariant, the Bott-Chern complexity, that measures how far a compact Kähler space is from being projective. They prove that sufficiently small values force projectivity and that the sharp non-projective Calabi-Yau examples are singular K3 surfaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The extremal case in Theorem 4.1 depends on unverified hypotheses in [16, Thm 0.3] and [20, Lemma 2.6], and the reduction to a dlt generalized pair is not fully justified.","rationale":"The reader correctly identified the dependence on [16, Theorem 0.3] and [20, Lemma 2.6] as the weakest link; I agree that the extremal case of Theorem 4.1 is not independently verifiable from the manuscript. My stress-test additionally flags a concrete internal issue: the passage from B'' to G'' by 'increasing to one' assumes f-exceptional coefficients are at most 1, but positive discrepancies can produce coefficients >1 in the log pullback of a dlt pair. If that happens, the asserted effective vertical divisor F is not an immediate consequence of the construction, and the termination argument via [20, Lemma 2.6] would need to be replaced or adapted. This does not make me doubt the main claim as stated, since the argument is plausibly repairable and the examples are consistent, but it does mean the proof as written is not yet complete enough for an unconditional accept. A conditional accept, pending verification of the two cited results and a clarification of the boundary-coefficient step, is the appropriate verdict.","tokens_in":12991,"tokens_out":26128,"duration_ms":297489,"concrete_test":"Verify verbatim the hypotheses of [16, Theorem 0.3] and [20, Lemma 2.6] against the exact objects used in Theorem 4.1: a compact Kähler (possibly non-projective) surface base Z, a generalized sub-pair with vertical negative coefficients, a nef divisor M'', and a (K+G+M)-MMP over Z. In the same pass, compute the log pullback for a two-step blow-up in a general fiber of a dlt pair to test whether f-exceptional coefficients of B'' can exceed 1; if such coefficients occur, give a justified replacement for the 'increasing to one' step that still yields a vertical effective representative F. If both checks pass, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 4.1(2) is the only place where the promised K3 base W is produced. After resolving the MRC fibration, the proof forms the log pullback B'' of the dlt boundary and applies the canonical bundle formula to the generalized sub-pair (X'',B''+M'') with M'' nef and B'' allowed negative coefficients. The cited [16, Theorem 0.3] must therefore cover sub-pairs over a compact Kähler surface with vertical negative coefficients; if it only covers pairs with effective boundaries or projective bases, the displayed (Z,B_Z+M_Z) is not obtained. The later MMP that contracts the vertical exceptional divisors and produces Z' is terminated by [20, Lemma 2.6], whose stated hypotheses may not match an arbitrary projective morphism over a non-projective Kähler surface. There is also an internal point: 'increasing to one' all f-exceptional coefficients of B'' is not meaningful if some discrepancies are positive, giving coefficients >1 in B'', and then K_X''+G''+M'' ∼_Q F ≥ 0 with F vertical does not obviously follow. Since every subsequent step (semiampleness, K∼Q 0 for Z', and toric fibers) rests on this reduction, the equality-case construction is the least secure part of the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Bott-Chern complexity c_BC(X,B) = dim X + h^{1,1}_{BC}(X) - |B| for compact Kähler pairs, paralleling the projective notion of complexity with the Bott-Chern cohomology in place of the Néron–Severi rank. The main results are: (1) if (X,B) is log canonical and -(K_X+B) is nef, then c_BC(X,B) >= 0; (2) if c_BC(X,B) < 3, then X is projective; (3) if c_BC(X,B) = 3 and X is non-projective, then a bimeromorphic model of X carries an MRC fibration whose general fiber is a projective toric variety and whose base is a singular non-projective K3 surface with Picard rank zero and h^{1,1}_{BC}=1. These results are proved via a fine-complexity invariant, bounds on the dimension of the MRC base, and the canonical bundle formula for generalized Kähler sub-pairs. Two examples of non-projective singular K3 surfaces with c_BC=3 are constructed using deformation theory and classification of extremal elliptic K3 surfaces.","tokens_in":13218,"tokens_out":9257,"duration_ms":96101,"significance":"If the main theorem is correct, this is a valuable and novel connection between the complexity program, Kähler geometry, and projectivity criteria. The invariant is natural and likely to find further applications, and the extremal non-projective examples are concrete and convincing. The paper is well-structured and builds on recent substantial advances in the Kähler MMP. However, the proof of the equality case (Theorem 4.1(2)) depends on two recent preprints, [16] and [20], whose exact hypotheses are not stated or verified in the manuscript; this is a load-bearing point for the production of the K3 base. The paper would be significantly strengthened by making those dependencies explicit and justifying their applicability.","major_comments":[{"comment":"The proof applies the canonical bundle formula for generalized Kähler sub-pairs to (X'',B''+M'') over the compact Kähler surface Z, and later terminates the MMP over Z using [20, Lemma 2.6] and [22, Lemma 2.9]. These are load-bearing applications: they produce the surface (Z,B_Z+M_Z) and the model Y' after contraction. The manuscript does not state the precise hypotheses of [16, Theorem 0.3], and it is not clear that it covers sub-pairs with vertical negative coefficients over a non-projective Kähler base. Moreover, [20] is announced for Kähler varieties with projective Albanese map, while [22] is an algebraic statement; neither obviously applies to an arbitrary compact Kähler surface base. The authors should state the exact theorems they are invoking and prove that the hypotheses are satisfied in this situation, or give alternative arguments.","section":"Theorem 4.1(2), proof in §4"},{"comment":"The assertion \"KY' + GY' + MY' is Q-linearly equivalent to an effective divisor which is exceptional over X'. Thus, we conclude that KY' + GY' + MY' is Q-linearly trivial\" is too quick. While it is true that a semiample divisor cannot be Q-linearly equivalent to a nonzero effective exceptional divisor, this deserves a one-sentence justification, especially because the argument also relies on the preceding semiampleness of the same divisor. Adding this justification would make the proof more transparent.","section":"Theorem 4.1(2), proof in §4, final paragraph"}],"minor_comments":[{"comment":"The notation \"dim H^{1,1}_{BC}(X)\" is used where \"h^{1,1}_{BC}(X)\" is standard; please unify with the body of the paper.","section":"Abstract and Introduction"},{"comment":"The notation c(X,B) is used both for the fine complexity and for the projective complexity from the literature; consider a distinct notation, for instance c_f(X,B), to avoid ambiguity.","section":"Definitions 2.7 and 2.8"},{"comment":"The proof says \"if c(X,B)<2, then X is a projective variety and so [1, Theorem 1.2] applies to prove (1) and (2)\", but parts (1) and (2) concern c(X,B)>=0 and c(X,B)<1. Please spell out that the non-negativity and toric conclusions follow from the projective result after the projectivity is established.","section":"Proof of Theorem 1.4"},{"comment":"The notation B^{=1}_Z and B''^{=1} is used without definition; please define it as the sum of the components with coefficient exactly one.","section":"Section 4, after diagram"},{"comment":"The deformation argument is sketched; it would be helpful to state explicitly that the general fiber X_c has Picard lattice equal to L (by a standard semi-continuity argument), which is what makes the contraction possible.","section":"Examples 5.1 and 5.2"}],"recommendation":"major_revision","confidential_remarks":"The main proof depends essentially on [16], a preprint co-authored by the first author of the current manuscript, and on [20], a 2025 preprint by Y.-T. Huang. Given this self-citation pattern, I recommend that the editor obtain verification from a referee who can check the hypotheses of these two preprints in the exact setting of Theorem 4.1(2). The concern is not about novelty but about the completeness of the proof as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper. It defines the Bott-Chern complexity for compact Kähler pairs, proves that Calabi–Yau pairs have nonnegative complexity, and that non-projective ones have complexity at least 3. The threshold is sharp: they build singular non-projective K3s with Picard rank zero and h^{1,1}_{BC}=1, so c_BC=3. That is a citable, genuinely useful result, and the invariant is new.\n\nThe main proof is structurally coherent. They reduce to the fine complexity, bound the base of the MRC fibration using known complexity results on the fiber, and then use the canonical bundle formula to pin down the base in the equality case. The argument is deductive, with no fitted parameters and no circularity. The exposition is clear and the citation pattern looks honest.\n\nThe soft spots are where you would expect. The extremal step in Theorem 4.1(2) invokes Theorem 0.3 of [16] (Hacon–Paun) and Lemma 2.6 of [20] (Huang) for the canonical bundle formula for generalized sub-pairs over a non-projective Kähler surface and for MMP termination. I cannot verify from this text that those hypotheses cover exactly the situation with negative coefficients on vertical exceptional divisors. If they do, the argument goes through; if not, the K3 base might not be produced. That is the load-bearing point, and a referee should check it carefully.\n\nI also wish the examples were spelled out a bit more -- the Shimada–Zhang table entry and the deformation/contraction argument are compressed. The examples are believable, but the reader should not have to take the classification on faith. The stress-test worry about coefficients exceeding 1 after \"increasing to one\" seems off: for log canonical pairs, discrepancies are at least -1, so those coefficients land in [0,1]. That specific point is not a problem.\n\nWho is this for? Anyone working in Kähler geometry or birational complexity. I would send it to a serious referee: the main theorem is new and plausible, and the unverified preprints are checkable. I would cite it for the invariant and the examples.","headline":"A new, clean invariant for Kähler pairs with a sharp projectivity cutoff, but the equality case leans on preprints I can't fully check.","tokens_in":13741,"tokens_out":3948,"would_cite":true,"duration_ms":38064,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32J27","14E30","14M25","14J28"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces the Bott-Chern complexity of compact Kähler pairs and proves that values below 3 force projectivity, while the non-projective equality case is a toric fibration over a singular K3 surface.","keywords":["Bott-Chern cohomology","Kähler varieties","Calabi-Yau pairs","complexity","projective toric varieties","K3 surfaces","minimal model program","projectivity criterion"],"falsifier":"Search for a strongly $\\mathbb{Q}$-factorial compact Kähler log canonical pair $(X,B)$ with $-(K_X+B)$ nef and Bott-Chern complexity strictly below 3 for which $X$ is not projective; the theorem asserts that no such pair exists, so any such pair would settle the claim false.","tokens_in":12794,"feed_emoji":"📐","tokens_out":11850,"duration_ms":111267,"temperature":0.7,"pith_summary":"This paper introduces an invariant for compact Kähler pairs, the Bott-Chern complexity, which combines the dimension of the variety, the size of its first Bott-Chern cohomology group, and the total coefficient of a boundary divisor. The central threshold result says that under a standard mild-singularity assumption, if the anticanonical divisor is nef and the Bott-Chern complexity is below 3, the variety must be projective. If the complexity equals 3 and the variety is not projective, then up to bimeromorphic modification the variety is a fibration whose general fibers are projective toric varieties over a singular non-projective K3 surface with Picard rank zero and $h^{1,1}_{BC}=1$. The value 3 is optimal, realized by non-projective singular K3 surfaces with empty boundary. A sympathetic reader should care because this gives a numerical, cohomological criterion for projectivity in the Kähler setting, where such criteria are scarce.","feed_headline":"Complexity below 3 forces a Kähler variety to be projective","feed_subtitle":"Non-projective Calabi-Yau pairs must score at least 3; equality forces a toric fibration over a singular K3 surface.","key_machinery":"The carrying object is the Bott-Chern cohomology $H^{1,1}_{BC}(X)$, the space of real $(1,1)$-forms with local potentials modulo $d d^c$-exact forms, which plays the role of the Néron–Severi space in the Kähler setting; the Bott-Chern complexity is $\\dim X + h^{1,1}_{BC}(X) - |B|$ minimized over decompositions of $B$ into $\\mathbb{Q}$-Cartier divisors. The proof also uses the fine complexity, where the span of the components of $B$ in numerical $\\mathbb{R}$-Weil divisors replaces $H^{1,1}_{BC}$, along with two structural inputs: the dimension of the base of the maximally rationally connected fibration is bounded by the fine complexity, and the canonical bundle formula for generalized Kähler sub-pairs with possibly negative boundary coefficients governs the extremal case. The examples rely on the classification of extremal elliptic K3 surfaces to produce singular K3 surfaces with Picard rank zero and $h^{1,1}_{BC}=1$.","core_discovery":"The paper's central claim is that the Bott-Chern complexity $c_{BC}(X,B)$, defined as the minimum over decompositions of $B$ into $\\mathbb{Q}$-Cartier components of $\\dim X + h^{1,1}_{BC}(X) - |B|$, controls projectivity for strongly $\\mathbb{Q}$-factorial compact Kähler log canonical pairs with $-(K_X+B)$ nef. Theorem 4.1 states that $c_{BC}(X,B)<3$ forces $X$ to be projective; if $c_{BC}(X,B)=3$ and $X$ is not projective, then there is a dlt modification $(X',B')\\to (X,B)$, a small modification $Y'\\to X'$, and an MRC fibration $Y'\\to W$ whose general fiber is a projective toric variety and whose base $W$ is a singular non-projective K3 surface of Picard rank zero and $h^{1,1}_{BC}(W)=1$. Consequently, Calabi-Yau compact Kähler pairs have nonnegative Bott-Chern complexity, and non-projective ones have complexity at least 3; the examples constructed by contracting configurations of curves in degenerations of extremal elliptic K3 surfaces show the bound is attained.","pith_inferences":["One testable extension is a converse classification: check whether every Kähler fibration with projective toric general fibers over a singular K3 surface of Picard rank zero and $h^{1,1}_{BC}=1$ attains Bott-Chern complexity exactly 3.","Because the main hypothesis is only that $-(K_X+B)$ is nef, the threshold may extend beyond Calabi-Yau pairs to log Fano type Kähler pairs; a natural test is to allow $B$ to have negative coefficients and track how the canonical bundle formula absorbs them.","The examples are produced from two extremal elliptic K3 fibrations; surveying the full classification of such fibrations could reveal whether the equality value 3 is isolated or occurs in a family of singular K3 models with $h^{1,1}_{BC}=1$."],"forward_implications":["If the main theorem is correct, any Calabi-Yau compact Kähler pair has nonnegative Bott-Chern complexity, and a non-projective one has complexity at least 3.","For strongly $\\mathbb{Q}$-factorial log canonical pairs with nef anticanonical divisor, Bott-Chern complexity below 3 is a projectivity criterion: no non-projective example can exist below that threshold.","In the minimal non-projective case, a bimeromorphic model is a fibration with projective toric general fibers over a singular non-projective K3 surface with Picard rank zero and $h^{1,1}_{BC}=1$.","The fine-complexity analogues hold: fine complexity below 1 forces a projective toric variety, and fine complexity below 2 forces projectivity together with $H^{1,1}_{BC}(X)=\\mathrm{Pic}(X)_{\\mathbb{R}}$.","The bound 3 is sharp, since singular non-projective K3 surfaces with empty boundary realize exactly $c_{BC}=3$."],"supporting_citations":[{"why":"Supplies the non-negativity of complexity for projective pairs and the toricity criterion $c<1$, applied to the general fibers.","marker":"[1]"},{"why":"Establishes the maximally rationally connected fibration for compact Kähler manifolds, used to define and bound the base of the fibration.","marker":"[3]"},{"why":"Provides projectivity criteria for Kähler morphisms, used to make the MRC fibration projective and to handle base dimension one.","marker":"[4]"},{"why":"Gives invariance of strong $\\mathbb{Q}$-factoriality and the equality $H^{1,1}_{BC}=\\mathrm{Pic}(X)_{\\mathbb{R}}$ for rationally connected fibrations.","marker":"[5]"},{"why":"Supplies the canonical bundle formula and dlt modifications for generalized Kähler pairs, the central mechanism of the equality case.","marker":"[16]"},{"why":"Defines Bott-Chern cohomology and the nef and pseudoeffective conventions for Kähler varieties, fixing the invariant's grounding.","marker":"[19]"},{"why":"Provides the termination of the relevant MMP over a Kähler surface, used to contract vertical divisors in the extremal case.","marker":"[20]"},{"why":"Shows that non-uniruled compact Kähler manifolds have pseudo-effective canonical bundle, controlling the MRC base.","marker":"[29]"},{"why":"Classifies extremal elliptic K3 surfaces, from which the non-projective singular K3 examples are constructed.","marker":"[30]"}],"fun_headline_variants":["Bott-Chern complexity <3 forces Kähler projectivity","Non-projective Calabi-Yau pairs: Kähler complexity ≥3","Complexity under 3 implies Kähler projectivity","Bott-Chern complexity 3: the non-projective Kähler threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equality-case argument leans on a technical adjunction formula and on termination of a minimal-model program for generalized Kähler pairs whose boundary may have negative coefficients, both cited to recent preprints; if these apply only in narrower settings, the description of the base as a K3 surface could fail.","fun_headline_variants_meta":{"raw":{"variants":["Bott-Chern complexity <3 forces Kähler projectivity","Non-projective Calabi-Yau pairs: Kähler complexity ≥3","Complexity under 3 implies Kähler projectivity","Bott-Chern complexity 3: the non-projective Kähler threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001093,"raw_usage":{"total_tokens":4549,"prompt_tokens":918,"completion_tokens":3631,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":3553}},"tokens_in":534,"tokens_out":3631,"duration_ms":26185,"temperature":1.0,"reasoning_tokens":3553,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:35:19.594512+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a strongly $\\mathbb{Q}$-factorial compact Kähler log canonical pair $(X,B)$ with $-(K_X+B)$ nef and Bott-Chern complexity strictly below 3 for which $X$ is not projective; the theorem asserts that no such pair exists, so any such pair would settle the claim false.","supporting_citations":[{"cited_title":"On the Existence of Good Minimal Models for K\\\"ahler Varieties with Projective Albanese Map","cited_arxiv_id":"2502.18800","evidence_quote":"Provides the termination of the relevant MMP over a Kähler surface, used to contract vertical divisors in the extremal case."},{"cited_title":"Shimada and D.-Q","cited_arxiv_id":null,"evidence_quote":"Classifies extremal elliptic K3 surfaces, from which the non-projective singular K3 examples are constructed."}],"review_version":1}