Pith. sign in

REVIEW 3 major objections 6 minor 29 references

An Aubin-Yau theorem for transversally K\"ahler foliations

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A transversally Kähler foliation with negative basic first Chern class admits a unique transversally Kähler, transversally Einstein metric with Einstein constant -1.

desk verdict A plausible foliated Aubin-Yau theorem that is not ready as stated: compactness is missing from the hypotheses and the uniqueness proof contains a false determinant identity, but the core approach is sound and worth refereeing after revision. read the letter →

arxiv 2506.03987 v1 pith:N4SQ3APH submitted 2025-06-04 math.DG math.APmath.CV

Vlad Marchidanu This is my paper
classification math.DGmath.APmath.CV MSC 53C5532J2758J0532M25
keywords transversallyKählerfoliationAubin-YautheorembasiccohomologyhomologicalorientabilitycomplexMonge-AmpèreequationEinsteinmetricVaismanmanifoldtransverse∂∂̄-lemma
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

This paper establishes a transverse version of the Aubin-Yau theorem for foliations: on a homologically orientable, transversally Kähler foliation whose basic first Chern class is negative, there exists a unique transversally Kähler metric that is also transversally Einstein with constant -1. The proof follows the classical continuity-method strategy, reducing the geometric problem to solving a complex Monge-Ampère equation for basic functions. A transverse version of the $\partial\bar{\partial}$-lemma is the key that turns the cohomological condition into a differential equation. As an application, the author gives a shorter proof of the known Aubin-Yau theorem for Vaisman manifolds.

What carries the argument

The central object is the transverse complex Monge-Ampère operator $\mathrm{Cal}_T^-(u)=\log\left(\frac{(\omega_0+i\partial\bar{\partial}u)^q}{\omega_0^q}\right)-u$ on basic functions, where $\omega_0$ is a transversally Kähler form and $q$ is the transverse complex dimension. The paper also relies on a transverse $\partial\bar{\partial}$-lemma (Lemma 3.3, cited to [10, Prop. 3.5.1]): on a homologically orientable transversally Kähler foliation, any two cohomologous basic $(1,1)$-forms differ by $i\partial\bar{\partial}$ of a basic function. This lemma converts the cohomological hypothesis on $c_1(\nu\mathcal{F})$ into the equation $\mathrm{Cal}_T^-(u)=f$. The analytic machinery consists of basic Hölder spaces, a transversally elliptic Laplacian $\Delta_T$, and the isomorphism property of $\Delta_T+\mathrm{Id}$, which supplies the linearized invertibility needed for the continuity method.

What would settle it

Find a compact homologically orientable transversally Kähler foliation with negative basic first Chern class for which the transverse Monge-Ampère equation has no smooth basic solution, or two distinct solutions; alternatively, exhibit cohomologous basic (1,1)-forms on such a foliation that do not differ by $i\partial\bar{\partial}$ of a basic function, which would refute Lemma 3.3 and the proof's bridge to the equation.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5.1: if $(M,\mathcal{F})$ is a homologically orientable, transversally Kähler foliation and the basic first Chern class $c_1(\nu\mathcal{F})\in H^2_{\mathrm{bas}}(M;\mathcal{F})$ is negative, then there exists a unique transversally Kähler metric whose transverse Ricci form equals minus the metric form, i.e., a transversally Einstein metric with Einstein constant $-1$. The paper proves this by showing that the problem is equivalent to the existence and uniqueness of a basic function $u$ solving the complex Monge-Ampère equation $\mathrm{Cal}_T^-(u)=f$ for any given basic $f$, where $\mathrm{Cal}_T^-(u)=\log\left(\frac{(\omega_0+i\partial\bar{\partial}u)^q}{\omega_0^q}\right)-u$. Existence is obtained via the Schauder continuity method with a priori estimates adapted from the classical proofs; uniqueness follows from the maximum principle applied to the linearized equation. The application to Vaisman manifolds yields a new, simpler proof of a known result on special Vaisman metrics.

Load-bearing premise

The proof depends on the transverse $\partial\bar{\partial}$-lemma for homologically orientable transversally Kähler foliations: any two cohomologous basic (1,1)-forms must differ by $i\partial\bar{\partial}$ of a basic function; if that lemma fails or needs extra hypotheses, the reduction of the geometric problem to the complex Monge-Ampère equation collapses.

Editorial extensions

If this is right

  • Establishes existence and uniqueness of transversally Kähler–Einstein metrics with constant $-1$ for negative basic first Chern class, generalizing the classical Aubin–Yau theorem to foliated geometry.
  • Provides a self-contained analytic toolkit (basic Hölder spaces, transverse Laplacian, a priori estimates) that can be reused for other transverse geometric PDEs.
  • Gives a new, simpler proof of the Aubin–Yau theorem for Vaisman manifolds, replacing a Weitzenböck-type argument with the foliated theorem.
  • Clarifies that homological orientability is the structural condition under which the transverse $\partial\bar{\partial}$-lemma holds, thereby identifying a natural hypothesis for future transverse Calabi–Yau theorems.

Reading between the lines

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

  • The same continuity-method framework may extend to transversally Kähler foliations with zero basic first Chern class, yielding a transverse Calabi–Yau theorem, provided the transverse $\partial\bar{\partial}$-lemma and estimates adapt; the paper does not address this.
  • Because the a priori estimates are local, the proof may carry over to non-compact or even singular foliations under suitable growth conditions, although the compactness and finite atlas arguments would need modification.
  • The proof indirectly highlights the transverse $\partial\bar{\partial}$-lemma as the main obstruction: testing Lemma 3.3 on explicit foliations (e.g., non-quasi-regular Vaisman manifolds) could reveal how much homological orientability can be relaxed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript proves a transverse Aubin-Yau theorem: on a homologically orientable, transversally Kähler foliation with negative basic first Chern class, it claims existence and uniqueness of a transversally Kähler, transversally Einstein metric with Einstein constant -1. The proof reduces the geometric problem to a complex Monge-Ampère equation for basic functions, solves this equation by the continuity method using classical a priori estimates imported from Blocki's exposition, and then applies the result to obtain a new proof of the Vaisman Aubin-Yau theorem. Sections 2-4 develop the needed foliation theory, basic Hölder/Sobolev spaces, and elliptic regularity for the transverse Laplacian.

Significance. If the theorem is correct after adding the necessary hypotheses, it fills a genuine gap: the literature appears to contain only brief claims of a transverse Calabi-Yau theorem, not a full Aubin-Yau theorem for transversally Kähler foliations. The explicit reduction to a Monge-Ampère equation for basic functions and the Vaisman application are useful and well-motivated. The paper is also honest about which estimates are imported and which steps are local. However, the main theorem as stated omits a compactness hypothesis that is used essentially in the proof, and the uniqueness proof contains a false algebraic identity. These issues are load-bearing, though both appear repairable.

major comments (3)
  1. [§5, Theorem 5.1 and §4, Theorems 4.7-4.8, Proposition 4.13] The statement of Theorem 5.1 does not assume M compact, but the proof requires compactness at several essential points. Theorem 4.7 is stated without a compactness hypothesis, yet its proof begins "Since M is compact" and uses the Rellich-Kondrachov compact embedding and the Fredholm alternative for L^2_bas(M;F). Proposition 4.5, used in the closedness argument, is stated only for compact manifolds, and Proposition 4.13 obtains a uniform constant by taking the maximum of local estimates over the foliated atlas, which only gives a finite constant if the atlas is finite. The maximum principle for basic functions used in Step 2 of Theorem 5.1 also requires that basic functions attain their extrema, which need not hold on a noncompact M or noncompact leaf space. As written, the main theorem is unproven. Adding "M compact" to Theorem 5.1, Theorem 4.8, and Theorem 4.7 would likely repair the argument, but this is a substantive missing hypothesis that should be stated explicitly from the beginning.
  2. [§5, Step 2, Eqs. (10)-(12)] The uniqueness proof contains a false algebraic identity. Equation (11) rewrites the left-hand side of (10) as log(det(g_x) + det(h_x)) - log(det(g_x)), but det(g+h) is not equal to det(g) + det(h) in dimension greater than one. Equation (12) then replaces the logarithm by the sum of the eigenvalues of h_x; the correct identity after diagonalization is log det(I+H) = sum_i log(1 + lambda_i), not sum_i lambda_i. Consequently, the displayed derivation of max_M u ≤ 0 from (12) is invalid. The intended maximum-principle argument is repairable: at a maximum point h_x has nonpositive eigenvalues, so each log(1 + lambda_i) is nonpositive and the equation gives u(x_max) ≤ 0. But the equations printed in the proof must be corrected.
  3. [§4.3, Proposition 4.10] In the Fréchet derivative computation, the paper invokes the Kähler identity Delta_d^omega = 2 Delta_∂bar^omega and then replaces -Delta_∂bar - Id by -Delta_d - Id. With the identity as stated, the correct replacement is -(1/2)Delta_d - Id. The conclusion that the derivative is an isomorphism can still be obtained by applying Theorem 4.7 to the operator Delta_T + 2 Id instead of Delta_T + Id, but the displayed formula in the proof is not correct and the application as written is not justified.
minor comments (6)
  1. [§5, Step 1] After deriving ∂bar∂bar(Cal_T(u) - f) = 0, the text states that it suffices to solve Cal_T(u) = f. One should justify that the additive constant can be removed by adding a constant to u, and note that this normalization uses compactness or a basic maximum principle.
  2. [§4.3, Theorem 4.8] The statement "Let (M,F) be a transversally Kähler" is incomplete; it should read "transversally Kähler foliation." The same statement also appears to need a compactness hypothesis, as discussed in the major comments.
  3. [§4.3, Proposition 4.13] In the last line of the proof, the operator is written as Cal_B(u); this should be Cal_T(u).
  4. [Definition 3.8] The notation c1(νF) ∈ H^2_bas(M;F) is imprecise, since c1(νF) is naturally a class in H^2(M). Please clarify the choice of lift in the basic cohomology group and state explicitly that the representative ω is closed and basic.
  5. [§4.3 and Section 5] The abstract and introduction describe the proof as self-contained, but the key C0, C2, and C^{2,α} a priori estimates are imported from [4], specifically Theorems 4.12 and the estimates preceding it. This is acceptable if framed clearly as using classical local estimates, but the wording "self-contained" is stronger than what the proof delivers.
  6. [Section 6.2, Theorem 6.12] In the uniqueness proof, the step "dd^c u = 0, whence from the maximum principle u must be constant" requires M to be compact; if Theorem 5.1 is amended with a compactness hypothesis, this should be stated explicitly here as well.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a genuine extension proved from external analytical estimates and a cited transverse dbar lemma, with no fitted inputs or load-bearing self-citations.

full rationale

The derivation chain is not circular. Theorem 5.1 reduces the existence of a transversally Kaehler-Einstein metric to solving the complex Monge-Ampere equation Cal_T(u)=f, equation (9). The reduction uses Lemma 3.3, a transverse dbar lemma quoted from [10], to pass between cohomologous basic (1,1)-forms; this is an external structural result, not a restatement of the theorem. Existence is proved by the continuity method: openness (Proposition 4.10) uses the linear isomorphism Theorem 4.7 and regularity from [12], and closedness (Proposition 4.13) imports Yau's a priori estimates from [25]/[4] as Theorem 4.12. These estimates are stated as external, parameter-free results with their own hypotheses, and the foliated theorem is not used to prove them. Uniqueness is argued from the maximum principle, independently of the existence input. The Vaisman application (Theorem 6.12) cites [13, Theorem 6.3] only as the previously known result being reproved, not as a premise, and [14, Proposition 1] is used only for the final Vaisman criterion. There are no fitted parameters, no prediction of quantities defined by the inputs, and no self-citations by the author that carry logical weight. The most serious concerns in the paper, namely the omitted compactness hypothesis in Theorem 5.1 and the algebraic identity in equation (11), are correctness issues rather than circularities, so they do not affect the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants or ad hoc free parameters appear. The paper relies on imported theorems, listed in axioms, and on literature references.

assumptions (4)
  • domain assumption Transverse d-bar lemma: on a homologically orientable transversally Kähler foliation, any two cohomologous basic (1,1) forms differ by i∂∂bar of a basic function.
    Lemma 3.3, cited to [10, Proposition 3.5.1], is the key reduction used in Step 1 of Theorem 5.1 and in the Vaisman application, but it is not proved in this paper.
  • standard math Yau's a priori estimates for the complex Monge-Ampère equation on open subsets of C^q (Theorem 4.12, cited to [25] and [4]).
    Used in Proposition 4.13 to obtain uniform C^{2,α} bounds for the continuity method; the estimates are local and imported from classical Kähler geometry.
  • standard math Gårding inequality for the basic Laplacian on W^{1,2}_bas and Rellich-Kondrachov compactness for basic Sobolev spaces.
    Used in Theorem 4.7 to prove the linearized operator Δ_T + Id is an isomorphism; the proof delegates these to [11] and [9].
  • standard math Exact sequence for Bott-Chern cohomology of Vaisman manifolds, used to show the hypotheses of Theorem 6.12 are equivalent to Istrati's [13, Theorem 6.3].
    Cited to [13, equation (9)] in the Vaisman application.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Aubin-Yau theorem for transversally K\"ahler foliations." pith.science (2026). https://pith.science/paper/N4SQ3APH

@misc{pith2026250603987,
  author       = {Pith},
  title        = {Pith review of: An Aubin-Yau theorem for transversally K\"ahler foliations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N4SQ3APH}},
  note         = {Machine review of arXiv:2506.03987}
}
read the original abstract

Transversally K\"ahler foliations are a generalisation of K\"ahler manifolds, appearing naturally in the complex non-K\"ahler setting. We give a self-contained proof of how the classical methods used in the proof of the Aubin-Yau theorem adapt to the transversally K\"ahler case under the homological orientability condition. We apply this result to obtain a new, simpler proof of the already known Vaisman Aubin-Yau theorem.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [10]

    El Kacimi-Alaoui, Op´ erateurs transversalement elliptiques sur un feuilletage riemannien et applications , Compositio Math

    A. El Kacimi-Alaoui, Op´ erateurs transversalement elliptiques sur un feuilletage riemannien et applications , Compositio Math. 73 (1990), no. 1, 57-106. (Cited on pages 3 and 11.)

    work page 1990

  2. [13]

    Istrati, Vaisman manifolds with vanishing first Chern class , J

    N. Istrati, Vaisman manifolds with vanishing first Chern class , J. Eur. Math. Soc. (2025). (Cited on pages 3 and 28.)

    work page 2025

  3. [1]

    ´Alvarez-L´ opez,The basic component of the mean curvature of Rieman- nian foliations, Ann

    J. ´Alvarez-L´ opez,The basic component of the mean curvature of Rieman- nian foliations, Ann. Global. Anal. Geom. 10 (1992), 179-194. (Cited on page 9.)

    work page 1992

  4. [2]

    Aubin, ´Equations du type Monge-Amp` ere sur les vari´ et´ es Riemanni- ennes compactes, C.R

    T. Aubin, ´Equations du type Monge-Amp` ere sur les vari´ et´ es Riemanni- ennes compactes, C.R. Acad. Sci. Paris 283 (1976), 119-121. (Cited on page 2.)

    work page 1976

  5. [3]

    Besse, Einstein Manifolds

    A. Besse, Einstein Manifolds. Ergebnisse der Mathematik und ihrer Gren- zgebiete 10. Springer Verlag Berlin, Berlin, 1981. (Cited on page 2.)

    work page 1981

  6. [4]

    Z. B locki. The Calabi-Yau Theorem. Chapter 5 in V. Guedj (Ed.). Com- plex Monge–Amp´ ere Equations and Geodesics in the Space of K¨ ahler Metrics. Lecture Notes for Winter School in Complex Analysis, Toulouse,

    work page

  7. [5]

    Calabi, On K¨ ahler manifolds with vanishing canonical class, Algebraic geometry and topology, a symposium in honor of S

    E. Calabi, On K¨ ahler manifolds with vanishing canonical class, Algebraic geometry and topology, a symposium in honor of S. Lefschetz (1955), 78-89. (Cited on page 2.)

    work page 1955

  8. [6]

    Conlon, Transversally parallelisable foliations of codimension 2, Trans

    L. Conlon, Transversally parallelisable foliations of codimension 2, Trans. Am. Math. Soc. 194 (1974), 79-102. (Cited on page 6.)

    work page 1974

Show all 29 references
  1. [7]

    A. C. da Silva, Lectures on Symplectic Geometry. Lecture Notes in Math- ematics 1764. Springer, 2001. (Cited on page 10.)

    2001

  2. [8]

    Dieudonn´ e, Foundations of Modern Analysis

    J. Dieudonn´ e, Foundations of Modern Analysis. New York: Academic Press, 1960. (Cited on page 20.)

    1960

  3. [9]

    L. C. Evans, Partial differential equations. Second edition. Graduate Studies in Mathematics 19. American Mathematical Society, 1998. (Cited on pages 16 and 17.)

    1998

  4. [11]

    Kamber, Ph

    F. Kamber, Ph. Tondeur, De Rham-Hodge theory for Riemannian foli- ations. Math. Ann. 277 (1987), 415-431. (Cited on pages 16 and 17.) – 30 – Vlad Marchidanu An Aubin-Yau theorem for transversally K¨ ahler foliations

    1987

  5. [12]

    Gilbarg, N

    D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften

  6. [14]

    Istrati, Existence criteria for special locally conformally K¨ ahler met- rics, Annali di Matematica Pura ed Applicada (1923 -) 198 (2019), 335-

    N. Istrati, Existence criteria for special locally conformally K¨ ahler met- rics, Annali di Matematica Pura ed Applicada (1923 -) 198 (2019), 335-

    2019

  7. [15]

    Molino, Connexions et G-structures sur les vari´ et´ es feuillet´ ees, Bull

    P. Molino, Connexions et G-structures sur les vari´ et´ es feuillet´ ees, Bull. Sci. Math. 92 (1968), 59-63. (Cited on page 6.)

    1968

  8. [16]

    Molino, Riemannian foliations

    P. Molino, Riemannian foliations. Birkh¨ auser, 1988. (Cited on pages 4, 5, 6, 7, and 14.)

    1988

  9. [17]

    Moroianu, Lectures on K¨ ahler geometry

    A. Moroianu, Lectures on K¨ ahler geometry. London Mathematical So- ciety Student Texts 69. Cambridge University Press, Cambridge, 2007. (Cited on pages 11 and 21.)

    2007

  10. [18]

    J. M. Lee, Introduction to Smooth Manifolds. Graduate Texts in Math- ematics, 218. Springer Verlag, New York, 2012. (Cited on page 4.)

    2012

  11. [19]

    Ornea, M

    L. Ornea, M. Verbitsky, Principles of Locally Conformally K¨ ahler Ge- ometry. Progress in Mathematics 354. Birkh¨ auser Cham, 2024. (Cited on page 27.)

    2024

  12. [20]

    E. Park, K. Richardson, The basic Laplacian of a Riemannian foliation , Amer. J. Math. 118 (1996), 1249-1275. (Cited on page 8.)

    1996

  13. [21]

    Rudin, Functional Analysis

    W. Rudin, Functional Analysis. Second edition. International Series in Pure and Applied Mathematics 8. New York, McGraw-Hill Math, 1991. (Cited on page 18.)

    1991

  14. [22]

    Sz´ ekelyhidi, An Introduction to Extremal Metrics

    G. Sz´ ekelyhidi, An Introduction to Extremal Metrics. Graduate Studies in Mathematics 152. Providence, 2014. (Cited on page 2.)

    2014

  15. [23]

    Tondeur, Geometry of Foliations

    P. Tondeur, Geometry of Foliations. Monographs in Mathematics 90. Birkh¨ auser Verlag, Basel, 1997. (Cited on pages 5 and 6.) – 31 – Vlad Marchidanu An Aubin-Yau theorem for transversally K¨ ahler foliations

    1997

  16. [24]

    Tsukada, Holomorphic forms and holomorphic vector fields on com- pact generalized Hopf manifolds , Compositio Math

    K. Tsukada, Holomorphic forms and holomorphic vector fields on com- pact generalized Hopf manifolds , Compositio Math. 93 (1) (1994), 1-22. (Cited on page 26.)

    1994

  17. [25]

    Yau, On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Amp´ ere equation, Com

    S.-T. Yau, On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Amp´ ere equation, Com. Pure and Appl. Math 31 (1978), 339-411. (Cited on pages 2 and 22.)

    1978

  18. [26]

    Vaisman, Generalized Hopf manifolds , Geom

    I. Vaisman, Generalized Hopf manifolds , Geom. Dedicada 13 (1982), 231-255. (Cited on page 26.)

    1982

  19. [27]

    Simion Stoilow

    M. Verbitsky, Theorems on the vanishing of cohomology for locally con- formally hyper-K ¨hler manifolds , Proc. Steklov Inst. Math. no. 3 ( 246), 54-78 (2004). (Cited on pages 26 and 27.) Vlad Marchidanu University of Bucharest, F aculty of Mathematics and Informatics, 14 Acad...

    2004

  20. [224]

    (Cited on pages 13, 16, 17, and 18.)

    Springer-Verlag, Berlin, 1983. (Cited on pages 13, 16, 17, and 18.)

    1983

  21. [2005]

    (Cited on pages 2 and 22.)

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.