{"id":"21766224-d526-4501-ac93-edcea2de0f4a","arxiv_id":"2501.04057","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The complex analytic analog of the minimal model program for normal pairs along the log canonical locus holds: under a semi-ampleness hypothesis on the non-lc locus, an MMP sequence exists and terminates at a good minimal model.","lead":"This paper proves that a procedure for simplifying complex analytic spaces, the minimal model program, works for normal pairs even when singularities are worse than log canonical, under a semi-ampleness condition along the non-lc locus. It is the complex analytic version of the author's earlier algebraic theorem, and it fills a key missing case in the developing analytic minimal model theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem's proof is deferred to [H23] and to unpublished analytic preprints; the decisive reliance is on analytic special termination and lc flips, which are not proved here.","rationale":"The reader's verdict is CONDITIONAL with MODERATE confidence, and the weakest assumption identified is the reliance on unpublished analytic foundations. I agree that this is the single most load-bearing concern. The paper's own introduction says that only Theorem 3.12 and Lemma 4.5 receive detailed proofs because the algebraic argument fails for lack of technical results; the main theorem (Theorem 4.7) is then asserted to follow from [H23, Proof of Theorem 5.3] 'with no changes.' That means the central claim stands or falls with the analytic analogs of the MMP machinery for lc pairs. The proof of Theorem 4.6 makes the dependency explicit: it needs special termination, adjunction for quasi-log complex analytic spaces, and the base-point-free theorem, citing [EH24a, Subsection 3.6], [F22b, Theorem 4.4], and Theorem 2.16. Lemma 3.11 Step 1 uses lc flips from [F25b, Theorem 1.7], and Step 3 uses existence of good minimal models for lc pairs from [EH24a, Theorem 1.1] and [F25b, Theorem 1.5]. These are not proved in this paper and are preprints. I did not find a specific internal inconsistency in the lemmas the paper does prove; the gluing construction in Lemma 3.11 Step 1 is plausible, and the Q-approximation in Lemma 4.5 relies on published [F25a] plus [F22a]. The concern is thus not a local computation error but a global dependence on external results of at least comparable difficulty to the main theorem. If those preprints are correct, the paper is a meaningful extension from lc to normal pairs; if any of them has a gap, the main theorem is unsupported. Hence the verdict should remain CONDITIONAL: accept only after the cited analytic foundations are verified. For a definitive check, one should trace the proof step-by-step and confirm every algebraic result in [H23, Theorem 5.3] has a cited analytic analog; a missing or weaker analog would falsify the claim.","tokens_in":25251,"tokens_out":13060,"duration_ms":115777,"concrete_test":"Retrieve [H23], [EH24a], and [F25b], and produce a step-by-step correspondence between the proof of [H23, Theorem 5.3] and the analytic citations in this paper. For each invocation of special termination, adjunction, lc flips, or good minimal models over a point, verify the cited analytic theorem is stated at the needed level of generality (normal non-Q-factorial spaces, Stein bases, non-compact lc centers). If any step in [H23, Theorem 5.3] has no matching analytic theorem, or if the matching theorem is still a preprint whose proof has not been independently checked, the main theorem should be marked conditional or unverified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.7, the paper's central claim, is proved by the sentence 'The argument of [H23, Proof of Theorem 5.3] works with no changes,' and the introduction states the proofs are 'almost the same as in the algebraic case.' The two results the author actually proves (Theorem 3.12, Lemma 4.5) are presented as the only places where the algebraic argument 'does not work well.' Everything else is inherited from [H23] together with analytic analogs of special termination ([EH24a, Subsection 3.6]), adjunction for quasi-log complex analytic spaces ([F22b, Theorem 4.4]), the base-point-free theorem (Theorem 2.16, whose proof itself cites [F22b]), lc flips ([F25b, Theorem 1.7]), and existence of good minimal models over a point or Stein germ ([EH24a, Theorem 1.1 and 1.2], [F25b, Theorem 1.5]). Lemma 3.11 Step 3 explicitly invokes [EH24a, Theorem 1.1] and [F25b, Theorem 1.5] to get a good minimal model for an lc pair over a Stein neighborhood, and Step 1 invokes [F25b, Theorem 1.7] to construct each flip. None of these theorems is proved in this paper, and [EH24a], [EH24b], and [F25b] are preprints. If any one of them is incomplete—in particular, if special termination does not hold in the non-Q-factorial, non-compact analytic category, or if the lc flip theorem requires Q-factoriality that is not preserved—then the induction in [H23, Theorem 5.3] has no analytic counterpart and Theorem 4.7 lacks a proof. The burden is on the external results, but the paper makes no attempt to isolate or verify them.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a complex analytic analog of the author's algebraic minimal model program for normal pairs along the log canonical locus. The main theorem (Theorem 4.7, stated as Theorem 1.1) asserts that, for a projective morphism from a normal analytic variety to a Stein space with a compact subset W satisfying property (P), and for a normal pair (X,∆) with a π-ample divisor A making K_X+∆+A globally R-Cartier and π-pseudo-effective, if the relative non-nef locus avoids Nlc(X,∆) over W and the restriction to Nlc(X,∆) is semi-ample over a neighborhood of W, then after shrinking Z around W there is a finite sequence of steps of a (K_X+∆+A)-MMP represented by bimeromorphic contractions with ρ-drop 1, avoiding Nlc(X,∆), ending in a semi-ample model. The paper also derives corollaries on stable base loci and finite generation, and a stack-theoretic formulation in Section 5. The two arguments developed in detail are Theorem 3.12, a quasi-log version of the MMP step with a bimeromorphic contraction, and Lemma 4.5, which reduces λ_i-limit statements for MMPs to the klt case. Most other results are proved by asserting that the algebraic arguments from [H23] 'work with no changes,' relying on complex analytic analogs of special termination, adjunction, lc flips, and existence of good minimal models, cited from preprints [EH24a], [EH24b], [F22b], and [F25b].","tokens_in":25640,"tokens_out":7315,"duration_ms":61465,"significance":"If the cited analytic foundations are correct, Theorem 4.7 is a substantial extension of the MMP beyond the log canonical category in the complex analytic setting, and the corollaries on non-vanishing of stable base loci and finite generation of relative canonical rings would be new and useful. The paper is clearly written and contains two genuinely analytic arguments of independent interest, Theorem 3.12 and Lemma 4.5, with detailed proofs. However, the central theorem is not proved in the manuscript: it is reduced verbatim to the algebraic proof in [H23] plus a list of unpublished analytic statements. The significance of the paper is therefore conditional on the correctness and availability of those external results. The reader cannot verify the main theorem from this paper alone, and the paper does not isolate which properties of the analytic setting (e.g., Q-factoriality over W, special termination over Stein neighborhoods) are used in each step.","major_comments":[{"comment":"The proof of the main theorem is the single sentence 'The argument of [H23, Proof of Theorem 5.3] works with no changes.' This is load-bearing because the algebraic proof of [H23, Theorem 5.3] uses special termination, adjunction, lc flips, and existence of good minimal models, whose complex analytic analogs are not proved here but cited from [EH24a, Subsection 3.6], [F22b, Theorem 4.4], and [F25b, Theorems 1.5 and 1.7], all of which are preprints. The paper proves Theorem 3.12 and Lemma 4.5 in detail, but the remaining steps of Theorem 4.7 are not verified to transfer to the analytic setting, for example the gluing of MMP steps over shrinkings and the preservation of Q-factoriality over W. The author should either provide complete analytic proofs of the cited statements or state them as explicit assumptions with precise references; as written, Theorem 4.7 is conditional.","section":"Theorem 4.7 (proof)"},{"comment":"The construction of each flip invokes [F25b, Theorem 1.7] to produce an lc pair (X_1, Δ_1) with a small bimeromorphic morphism to V''_1 such that K_{X_1}+Δ_1 is ample over V''_1. The hypotheses of [F25b, Theorem 1.7] are not stated, and the paper does not check that (X''_1, Δ''_1) satisfies them after shrinking, in particular whether Q-factoriality or dlt-ness over W is required and whether it is preserved. Since the MMP in Definition 3.3 allows arbitrary normal analytic X_i and the non-biholomorphic locus is only required to avoid Nlc(X,Δ), a gap here would invalidate every step. Please state the precise theorem used and verify its hypotheses are satisfied at each step of the MMP.","section":"Lemma 3.11, Step 1"},{"comment":"The proof of Theorem 4.6 relies on 'the complex analytic analog of the special termination of MMP ([F07b])' cited to [EH24a, Subsection 3.6]. Special termination is a delicate statement that can fail in the non-Q-factorial, non-compact analytic category, and the paper's own Lemma 3.8 shows that controlling exceptional divisors over W requires additional arguments. The author should state the special termination theorem with its hypotheses (e.g., whether the pair is Q-factorial over W, whether the boundary is dlt, whether termination holds over a neighborhood of W) and either prove it or give a precise reference to an available preprint with the proof. Without this, the induction in Theorem 4.7 has no verified analytic foundation.","section":"Theorem 4.6 (proof)"},{"comment":"Lemma 4.5 reduces the λ_i-limit statement to Theorem 4.4 by applying [F25a, Proof of Lemma 4.25] to obtain a Q-divisor perturbation (Y, Δ') → [X, ω'] and then Lemma 2.20 to obtain a normal pair (X, Γ) with K_X+Γ ∼_{R,Z} ω + (1/2)A. The line 'We may regard X_1 → ... as a sequence of steps of a (K_X+Γ+1/2A)-MMP over Z around W with scaling of 1/2A' is nontrivial: the original MMP is for the quasi-log space (ω+A), while Theorem 4.4 applies to normal pairs, and one must check that the extremal contractions of the two MMPs coincide after the perturbation. The author should justify this identification, for example by showing the same extremal rays are contracted and the same λ_i values are obtained.","section":"Lemma 4.5"},{"comment":"The paper is almost entirely a translation of the author's algebraic paper [H23], with the main theorem and many auxiliary results proved by assertions that 'the argument works with no changes.' While this can be acceptable when the analytic analogs are published, here the decisive inputs are preprints by the same research group ([EH24a], [EH24b], [F25b]) and unpublished notes ([F22b]). The author should provide a table or list of each cited analytic result with its exact statement and location, and indicate which are published or under review. Without this, the reader cannot separate the author's contribution from the external preprints, and the main theorem is not verifiable from the manuscript itself.","section":"Introduction / References"}],"minor_comments":[{"comment":"The conclusion states 'there exists a sequence of a (K_X + ∆)-MMP over Z around z with scaling of H', but the setup and proof concern a (K_{X_1}+B_1)-MMP, i.e., a (K_X+∆+A)-MMP; the statement should be corrected to match the proof.","section":"Theorem 5.1"},{"comment":"The sentence 'ρ(X/Z; W ) − ρ(X/Z; W ) = 1' appears to contain a typo; it should presumably be ρ(X/Z; W ) − ρ(V/Z; W ) = 1 or ρ(X/Z; W ) − ρ(X'/Z; W ) = 1.","section":"Remark 3.4"},{"comment":"The condition 'NNef(K_X + ∆ + A/Z) ∩ Nlc(X, ∆) ∩ = ∅' is missing the relevant subset; it should specify the intersection with π^{-1}(W) or with the whole fiber over Z, consistent with the other statements.","section":"Corollary 4.12"},{"comment":"The phrase 'We set W_j as the inverse image of W to X_j' is ambiguous; it should say 'the inverse image of W under the morphism X_j → Z' or 'the inverse image of W in V', since X_j is obtained by base change over Z_j.","section":"Lemma 3.8"},{"comment":"The proof of Theorem 2.10 says 'the argument of [H23, Proof of Theorem 2.14] works with no changes,' but the definition of the intersection number (D·C) and the reduction to a Stein neighborhood may require analytic justifications; adding a one-sentence reference to [F22a] for the numerical intersection would help the reader.","section":"Theorem 2.10"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is heavily self-citational: the main theorem reduces to the author's [H23] and to several preprints by the same group ([EH24a], [EH24b], [F25b]). There is no logical circularity, since [H23] is a separate algebraic paper, but the verifiability of the main result depends on these preprints being publicly available and correct. The editor may wish to confirm that [EH24a], [EH24b], [F22b], and [F25b] are posted on the arXiv (or otherwise available) with complete proofs, and to consider whether the journal's standards for self-containedness of a main theorem allow a proof by reference to unpublished work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hashizume proves the analytic analog of his algebraic MMP for normal pairs along the lc locus: under the standard Stein/property (P) setup, if the non-nef locus avoids Nlc and the restriction to Nlc is semi-ample, then after shrinking the base there is a finite MMP, disjoint from Nlc, ending in a good minimal model. The main theorem is genuinely new for analytic spaces, and the paper is honest about what is new: the gluing argument that does not transfer verbatim (Lemma 3.11 and Theorem 3.12) is written out in detail, and so is the termination lemma (Lemma 4.5) that uses the perturbation trick. Those are real technical contributions, not cosmetic.\n\nThe soft spot is exactly where the reader put it. Theorem 4.7 is proved by \"the argument of [H23, Proof of Theorem 5.3] works with no changes,\" and the surrounding theorems lean on a stack of analytic results—special termination, lc flips, existence of good minimal models over a Stein germ—cited from preprints [EH24a], [EH24b], [F25b], and [F22b]. None of those is proved here. If any of them has a gap, the main theorem collapses. That is a load-bearing reliance, not a cosmetic one. But it is not a circularity: [H23] is the published algebraic theorem, and the analytic foundations are being developed by the same school. For a field where \"cite the analytic analog\" is the norm, this is a legitimate—if fragile—way to write a paper.\n\nThe citation pattern is heavy on the author's own work and collaborators, but that is because the paper is explicitly the analytic version of [H23]; the self-citation is functional, not padding. The writing is clear and the paper does the right thing: it tells the reader which parts are verbatim and which are not.\n\nVerdict: this deserves a real referee. The referee should be asked to check the external preprints, especially [EH24a, Subsection 3.6] and [F25b]—not to re-prove them, but to confirm they are in the published pipeline. If they are, the paper is correct and should be accepted. If they are not, the author should add a dependency diagram and either prove or explicitly isolate the analytic special termination input. I would cite it, and I'd bring it to a reading group focused on analytic MMP.","headline":"A genuinely new analytic MMP statement whose proof is mostly inherited from the author's algebraic paper and unpublished analytic foundations; the two truly new lemmas are real work.","tokens_in":26151,"tokens_out":2937,"would_cite":true,"duration_ms":25738,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite minimal model program exists for complex analytic normal pairs under a disjointness condition.","keywords":["minimal model program","normal pair","complex analytic space","log canonical locus","non-nef locus","quasi-log complex analytic space","good minimal model","relative MMP"],"falsifier":"A concrete way to test the claim is to search for a normal pair over a Stein base satisfying all hypotheses of Theorem 4.7 whose canonical ring is not locally finitely generated, or to exhibit a failure of one of the cited analytic inputs, such as an lc pair over a Stein space with no flip and no good minimal model over a point; either would make the main theorem false.","tokens_in":25042,"feed_emoji":"📐","tokens_out":13277,"duration_ms":108933,"temperature":0.7,"pith_summary":"Over a projective morphism from a normal complex analytic space to a Stein space, this paper proves that a pair whose singularities may be worse than log canonical still has a good minimal model, provided the set where the adjoint divisor fails to be nef (the non-nef locus) misses the set where the pair fails to be log canonical (the non-lc locus), and provided the adjoint divisor is semi-ample when restricted to the non-lc locus. After shrinking the base around a compact set, one can run a finite minimal model program whose steps drop the relative Picard number by one, avoid the non-lc locus, and end with a semi-ample canonical divisor. This is the complex analytic analogue of the author's earlier algebraic theorem, and the proofs follow the algebraic pattern, with analytic foundations supplied by recent work on lc pairs and quasi-log spaces. A reader should care because it extends one of the main structural tools of birational geometry to analytic families with singularities that are not log canonical, and it yields concrete consequences such as base-point-freeness along the non-lc locus and finite generation of the canonical ring.","feed_headline":"Complex normal pairs admit a finite MMP ending semi-ample","feed_subtitle":"When the non-nef and non-lc loci stay apart, a good minimal model exists over Stein bases.","key_machinery":"The central machinery is the analytic theory of quasi-log complex analytic spaces — spaces carrying a globally $\\mathbb R$-Cartier divisor and a collection of qlc centers, formally imitating the structure of a simple normal crossing pair — together with the relative non-nef locus $\\operatorname{NNef}(D/Z)$, the union of centres of prime divisors with positive asymptotic vanishing order with respect to $D$. The proof reduces the theorem to standard MMP ingredients — dlt blow-up (a bimeromorphic modification making the pair divisorially log terminal), the cone and contraction theorem, the base-point-free theorem, special termination, adjunction, and analytic existence of flips and good minimal models over a point — and then repeats the algebraic argument of [H23] almost verbatim. The main technical step needing new work is Lemma 3.11, which constructs an MMP by gluing local good minimal models over varying open subsets of the intermediate space, because global gluing of MMPs is not automatic for analytic spaces.","core_discovery":"The central claim is Theorem 4.7: if $\\pi\\colon X\\to Z$ is a projective morphism from a normal analytic variety $X$ to a Stein space $Z$, $W\\subset Z$ is a compact subset satisfying condition (P), $(X,\\Delta)$ is a normal pair, and $A$ is a $\\pi$-ample $\\mathbb R$-divisor such that $K_X+\\Delta+A$ is globally $\\mathbb R$-Cartier and $\\pi$-pseudo-effective, then under the disjointness condition $\\operatorname{NNef}(K_X+\\Delta+A/Z)\\cap \\operatorname{Nlc}(X,\\Delta)\\cap \\pi^{-1}(W)=\\varnothing$ and the semi-ampleness of $(K_X+\\Delta+A)|_{\\operatorname{Nlc}(X,\\Delta)}$ over a neighbourhood of $W$, one can shrink $Z$ around $W$ and run a finite $(K_X+\\Delta+A)$-MMP over $Z$ around $W$ whose steps are represented by bimeromorphic contractions with relative Picard number dropping by one, whose non-biholomorphic locus avoids $\\operatorname{Nlc}(X,\\Delta)$, and which terminates with $K_{X_m}+B_m$ semi-ample over $Z$. If $X$ is $\\mathbb Q$-factorial over $W$, every intermediate space is also $\\mathbb Q$-factorial over $W$.","pith_inferences":["If the cited analytic foundations are verified, the verbatim reduction to [H23] means the main theorem should be checkable line by line; until then, the result is formally conditional on those preprints.","The semi-ampleness hypothesis on the restriction to the non-lc locus is likely removable: an analytic abundance theorem for lc pairs would supply it automatically, leaving only the disjointness condition as the hypothesis.","Because each MMP step is obtained by gluing local outputs over open subsets of the base, the same method could patch outputs over finite Stein covers to obtain MMPs over bases that are not Stein; the stack-theoretic formulation in Section 5 is a first step in that direction."],"forward_implications":["For any normal pair and $\\pi$-ample divisor $A$ satisfying the disjointness and semi-ampleness hypotheses, a finite $(K_X+\\Delta+A)$-MMP over $Z$ around $W$ exists and ends at a good minimal model, with every step avoiding the non-lc locus.","The relative stable base locus ${\\rm Bs}|K_X+\\Delta+A/Z|_{\\mathbb R}$ is disjoint from $\\operatorname{Nlc}(X,\\Delta)\\cap \\pi^{-1}(W)$, so the adjoint linear system is base-point-free along the non-lc locus.","When $\\Delta$ and $A$ are $\\mathbb Q$-divisors, the canonical ring $\\bigoplus_{m\\ge 0}\\pi_*\\mathcal O_X(\\lfloor m(K_X+\\Delta+A)\\rfloor)$ is locally finitely generated.","Running a $(K_X+\\Delta)$-MMP with scaling of a $\\pi$-ample divisor produces a possibly infinite sequence whose numerical nef thresholds tend to zero and whose steps never meet the non-lc locus; with the additional hypotheses of Theorem 4.7 the sequence terminates at a good minimal model.","For algebraic or analytic stacks, a $(K_X+B)$-MMP with scaling of a $\\pi$-ample line bundle exists smooth-locally on the base and terminates at a good minimal model when the local hypotheses hold."],"supporting_citations":[{"why":"The algebraic theorem whose proof is transplanted to the analytic setting; supplies the original statement and the argument skeleton for Theorem 4.7 and its corollaries.","marker":"[H23]"},{"why":"Foundations of the analytic MMP for lc pairs, including special termination and existence of good minimal models over a point, used in constructing each MMP step.","marker":"[EH24a]"},{"why":"Establishes quasi-log complex analytic spaces, the cone and contraction theorem, and the vanishing and torsion-free results used to run the MMP.","marker":"[F22b]"},{"why":"Cited for analytic existence of lc flips and good minimal models over a point, needed to produce the flips in each step.","marker":"[F25b]"},{"why":"Provides the dlt blow-up, property (P), and the basic divisor and morphism conventions in the analytic category.","marker":"[F22a]"},{"why":"Supplies the stack-level MMP framework used in Section 5 to remove the shrinking of the base.","marker":"[EH24b]"},{"why":"Original algebraic special termination result whose complex analytic analogue is invoked through [EH24a, Subsection 3.6] in the proof of Theorem 4.6.","marker":"[F07b]"},{"why":"Algebraic quasi-log foundations, including adjunction and base-point-free statements, that are adapted to the analytic setting in this paper.","marker":"[F17]"}],"fun_headline_variants":["Analytic pairs get finite MMP when non-nef and non-lc loci avoid","Finite MMP for analytic pairs when non-nef and non-lc stay disjoint","Stein-base analytic MMP terminates with semi-ample model","Analytic pairs: finite MMP with semi-ample output over Stein bases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the complex analytic versions of the standard minimal-model-program ingredients — special termination, adjunction, flips, and existence of good models over a point — are all correct as stated in the cited papers, several of which are preprints, and that the algebraic proof from the earlier paper transfers verbatim.","fun_headline_variants_meta":{"raw":{"variants":["Analytic pairs get finite MMP when non-nef and non-lc loci avoid","Finite MMP for analytic pairs when non-nef and non-lc stay disjoint","Stein-base analytic MMP terminates with semi-ample model","Analytic pairs: finite MMP with semi-ample output over Stein bases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000767,"raw_usage":{"total_tokens":3348,"prompt_tokens":838,"completion_tokens":2510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":2430}},"tokens_in":454,"tokens_out":2510,"duration_ms":16413,"temperature":1.0,"reasoning_tokens":2430,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:52:06.767696+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the claim is to search for a normal pair over a Stein base satisfying all hypotheses of Theorem 4.7 whose canonical ring is not locally finitely generated, or to exhibit a failure of one of the cited analytic inputs, such as an lc pair over a Stein space with no flip and no good minimal model over a point; either would make the main theorem false.","supporting_citations":[],"review_version":1}