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Singular Calabi-Yau metrics

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2508.19438 v1 pith:HMY7QXJD submitted 2025-08-26 math.CV math.DG

Eleonora Di Nezza , Chinh H. Lu This is my paper · ORCID
classification math.CVmath.DG MSC 32W2032Q2032Q2553C55
keywords HermitianmanifoldscomplexMonge-AmpèreequationsingularCalabi-YaumetricslogterminalsingularitiesBott-CherncohomologyRicci-flatcurrentspluripotentialtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [§2.7, Lemma 2.24] 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'.
  2. [§5.4, Theorem 5.3] 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.
minor comments (5)
  1. [§7, Theorem 7.2] 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.
  2. [§5.4, proof of Theorem 5.3] The displayed formula contains 'j−jψ_j', which appears to be a typo for 'j^{-1}ψ_j'.
  3. [§4.2, Theorem 4.3] In the statement, 'ocs_X(u)' should be 'osc_X(u)'.
  4. [§5.4, proof of Theorem 5.4] The phrase 'By the exact the same arguments' should read 'By exactly the same arguments'.
  5. [§6, Theorem 6.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survey's derivation chain is independent; self-citations are proper attribution, not logical loops.

full rationale

The paper is an expository survey. Its central result, Theorem 7.2, is derived as a consequence of the complex Monge-Ampère existence theorems 6.5 and 6.6, which in turn rest on the domination principle (Theorem 2.26), the L∞ estimate (Theorem 4.2 and Lemma 4.1), and the local comparison principle (Lemma 2.24). Each of these is proved in the text from the pluripotential machinery of Section 2: the definition of θ-psh functions, the Monge-Ampère operator for non-closed forms (2.5), the maximum principle (Lemma 2.23), the orthogonal relation (Theorem 2.19), and Corollary 2.15. In Lemma 2.24, the proof uses the envelope w = P(u−v) and shows (ddcw)^n = 0 via Lemma 2.23 and the orthogonal relation; this is a genuine reduction, not a restatement of the assumption. Citations to [GL22], [GL23], [BGL24], and [DDNL21] point to prior published or arXiv work by the authors and coauthors, but these are not invoked as a substitute for the derivation, and no parameter is fitted to the quantity predicted in Theorem 7.2. The only blemishes are expository: the cross-reference to 'Theorem 2.18' in Lemma 2.24 should be Theorem 2.19, and Lemma 2.23's proof is only sketched with a pointer to [GL22, Lemma 1.2]. These do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

This is a survey compiling known results. No free parameters or invented entities. It rests on standard pluripotential theory plus the recent papers it cites, several co-authored by C. H. Lu, which are treated as established background rather than independently re-derived.

assumptions (6)
  • domain assumption Yau's theorem on the Calabi conjecture (Yau78)
    Motivates the Monge-Ampère framework in Section 1; used as the historical baseline.
  • standard math Bedford-Taylor theory for (ddc)^n of bounded psh functions, including continuity along decreasing sequences (Theorem 2.9)
    Foundational pluripotential theory the notes rely on throughout.
  • domain assumption Correctness of the cited results in [GL23], [BGL24], [DDNL21], [GLZ19], several with an author overlapping this paper
    The survey's proofs of L∞ estimates, domination principle, and envelopes follow these papers; if any has an error, the presentation inherits it.
  • standard math Hironaka log-resolution and adapted volume forms for log-terminal singularities
    Used in Section 7 to reduce the singular Hermitian Calabi-Yau problem on V to the resolution X.
  • standard math Gauduchon's theorem on existence/uniqueness of Gauduchon metric in conformal class
    Used in Theorem 5.3 (Section 5.4) and Theorem 6.5.
  • standard math Demailly regularization of quasi-psh functions (Dem94)
    Used in Section 6, Theorem 6.6.

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Pith. "Pith review of Singular Calabi-Yau metrics." pith.science (2026). https://pith.science/paper/HMY7QXJD

@misc{pith2026250819438,
  author       = {Pith},
  title        = {Pith review of: Singular Calabi-Yau metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMY7QXJD}},
  note         = {Machine review of arXiv:2508.19438}
}
read the original abstract

These are notes of lectures given by the first named author during the CIME Summer school Calabi-Yau varieties. We survey known results concerning the complex Monge-Amp\`ere equations in Hermitian contexts obtained by many authors during the last fifteen years.

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