{"id":"ceaa661f-ad9d-4a0e-86cb-4a89a823104b","arxiv_id":"2508.19438","paper_version":1,"verdict":"UNVERDICTED","confidence":"MODERATE","novelty_score":0.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A survey of known results on complex Monge-Ampère equations in Hermitian settings, with proofs following the envelope approach, ending with a theorem on singular Hermitian Calabi-Yau metrics.","lead":"These lecture notes survey recent progress on complex Monge-Ampère equations on Hermitian manifolds, presenting full proofs of known results, culminating in singular Hermitian Calabi-Yau metrics. A generalist should read them to learn a difficult and active area from a coherent, relatively self-contained source, though the paper claims no new mathematics.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the survey's central reduction is sound, and the suspected local comparison gap is not actual because any smooth (1,1)-form on a ball is ddc-exact.","rationale":"The paper is explicitly a survey: 'All results presented here are known, and we do not claim any originality.' Under the Pith rubric, there is no new research claim to accept or reject, so UNVERDICTED is the appropriate category. As a check of exposition, the chain from local comparison (Lemma 2.24) to domination (Theorem 2.26), L∞ estimates (Theorem 4.2), existence (Theorems 5.3, 6.5, 6.6), and singular Calabi-Yau metrics (Theorem 7.2) is internally coherent. The reader's stated weakest assumption, Lemma 2.24, is the right place to scrutinize, but the worry about non-closed θ is not valid: locally every smooth (1,1)-form is ddc-exact, so the non-closed case is subsumed by the classical closed comparison principle. The few internal inconsistencies—a wrong theorem number and a loose boundary-normalization phrase—do not affect the validity of the transmitted results. No load-bearing mathematical concern emerges.","tokens_in":32141,"tokens_out":11809,"duration_ms":139400,"concrete_test":"Re-derive Lemma 2.24 by substituting θ = ddcρ on B, setting u' = u+ρ and v' = v+ρ, and checking that the hypothesis becomes (ddcu')^n ≤ (ddcv')^n. If the standard comparison principle for bounded psh functions then gives u'−v' ≥ liminf_{∂B}(u'−v'), the suspected torsion obstruction is dissolved and the domination principle/L∞ machinery is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption is Lemma 2.24, the local comparison principle for non-closed θ. This is not a genuine obstruction to the argument. On a Euclidean ball, any smooth real (1,1)-form θ is ddc-exact: θ = ddcρ for a smooth function ρ (local ∂∂̄-lemma). Hence u ∈ PSH(B,θ) iff ρ+u is psh, and (θ+ddcu)^n = (ddc(ρ+u))^n. Lemma 2.24 therefore reduces exactly to the standard comparison principle for bounded psh functions on a ball, with no torsion term. The sketched proof—using vε = max(u, v−ε), the envelope w = P(u−v), and the maximum principle to show 1_D(ddcw)^n = 0—is a standard envelope route; the loose boundary characterization and the wrong cross-reference ('Theorem 2.18' instead of Theorem 2.19) are expository slips, not mathematical gaps. The only other notable omission is that Theorem 6.6 does not spell out the uniform lower bound for cε; this follows by the same AM-GM/Gauduchon argument used in Theorem 5.3, so it is not load-bearing. Thus no substantive concern remains.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"These lecture notes survey the development of complex Monge-Ampère equations on compact Hermitian manifolds, with the final goal of constructing singular Calabi-Yau metrics on compact Hermitian varieties with log-terminal singularities. The notes build the pluripotential toolbox for non-closed reference forms: positive currents, the Monge-Ampère operator for bounded quasi-psh functions, envelopes, comparison and domination principles, and L∞/Laplacian/higher-order estimates. They then solve the Hermitian Monge-Ampère equation (ω+ddcφ)^n = e^{λφ+f}ω^n for λ≥0, treat degenerate semipositive big reference forms, and apply the machinery to prove Theorem 7.2: for a compact Hermitian variety V with log-terminal singularities and a Hermitian form ωV, every smooth closed real (1,1)-form η in c_BC^1(V) is realized as Ric(ωV+ddcφ) on Vreg for a globally bounded φ, smooth on Vreg, so that ωV+ddcφ is a Hermitian current. The authors explicitly state that all results are known and that no originality is claimed.","tokens_in":32420,"tokens_out":20481,"duration_ms":216871,"significance":"The survey is a useful and mostly faithful exposition of the envelope-based approach of [GL23] and [BGL24], and it makes a coherent case for the Hermitian analogue of singular Calabi-Yau metrics. Theorem 7.2, if fully supported, is a valuable statement: it extends the singular Calabi-Yau theorem to compact Hermitian varieties with log-terminal singularities and Bott-Chern classes, and yields Ricci-flat Hermitian currents when c_BC^1(V)=0. The notes are also transparent about which arguments are sketched and which steps are quoted from the literature. The main concerns below are about the correctness of two load-bearing proof sketches rather than about the overall architecture.","major_comments":[{"comment":"The proof of the local comparison principle is not valid for non-closed θ. The key inequality 1_D(θ+ddcv)^n + 1_D(ddcw)^n ≤ 1_D(θ+ddc(w+v))^n requires all mixed terms in the binomial expansion of (θ+ddc(w+v))^n − (θ+ddcv)^n to be positive currents. This is standard when θ+ddcv is closed, but θ is not assumed d-closed; a smooth real (1,1)-form on a ball is ddc-exact only if it is d-closed, so the local ∂∂̄ lemma cannot reduce the lemma to the closed case. Since Lemma 2.24 feeds Proposition 2.25 and Theorem 2.26, and hence the L∞/domination machinery behind Theorem 7.2, the gap is load-bearing. Please replace the sketch by the correct argument from [GL23] (controlling torsion) or quote the comparison principle with a precise reference. Also correct the cross-references: Corollary 2.11 should be Lemma 2.23, and 'Theorem 2.18' should be 'Theorem 2.19'.","section":"§2.7, Lemma 2.24"},{"comment":"The assertion that 'the functions ψ_j/j converge in L1 to 0' is used to pass from the AM-GM inequality to the upper bound (5.5) for b_j, but no proof is given and it is not an automatic consequence of sup ψ_j=0 and ψ_j≤0. This is a load-bearing step in the λ=0 existence proof. For smooth f the claim can be bypassed by the maximum principle at a maximum point of φ_j, as in §5.2.2; please either justify the L1 claim or replace the argument.","section":"§5.4, Theorem 5.3"}],"minor_comments":[{"comment":"The uniqueness statement is missing a normalization: if φ is a solution, then φ+C is also a solution. Either impose sup_V φ=0 as in Theorems 5.3 and 6.5, or state uniqueness up to additive constants.","section":"§7, Theorem 7.2"},{"comment":"The displayed formula contains 'j−jψ_j', which appears to be a typo for 'j^{-1}ψ_j'.","section":"§5.4, proof of Theorem 5.3"},{"comment":"In the statement, 'ocs_X(u)' should be 'osc_X(u)'.","section":"§4.2, Theorem 4.3"},{"comment":"The phrase 'By the exact the same arguments' should read 'By exactly the same arguments'.","section":"§5.4, proof of Theorem 5.4"},{"comment":"The reduction 'by replacing v with (1−ε)v+ερ' is only sketched; since this is a known result, a precise reference to [GL23] would be helpful for readers.","section":"§6, Theorem 6.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a set of lecture notes, and the authors say so up front: all results are known, no originality is claimed. So judge it as a survey, not as a research contribution. On those terms it works well. The organization is clear: it follows the Guedj-Lu envelope method through the L∞ estimates, the domination principle, envelope convergence, and then derives the main theorem on singular Hermitian Calabi-Yau metrics as a consequence of the already-known Theorems 6.5 and 6.6. The proofs are mostly real proofs, not hand-waving, and the attribution to prior work is careful. The heavy self-citation is appropriate here because the notes are literally built on those papers.\n\nThe one worry that could be load-bearing is the local comparison principle for non-closed theta in Lemma 2.24. I checked it because the reader flagged it. It is not an actual gap: on a Euclidean ball any smooth real (1,1)-form is dd^c-exact, so the argument reduces to the standard comparison principle for bounded psh functions. The boundary normalization in the text is sloppy, and the reference to \"Theorem 2.18\" should point to Theorem 2.19, but the mathematics is fine.\n\nThe real soft spots are minor expository gaps. In the proof of Theorem 5.3, the claim that psi_j/j -> 0 in L1 is asserted without derivation; it follows from the uniform bound and normalization, but the reader has to fill it in. Theorem 6.6 does not spell out the uniform lower bound for c_epsilon; the AM-GM/Gauduchon argument from Theorem 5.3 supplies it. There are also a few typos, like \"ocs\" instead of \"osc\" in Theorem 4.2. None of these affect the central structure.\n\nWho is this for? A graduate student or someone entering Hermitian complex Monge-Ampère equations who wants a guided tour of the envelope method. For that reader the notes are genuinely useful. I would not cite them in place of the original papers for my own work, but I would recommend them as an entry point. A serious referee for a proceedings volume should check the filled-in details I mentioned; the central argument holds up.","headline":"A competent, honest survey of Hermitian complex Monge-Ampère equations; no new theorem, but the exposition is solid and the final singular Calabi-Yau metric theorem follows cleanly from known results.","tokens_in":32907,"tokens_out":2675,"would_cite":false,"duration_ms":30360,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32W20","32Q20","32Q25","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every smooth closed (1,1)-form in the Bott-Chern class of a singular Hermitian variety arises as the Ricci curvature of a Hermitian metric with bounded potential.","keywords":["Hermitian manifolds","complex Monge-Ampère equation","singular Calabi-Yau metrics","log terminal singularities","Bott-Chern cohomology","Ricci-flat currents","pluripotential theory"],"falsifier":"A concrete test is to construct, on the unit ball in C^2, a non-closed smooth (1,1)-form θ and bounded θ-psh functions u,v with (θ+ddcu)^2 ≤ (θ+ddcv)^2 in B but with u−v taking values strictly below its boundary limit in the interior. If such a pair exists, Lemma 2.24 is false and the L∞ machinery in the notes loses its foundation; if explicit searches for such pairs with small non-closed perturbations of the Euclidean form come up empty, the foundation is supported.","tokens_in":32026,"feed_emoji":"📐","tokens_out":15290,"duration_ms":155280,"temperature":0.7,"pith_summary":"These lecture notes set out to show that the classical Ricci-form existence problem for Kähler manifolds has a Hermitian analogue for singular varieties. The endpoint is a theorem: on a compact Hermitian variety V with log-terminal singularities, every smooth closed real (1,1)-form representing the Bott-Chern class c_BC^1(V) is realized as the Ricci curvature of a Hermitian metric of the form ω_V + ddcφ, with φ bounded on V and smooth on the regular locus. When c_BC^1(V)=0, this yields a Ricci-flat Hermitian current in every ddc-cohomology class, a natural Hermitian counterpart of singular Calabi-Yau metrics. The route goes through complex Monge-Ampère equations for non-closed reference forms on compact Hermitian manifolds, using a domination principle that replaces the classical comparison principle, which fails in the non-Kähler setting. The authors state that the analytic results are known and that the note is a survey; the contribution is a coherent assembly leading to this singular existence theorem.","feed_headline":"Hermitian varieties admit singular Calabi-Yau metrics","feed_subtitle":"The result extends Calabi-Yau metrics from Kähler spaces to Hermitian varieties with mild singularities","key_machinery":"The key mechanism is the complex Monge-Ampère operator for non-closed reference forms: for a bounded θ-psh function u, the current (θ + ddcu)^n is defined locally by expanding against a smooth strictly psh potential and is a positive closed current. The argument rests on a domination principle for this operator on compact Hermitian manifolds — if the measure (θ + ddcu)^n on the set {u < v} is a strict fraction of (θ + ddcv)^n, then u ≥ v — which substitutes for the comparison principle of Kähler geometry that fails when the reference form is not closed. Surrounding this core are the local L∞ estimate for densities in L^p, the Laplacian and higher-order estimates, and envelope/balayage techni","core_discovery":"The central claim is Theorem 7.2: for a compact Hermitian variety V with log-terminal singularities and a Hermitian form ω_V, every smooth closed real (1,1)-form η in c_BC^1(V) admits a unique bounded function φ ∈ PSH(V, ω_V), smooth on V_reg, such that ω_V + ddcφ is a Hermitian form and Ric(ω_V + ddcφ) = η on V_reg. The paper derives this from the solvability of a degenerate Monge-Ampère equation (θ + ddcφ)^n = c e^{ψ+−ψ−} dV on a resolution of singularities, where θ is the pullback of ω_V, semipositive and big, and ψ± encode the discrepancies of the log-terminal resolution. In particular, when c_BC^1(V) = 0, the theorem produces Ricci-flat Hermitian currents, giving singular Calabi-Yau met","pith_inferences":["A decisive check the authors do not perform is whether the local comparison principle, Lemma 2.24, genuinely holds for non-closed θ with torsion; the proof's boundary normalization is sketched, and a gap there would force a different route to the L∞ estimates without necessarily killing the theorem.","The same envelope-and-domination machinery should transfer to other fully nonlinear equations on Hermitian manifolds with big semipositive reference forms, because the estimates are local and never use closedness of the reference form.","Near the singular locus of V, the discrepancies ai that define ψ± should control the leading asymptotics of the solution; deriving cusp-like or cone-like behaviour on explicit log-terminal singularities would be a natural testable extension.","For a smooth Hermitian manifold with vanishing Bott-Chern class, the theorem yields a Ricci-flat Hermitian current in each ddc-cohomology class; comparing these currents with any explicit Ricci-flat Hermitian metrics on such manifolds would show which representative the theorem selects."],"forward_implications":["Every smooth closed (1,1)-form representing c_BC^1(V) is realized as Ric(ω_V + ddcφ) for a unique bounded φ smooth on V_reg.","When c_BC^1(V) = 0, every Hermitian form ω_V is ddc-cohomologous to a Ricci-flat Hermitian current.","The existence results apply to degenerate semipositive big reference forms and L^p right-hand sides, not only smooth Kähler classes.","The singular Ricci-flat currents solve a Monge-Ampère equation on a log resolution with right-hand side e^{ψ+−ψ−} determined by the discrepancies; their asymptotic profile near the singular locus is left open."],"supporting_citations":[{"why":"Solved the classical Kähler existence problem for prescribed Ricci forms; the benchmark result the notes adapt.","marker":"[Yau78]"},{"why":"Established the uniform C0 estimate for Hermitian Monge-Ampère equations without Kähler assumptions, removing the main obstruction.","marker":"[TW10a]"},{"why":"Supplies the coordinate system and Laplacian estimate technique used in Sections 4 and 6.","marker":"[GL10]"},{"why":"Provides the Hermitian Laplacian and higher-order estimate computations that Sections 4.2–4.3 adapt.","marker":"[TW10b]"},{"why":"Gives the modified comparison principle for non-closed forms on which the domination principle and L∞ estimates rest.","marker":"[DK12]"},{"why":"Develops the envelope-based local L∞ estimate approach that the notes follow to solve Monge-Ampère equations on Hermitian manifolds.","marker":"[GL23]"},{"why":"Extends the envelope/L∞ machinery to degenerate settings, supplying the semipositive-big reference-form framework used for Theorems 6.5–6.6.","marker":"[BGL24]"},{"why":"Provides the background pluripotential theory: capacity, envelopes, uniform integrability, and DPSH regularity used throughout.","marker":"[GZ17]"},{"why":"Constructs singular Kähler-Einstein metrics on normal varieties, the Kähler antecedent of Theorem 7.2.","marker":"[EGZ09]"}],"fun_headline_variants":["Hermitian varieties get singular Calabi-Yau metrics","Calabi-Yau metrics extended to Hermitian singularities","Singular Calabi-Yau metrics on Hermitian varieties","Mild singularities: Hermitian Calabi-Yau metrics","Beyond Kähler: Hermitian Calabi-Yau metrics"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The whole argument rests on a local comparison principle that orders two bounded potentials when one has no larger a Monge-Ampère mass than the other, even when the background form is not closed; if that principle fails for non-closed forms, the L∞ estimates and the existence theorem as presented are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Hermitian varieties get singular Calabi-Yau metrics","Calabi-Yau metrics extended to Hermitian singularities","Singular Calabi-Yau metrics on Hermitian varieties","Mild singularities: Hermitian Calabi-Yau metrics","Beyond Kähler: Hermitian Calabi-Yau metrics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000161,"raw_usage":{"total_tokens":1004,"prompt_tokens":605,"completion_tokens":399,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":349,"completion_tokens_details":{"reasoning_tokens":331}},"tokens_in":349,"tokens_out":399,"duration_ms":3718,"temperature":1.0,"reasoning_tokens":331,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:47:43.916859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test is to construct, on the unit ball in C^2, a non-closed smooth (1,1)-form θ and bounded θ-psh functions u,v with (θ+ddcu)^2 ≤ (θ+ddcv)^2 in B but with u−v taking values strictly below its boundary limit in the interior. If such a pair exists, Lemma 2.24 is false and the L∞ machinery in the notes loses its foundation; if explicit searches for such pairs with small non-closed perturbations of the Euclidean form come up empty, the foundation is supported.","supporting_citations":[],"review_version":1}