On Homogeneous K\"ahler Manifolds

Antonio De Nicola; Fabrizio Pugliese; Luca Vitagliano

REVIEW 4 major objections 3 minor 20 references

On Homogeneous K\"ahler Manifolds

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Homogeneous Kähler structures are exactly the ACM datasets of type H satisfying equations (2.11) and (2.19), and they reduce to Sasakian structures precisely when the Euler vector field is pre-geodesic.

desk verdict A useful clarification that pins down the exact Sasakian gap, but the referee should verify the unshown algebra in §2.2 and fix a handful of displayed slips. read the letter →

arxiv 2608.03785 v1 pith:EACPOB2E submitted 2026-08-04 math.DG

Antonio De Nicola , Fabrizio Pugliese , Luca Vitagliano This is my paper · ORCID

classification math.DG MSC 53C2553D1053C55

keywords homogeneousKählerstructuresSasakianco-KählerlinebundlesprincipalR^×-bundlesalmostcontactmetricAtiyahalgebroid

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 asks a precise question: the cone construction that turns a Sasakian manifold into a Kähler manifold was recently generalized to homogeneous Kähler structures on principal R^×-bundles, but those structures are strictly more general than Sasakian ones. The authors prove exactly how much more general. Using the line bundle associated to the principal bundle, they show every homogeneous Kähler structure is encoded by an almost contact metric dataset of type H — a tuple (φ, g_M, ν, ϕ, ξ, η, α, β) satisfying two equations — and the extra generality consists precisely of a 1-form ν plus possibly non-trivial line bundles. When the Euler vector field on the bundle is pre-geodesic, ν vanishes and the structure is exactly a Sasakian structure on the base. The same analysis with a different homogeneity condition yields the analogous statement for co-Kähler structures.

What carries the argument

The paper's workhorse is the homogenization correspondence between line bundles and principal R^×-bundles: sections of bundles built from L and the Atiyah algebroid DL correspond to homogeneous tensors on L̃. From a homogeneous Kähler structure one extracts an ACM dataset (φ, g_M, ν, ϕ, ξ, η, α, β); equations (2.11) and (2.19) are the integrability conditions for the complex structure and the Kähler form, respectively. The decisive mechanism is Lemma 2.7, which identifies the vanishing of the 1-form ν with the Euler vector field being pre-geodesic; the invariant analogue is Lemma 3.9, where constant u and flat connection ∇ are equivalent to E being parallel.

What would settle it

Compute ∇^LC_E E in Example 2.20 after deforming E by the vector field U: Lemma 2.7 predicts that whenever ν ≠ 0, the Euler vector field is not pre-geodesic, so the orthogonal distribution is non-integrable. Checking this directly — and likewise testing an invariant structure with constant u but non-flat connection — would confirm or break the bridge that identifies the Sasakian and co-Kähler gaps.

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Extended reading notes

Core claim

Theorem 2.16 establishes a one-to-one correspondence between homogeneous Kähler structures on the homogeneous manifold L̃ = L*∖0 and ACM datasets of type H satisfying equations (2.11) and (2.19). Corollary 2.18 sharpens this: if the line bundle is oriented and the Euler vector field is pre-geodesic, the correspondence reduces exactly to Sasakian structures on the base. The paper's central claim is that homogeneous Kähler geometry is Sasakian geometry plus two controlled relaxations: the contact structure may be non-coorientable, and the orthogonal distribution to the fibers need not be integrable. The invariant analogue, Theorems 3.11 and 3.13, does the same for co-Kähler structures.

Load-bearing premise

The identification of homogeneous Kähler structures with Sasakian structures passes through Lemma 2.7, which says the Euler vector field on the total space is pre-geodesic exactly when the 1-form ν vanishes; if that equivalence fails, the precise boundary between the two geometries shifts.

Editorial extensions

If this is right

Homogeneous Kähler structures are classified by base data on M together with a line bundle, not by auxiliary data living on the total space.
Sasakian structures on oriented bases are exactly the special case where the Euler vector field is pre-geodesic; non-trivial line bundles absorb non-coorientable contact structures.
The generalization beyond Sasakian is local: examples with trivial line bundle and ν ≠ 0 show the same geometry is already new in open subsets.
For the invariant homogeneity condition, co-Kähler structures are the special case with flat connection and constant u, and examples show genuine local generalizations exist.
Only three inequivalent homogeneity conditions exist for almost Hermitian structures on a homogeneous manifold, so the homogeneous and invariant cases exhaust the meaningful options (the third is a twisted variant treated in the appendix).

Reading between the lines

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

The dataset description suggests that deforming a Sasakian structure can be reformulated as solving equations in (φ, g_M, ν, ϕ, ξ, η) over M, where ν is a variable measuring the deviation from Sasakianity rather than a fixed background.
Because Lemma 2.7 ties ν = 0 to integrability of the orthogonal distribution, ν can be interpreted as the geometric obstruction to that distribution being integrable; this may connect to transverse-structure invariants in odd-dimensional geometry.
The invariant case with constant u but non-flat connection (Example 3.16) points to a class of 'transverse co-Kähler' objects whose curvature ρ encodes the non-integrability; classifying such objects could extend cosymplectic geometry in a new direction.
The exhaustive case analysis implies that within this principal-bundle framework there are essentially only the Sasakian and co-Kähler branches, so other odd-dimensional Kähler-like geometries would need different homogeneity actions or different structure groups.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: this note does what the title says. It identifies exactly where homogeneous Kähler structures are more general than Sasakian ones—the 1-form ν and the orientability of the line bundle—reformulates the dictionary in line-bundle language, and extends the same analysis to the invariant/co-Kähler case. On my reading the central claims are sound, but the proof has rough edges that a referee should have caught before publication.

The genuinely new material is the exhaustive case analysis in §1.3, the ACM dataset formulations (Definitions 2.5, 2.17, 3.12), the invariant-case theorems (3.4, 3.7, 3.11, 3.13), and the precise statement that the Sasakian gap is exactly ν plus non-trivial line bundles. The paper is honest about its overlap with [8] and its reliance on the authors' own homogenization framework; those are not flaws. The structural picture is coherent, and the computations that are shown hang together.

The soft spots are mostly presentation-level, with one proof gap. The integrability conditions (2.11)–(2.14) are load-bearing for Theorem 2.16, and they appear after a 'direct computation' with no derivation. I did not find an error, but the referee should ask for the details. The proof of Lemma 2.7 has a sign error: (2.9) shows +1/2 L_U eu, whereas the computation gives -1/2. The conclusion survives because only the zero locus matters, but the displayed proof is wrong. There are also apparent arithmetic slips in Examples 2.20 and 3.15: in 2.20, φ_s should be 4/5 rather than 4/3; in 3.15, u_s has the wrong coefficient for x_2. The qualitative points—ν≠0 and non-integrable distributions—are unaffected. Appendix A states Theorem A.1 without proof; that is disclosed, but for a paper whose whole method is explicit correspondences, an unproved analogue is unsatisfying.

Who is this for? People working on the contact/Sasakian side of the symplectic-to-contact dictionary. They will get a clear, mostly reliable map of the generalization. It deserves a serious referee; with the computation displayed and the constants fixed, it would be a publishable note. I would send it to review rather than desk reject, and I would ask for minor revision.

Referee Report

4 major / 3 minor

Summary. The paper studies homogeneous Kähler structures on principal R^x-bundles eL associated to a line bundle L, in the sense of [8]. It gives an explicit dictionary between homogeneous almost Hermitian structures on eL and ACM datasets (φ, g_M, ν, ϕ, ξ, η, α, β) on the base, and then translates the integrability conditions: Theorem 2.16 states a one-to-one correspondence between homogeneous Kähler structures and ACM datasets satisfying (2.11) and (2.19). For oriented line bundles with pre-geodesic Euler field, the correspondence reduces to ordinary Sasakian structures, so the extra generality beyond Sasakian geometry is captured by the 1-form ν and by non-trivial line bundles. Section 3 develops the parallel invariant case, with ACM datasets of type I and a corresponding characterization of invariant Kähler structures; when the Euler field is parallel and the connection is flat, these reduce to co-Kähler structures. The paper also classifies the possible homogeneity conditions and provides local examples with trivial line bundle but non-Sasakian/non-co-Kähler behavior.

Significance. If correct, the main result gives a precise and useful answer to the question left open in [8]: exactly how much more general homogeneous Kähler structures are than Sasakian structures. The line-bundle formulation is natural and the correspondence is explicit enough to be applied. The classification of homogeneity cases in Section 1.3 and the invariant-case parallel are valuable additions. The paper is also honest about its overlap with [8] and transparently builds on the authors' earlier dictionary. The examples, once corrected, support the claim that the generalized structures already occur even locally and with trivial line bundle.

major comments (4)

[§2.2, Theorem 2.10] The decomposition of the Nijenhuis torsion N_K into the four conditions (2.11)–(2.14) is the central algebraic input for Theorem 2.16, but it is introduced as 'a direct computation' with no derivation. The formulas are plausible and I found the ν=0 case internally consistent, but because the exact signs and the presence of dν terms are load-bearing for the claimed Sasakian gap, the computation should be made auditable. Please include the calculation at least for the ξ-component of N_K(∇X,∇Y) and the full component of N_K(I,∇X), or give an appendix with the complete derivation.
[§2.1, Lemma 2.7, Eq. (2.9)] The displayed identity (2.9) has a sign error. From L_E eG = eG one obtains eG(U, ∇_E E) = -eG(∇_U E, E) = -1/2 L_U eG(E,E), not +1/2. The zero-locus argument is unaffected, so the lemma's statement survives, but the proof as printed is incorrect. Please correct the sign and show the two-line computation.
[§2.5, Example 2.20 and Eq. (2.24)] In the deformed example E -> E + U, the norm squared of E+U on S^3 is 5/4, so φ_s = 4/5, not 4/3. Consequently the printed ν_s is also off by a factor; the correct value is ν_s = -(4/5) i*_{S^3} U^♭ (equivalently (4/5) i*_{S^3}(y_1 dx_1 - x_1 dy_1 - y_2 dx_2 + x_2 dy_2)). The qualitative conclusion ν_s ≠ 0 still holds. Separately, Eq. (2.24) contains an unexplained minus sign: from (2.7), β = φ^{-1} η, so one expects β_s = φ_s^{-1} η_s, not -φ_s^{-1} η_s.
[§3.5, Example 3.15] For E -> E + U with the same U as in Example 2.20, the squared norm of E+U on the hypersurface Σ is 1 - 2x_2 + x_1^2 + x_2^2 + y_1^2, not 1 - x_2 + x_1^2 + x_2^2 + y_1^2. The example still works, but the stated expression should be corrected.

minor comments (3)

[§3.2, Lemma 3.9] In the proof, the second occurrence of 'Condition (1) in the statement holds' should be 'Condition (2)' or 'condition (2) holds'. As written, the converse direction is mislabelled.
[§3.5] Minor wording: 'we provide two example' should be 'two examples'.
[§2.5] The notation φ_s and eφ is a little dense; a short sentence recalling that eφ = eG(E,E)^{-1} would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central correspondences are derived from definitions, and the authors' own prior dictionary is used only as a translation tool, not as the target result.

full rationale

Walking the derivation chain, the paper's main results (Theorems 2.16 and 3.11) are obtained by combining independently derived classification statements: Theorem 2.3 establishes the bijection between homogeneous almost Hermitian structures and ACM datasets of type H; Theorem 2.10 decomposes the vanishing Nijenhuis torsion into (2.11)-(2.14) by direct computation; Theorem 2.13 translates dΩ=0 into (2.19); Lemma 2.7 and Corollaries 2.12/2.15/2.18 connect the ν=0 case to pre-geodesic Euler fields and standard Sasakian structures. The invariant case parallels this with Theorems 3.1, 3.4, 3.7 and Corollaries 3.6/3.10/3.13. None of these steps assumes Theorem 2.16 or Corollary 2.18 as an input. The paper does rely on the authors' own prior homogenization dictionary [11,15,19,20], including the G ↔ (∇,φ,g_M,ν) equivalence from [15, Section 6.3]. However, these are parameter-free translation lemmas whose statements do not include the Kähler/Sasakian or co-Kähler classifications; the paper also states the relevant formulas explicitly. Under the stated rules, such self-citations are real evidence rather than circularity. The skeptic's concerns—the unshown 'direct computation' in Theorem 2.10 and the apparent sign issue in Eq. (2.9) of Lemma 2.7—are correctness/verification issues, not circularity: an algebraic error would invalidate the theorem, not make the conclusion equivalent to its inputs. No fitted parameters are renamed as predictions, no uniqueness theorem is imported from the authors, and no known result is simply relabeled. Therefore the paper is not circular; the main risk is computational verification, not circular reasoning.

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

No numbers are fitted to data anywhere in this paper; the α in |L|^α is a classification label, and quantities like φ_s and u_s are computed, not fitted. The axioms listed are the background framework (homogenization dictionary, Atiyah algebroid calculus, connected base) and two structural hypotheses specific to how the paper draws its boundary between homogeneous Kähler and Sasakian structures. No new entities (particles, forces, dimensions) are posited; the datasets introduced are bookkeeping definitions.

assumptions (4)

standard math Homogenization dictionary: equivalence between line bundles L→M and principal R^×-bundles, and identification of sections of f(L)⊗T^{p,q}DL with f-homogeneous tensors on L̃ (§1.1-1.2, from [11], [15])
Background framework cited to the authors' and collaborators' prior work; the entire paper is formulated inside it, but the target results are not assumed by it.
domain assumption Base manifold M is connected (stated at end of Introduction)
Global identifications in the correspondence theorems implicitly use connectivity of M.
domain assumption The f-homogeneity families f(r)=sign(r)|r|^α and f(r)=|r|^α exhaust the possible homogeneity weights, so CASEs A, B, C, D and their reductions give the three meaningful regimes (§1.3)
Follows from K^2=-1 and positive-definiteness; the reduction of α-cases via L′ = sign(L)⊗|L|^α is a normalization choice that the rest of the paper depends on.
ad hoc to paper Local reconstruction: any Kähler manifold with vector field E satisfying L_E G = G, L_E K = 0 and a transverse hypersurface Σ can be locally identified with a homogeneous manifold with Euler field E (§2.5, after (2.25))
This turns the deformed-vector-field examples into examples of homogeneous Kähler structures; plausible but not proven in the note.

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Cite this review

Pith. "Pith review of On Homogeneous K\"ahler Manifolds." pith.science (2026). https://pith.science/paper/EACPOB2E

@misc{pith2026260803785,
  author       = {Pith},
  title        = {Pith review of: On Homogeneous K\"ahler Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EACPOB2E}},
  note         = {Machine review of arXiv:2608.03785}
}

abstract

The cone $M \times \mathbb{R}_+$ over a Sasakian manifold $M$ is equipped with a canonical K\"ahler structure with specific homogeneity properties with respect to the $\mathbb{R}_+$ coordinate. This K\"ahler structure completely encodes the underlying Sasakian structure. Recently, Grabowski, Grabowska and Mohseni provided a broader conceptual framework for this phenomenon via homogeneous K\"ahler structures, i.e. K\"ahler structures on a principal $\mathbb{R}^\times$-bundle $P$ satisfying similar homogeneity properties. This approach successfully extends Sasakian geometry from cooriented contact manifolds (where $P$ is a trivial principal bundle) to non-necessarily coorientable contact structures (where $P$ is non-necessarily trivial). Homogeneous K\"ahler structures are genuinely more general than Sasakian structures and this note precisely characterizes the extent of this generalization. This is achieved through a detailed analysis of all the involved compatibilities in terms of the line bundle tautologically associated to $P$. We also show that modifying the homogeneity condition on the K\"ahler structure allows this framework to encompass co-K\"ahler structures and a natural generalization of those as well.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [8]

Grabowska K., Grabowski J., and Mohseni R., Sasaki structures on general contact manifolds, J. Sympl. Geom.(2025), accepted for publication; e-print:arXiv:2412.16697 [math.DG]

work page Pith review arXiv 2025

[2] [1]

Blair D.E., Riemannian geometry of contact and symplectic manifolds, Birkhäuser, Boston, 2002

work page 2002

[3] [2]

Contact Dual Pairs

Blaga A., Tortorella A. G., and Vizman C., Contact dual pairs,Int. Math. Res. Not. IMRN2020(2020), 8818–8877.arXiv:1903.05250 [math.DG]

work page Pith review arXiv 2020

[4] [3]

Higher omni-Lie algebroids

Bi Y., Vitagliano L., and Zhang T., Higher Omni-Lie Algebroids,J. Lie Theory29(2019), 881–899; e-print:arXiv:1812.09496 [math.DG]

work page Pith review arXiv 2019

[5] [4]

Boyer C.P., and Galicki K., Sasakian geometry,Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008

work page 2008

[6] [5]

Cappelletti-Montano B., De Nicola A., and Yudin I., A survey on cosymplectic geometry, Rev. Math. Phys.25(2013), 1343002 (55 pages); e-print:arXiv:1305.3704 [math.DG]. HOMOGENEOUS KÄHLER MANIFOLDS 31

work page Pith review arXiv 2013

[7] [6]

Grabowska K., Grabowski J., Contact Hamiltonian mechanics: the Tulczyjew triple,Adv. Theor. Math. Phys.28(2024), 599–654.arXiv:2209.03154 [math.SG]

work page Pith review arXiv 2024

[8] [7]

Grabowska K., and Grabowski J., Reductions: precontact versus presymplectic,Ann. Mat. Pura Appl.202 (2023), 2803–2839; e-print:arXiv:2211.16792 [math.SG]

work page Pith review arXiv 2023