$p$-K\"ahler structures on nilmanifolds and holomorphically parallelizable manifolds

Asia Mainenti; Ettore Lo Giudice

REVIEW 4 major objections 7 minor 34 references

$p$-K\"ahler structures on nilmanifolds and holomorphically parallelizable manifolds

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Pith's one-line read This paper proves the Alessandrini–Bassanelli conjecture on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds: any p-Kähler structure on these manifolds yields q-Kähler structures for all q ≥

desk verdict Proves the Alessandrini–Bassanelli conjecture on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds; the core argument is sound but one load-bearing step in the nilmanifold induction is unstated and must be fixed. read the letter →

arxiv 2607.29506 v1 pith:Y3OCTDL4 submitted 2026-07-31 math.DG

Ettore Lo Giudice , Asia Mainenti This is my paper · ORCID

classification math.DG MSC 53C5522E2522E46

keywords p-KählerstructuresAlessandrini-BassanelliconjecturenilmanifoldsnilpotentcomplexholomorphicallyparallelizablemanifoldssolvmanifoldsbalancedmetricsreductiveLiegroups

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 the Alessandrini–Bassanelli conjecture for two large families of compact complex manifolds: nilmanifolds with nilpotent complex structures, and holomorphically parallelizable solvmanifolds. The conjecture says that a closed transverse $(p,p)$-form—a $p$-Kähler structure—forces closed transverse $(q,q)$-forms for every $q \ge p$; if true in general, every $p$-Kähler manifold would admit a balanced metric. The authors prove it by passing to Lie algebras, where a structural induction shows the existence of enough closed $(1,0)$-forms to raise the degree of the form. Consequences include: no such solvmanifold can be $p$-Kähler for $p \le n/2$, a classification at the threshold, and new $(n-2)$-Kähler examples on quotients of reductive complex Lie groups without rank-one simple factors.

What carries the argument

The central mechanism is symmetrization plus degree-raising by wedging with closed $(1,0)$-forms. Lemma 2.9 lets a $p$-Kähler structure on a compact quotient $\Gamma\backslash G$ descend to the Lie algebra $g$. Proposition 4.4 then splits a $p$-Kähler nilpotent Lie algebra with quasi-nilpotent complex structure into an $a$-extension of a $(p-1)$-Kähler Lie algebra, driving the induction. In the solvable holomorphically parallelizable case, Lemma 5.2 produces a basis with enough closed $(1,0)$-forms, and Lemma 3.5 converts a transverse $p$-form plus those closed forms into a transverse $(p+1)$-form. In the reductive case, Lemma 5.14 shows exact decomposable 2-forms vanish when all simple factors have rank at least 2.

What would settle it

Search for a nilmanifold with nilpotent complex structure in the smallest dimension where the induction has not been checked—starting with 6 real dimensions—that admits a closed transverse $(2,2)$-form but no closed transverse $(3,3)$-form. Such a pair would be a counterexample to the main theorem; equivalently, check whether every 6-dimensional nilpotent Lie algebra with nilpotent $J$ and a $p$-Kähler form satisfies the quasi-nilpotent $a$-extension reduction of Proposition 4.4.

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Extended reading notes

Core claim

The paper's central claim is that the Alessandrini–Bassanelli conjecture holds on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds: a $p$-Kähler structure on any such manifold forces $q$-Kähler structures for all $q \ge p$, hence a balanced metric. The proof reduces to Lie algebras via symmetrization. In the nilpotent case it uses the fact that a $p$-Kähler nilpotent Lie algebra with quasi-nilpotent complex structure is an $a$-extension of a $(p-1)$-Kähler one, allowing induction; in the solvable case, structure equations force enough closed $(1,0)$-forms. Consequences include a sharp no-go bound $p \le n/2$ on solvmanifolds and a characterization of reductive $q$

Load-bearing premise

The proof assumes that a $p$-Kähler nilpotent Lie algebra with a nilpotent complex structure can always be reduced to a $(p-1)$-Kähler one by dividing out a central complex line, and that the base cases of the induction are correct; if that reduction fails for some nilpotent complex structure, the nilmanifold half of the main theorem loses its support.

Editorial extensions

If this is right

On nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds, the Alessandrini–Bassanelli conjecture holds: any p-Kähler structure implies balanced metrics.
Non-abelian holomorphically parallelizable solvmanifolds admit no p-Kähler structures for p ≤ n/2, and the bound is sharp; the lowest admissible degree is floor(n/2)+1.
The paper classifies which holomorphically parallelizable solvmanifolds with non-trivial center realize the lowest possible p-Kähler degree: in odd dimensions they are the complex Heisenberg-type algebras, and in even dimensions a two-derivation family with explicit structure equations.
Products of balanced manifolds that are (m−2)- and (n−2)-Kähler are (m+n−2)-Kähler and balanced, without the extra positivity assumptions previously needed.
Compact quotients of reductive complex Lie groups admit (n−2)-Kähler structures if and only if no simple factor has rank 1; in the semisimple rank at least 2 case, the structure can be chosen with vanishing Bott-Chern class.

Reading between the lines

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

The algebraic mechanism suggests a testable generalization: if the a-extension induction works for any quasi-nilpotent complex structure, the conjecture would extend to larger classes of solvmanifolds than those explicitly covered.
The paper shows that squares of transverse (2,2)-forms are always transverse, but for p ≥ 3 there exist transverse (p,p)-forms whose squares are not transverse. Whether any such form can be closed on a compact manifold remains open, so powers of p-Kähler structures are not yet settled and could provide a subtle counterexample to the general conjecture.
The reductive characterization points to a root-system mechanism: the rank-1 obstruction appears tied to the existence of exact decomposable 2-forms, so one could compute the Bott-Chern cohomology of the new examples to see which balanced metrics arise and whether the vanishing class is a general feature.

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Editorial analysis

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

Desk Editor's Note

read the letter

Short version: this paper does what it claims. It proves the Alessandrini–Bassanelli conjecture on all nilmanifolds with nilpotent complex structures and all holomorphically parallelizable solvmanifolds, plus a clean classification and new examples. The main argument is a Lie-algebra induction, and the biggest issue is one unstated step in that induction. I'd send it to review; a good referee can sort the gap.

What's new: previously the conjecture was known only for a restrictive class of h.p. nilmanifolds and for examples. Here the symmetrization argument plus the a-extension reduction covers two broad families. The sharp bound p ≤ n/2 in the solvable case, the characterization of extremal cases, and the (n−2)-Kähler examples on reductive quotients with vanishing Bott–Chern class are solid additions. The product theorem in Section 4 is also a nice result, and Lemma 3.5 (wedging with a Kähler form preserves transversality) is simple and useful. The paper is organized, and the writing is mostly clear.

Soft spots, in order:

1. Theorem 4.10 is load-bearing. It applies Proposition 4.4, whose hypothesis is 'quasi-nilpotent complex structure.' The paper moves from 'nilpotent' to 'quasi-nilpotent' without comment. The implication is true — in a J-adapted basis, the dual vector to φ^n is central, so the center contains a J-invariant 2-plane — but the argument is absent. This is a missing lemma, not a fatal flaw. It must be stated and proved.

2. Lemma 5.2 has presentation bugs. The form β is described as d of a (p̃−1,0)-form but treated as a (p̃,0)-form; and it cites (5.3) before (5.3) is proven. This is probably a degree typo and an ordering issue, but in the technical heart of the solvable case it needs to be cleaned up. Same for the cross-references around (5.5).

3. Theorem 5.15's rank-1 factor argument is actually fine: applying Lemma 2.14 with the complement as the ideal reduces to sl2 admitting a Kähler metric, impossible. But the reduction could be written more explicitly; as written it strains the reader.

The references to the authors' own earlier work are appropriate; [14] and [22] cover base cases and the h.p. nilmanifold case. No circularity.

Bottom line: this is a substantial, believable piece of work. The missing quasi-nilpotent lemma and the presentation issues are fixable. Don't desk-reject. Send to a referee, and make sure the referee checks the induction in Theorem 4.10 carefully.

Referee Report

4 major / 7 minor

Summary. The paper studies the Alessandrini–Bassanelli conjecture, which predicts that a compact complex manifold admitting a p-Kähler structure admits q-Kähler structures for every q ≥ p. The authors prove the conjecture for nilmanifolds with nilpotent complex structures and for holomorphically parallelizable solvmanifolds; they also classify which such solvmanifolds admit p-Kähler structures for low p, and construct (n−2)-Kähler examples on quotients of reductive complex Lie groups. The proof strategy is Lie-algebraic: after a symmetrization step, the problem is reduced to invariant forms on Lie algebras, where nilpotent/quasi-nilpotent complex structures and structure equations are used to produce higher-degree transverse closed forms by wedging with closed (1,1)-terms.

Significance. If the main theorem is correct, it is a substantial advance on a longstanding conjecture: it gives the first evidence that arbitrary p-Kähler structures force the existence of balanced metrics in two broad classes of non-Kähler compact complex manifolds. The paper also contains useful algebraic tools (Lemma 3.5 for constructing transverse forms, Proposition 4.5 for dimensional reduction along central ideals) and explicit sharp examples. The main arguments are mostly self-contained and the algebraic reductions are clearly organized; the reliance on the authors' earlier results [14,22] is explicit and does not presuppose the conjecture. However, the proof of the solvmanifold part contains a bidegree error in Lemma 5.2 that is load-bearing, and the nilmanifold induction has an unproved quasi-nilpotency step; these must be repaired before the main claims are established.

major comments (4)

[Lemma 5.2] The form β := d(φ1···φ̂u···φ̂v···φ̃p+1) is a (p̃−1,0)-form, not a (p̃,0)-form: it is the exterior derivative of a wedge of p̃−1 one-forms. Therefore Remark 2.8 / Proposition 2.15, which only forbid nonzero exact decomposable holomorphic (p̃,0)-forms on a p-Kähler algebra, cannot be used to conclude β=0. This conclusion is the step that forces the structure equations (5.3); without it Theorem 5.4 has no basis for constructing the (p+1)-Kähler form. The proof also refers to (5.3) before it is established, but the degree mismatch is the substantive gap.
[Theorem 4.10] The induction step invokes Proposition 4.4, whose hypothesis requires J to be quasi-nilpotent. For a nilpotent complex structure in the sense of Definition 2.10 the implication is true: in a J-adapted basis satisfying (2.1), the dual basis element e_n is central because every dφ^j omits φ^n, and similarly ē_n is central, so their real J-span is a J-invariant central 2-plane. But this argument is not given. Without it, the reduction to a (p−1)-Kähler a-extension is not justified and the induction does not close.
[Theorem 5.15] In the rank-1 obstruction, Lemma 2.14 is applied with k the complement of sl2C. This requires p > dim_C k, which holds for n ≥ 4 but not for the case g = sl2C (n = 3). The conclusion there would be that sl2C is 1-Kähler, i.e. Kähler; the needed contradiction is standard but not supplied. A separate sentence covering n = 3 is necessary.
[Proposition 5.16] The form Ω is declared to be transverse without proof. A sum of squares of holomorphic (n−2,0)-forms need not be transverse: for example on C^4, ψ = φ1∧φ2 gives ψ∧ψ = 0 when evaluated with η = φ2∧φ3. One must show that the space used to define Ω is large enough that for every nonzero decomposable 2-form η some summand has nonzero wedge with η. In addition, the assertion that σ_j can be chosen of pure type (n−3,0) from ψ_j = dσ_j requires justification; exactness in the total complex does not by itself imply a representative of that bidegree.

minor comments (7)

[Lemma 5.2] The phrase 'because by (5.3)' appears before (5.3) has been derived; the argument should be reordered.
[Theorem 4.1] In equation (4.2), the second line should read Vol_M ∧ \tilde{Ω}_{n−2}, not Vol_M ∧ \tilde{Ω}_{m−2}; the same typo appears in the surrounding text.
[Proposition 4.5] The notation π* is ambiguous: it is not clear whether this is pullback, pushforward, or contraction followed by identification with the quotient. Please clarify the map being used.
[Section 5.1] The sentence after (5.1) is incomplete: 'extends to an isomorphism (5.1) Λ^{k,0}g*_C ≃ Λ^k g*' has no closing period or continuation.
[Example 2.16] There is a duplicated word: 'admitting admitting p-Kähler structures'.
[Theorem 5.4] The proof uses Lemma 5.2, which is stated only for 1 < p < n−1. The case p = 1 is trivial because Kähler implies q-Kähler for all q, but it should be mentioned explicitly.
[Proposition 5.16] Typo: 'grater' should be 'greater'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the p-to-(p+1) construction is Lemma 3.5, and the main reductions are independent published Lie-algebra lemmas; self-citations are real evidence, with only a small unstated quasi-nilpotent implication.

full rationale

The paper's central claims are proven by induction at Lie-algebra level, and no prediction is fitted to data or defined in terms of its conclusion. The key new tool Lemma 3.5 is proved in-paper: if Ω is transverse (p,p) and φ_1,...,φ_k are independent (1,0)-forms with k ≥ n−p, then Ω ∧ i Σ φ_j∧\bar φ_j is transverse (p+1,p+1). Theorem 4.10 and Theorem 5.4 then produce a (p+1)-form by wedging the given p-Kähler form with closed (1,0)-forms whose existence is obtained from Lemma 4.9 and Lemma 5.2; this is a genuine construction, not a re-labeling. The nilmanifold induction imports Proposition 4.4 from [14] (Fino–Mainenti, one of the authors) and low-dimensional base cases from [14, Prop. 3.5] and [22, Thm. 4.2] (Lo Giudice). These citations are load-bearing, but their statements are parameter-free, are not the Alessandrini–Bassanelli conjecture, and are published external results; under the review rules they count as independent support. The one real presentation gap is in Theorem 4.10: Proposition 4.4 assumes a quasi-nilpotent complex structure, while the theorem only states J is nilpotent. The manuscript does not spell out that every nilpotent J is quasi-nilpotent; the implication is true (in a J-adapted basis the dual vectors e_n and \bar e_n are central and span a J-invariant 2-plane), but it is an unstated lemma. This affects exposition, not circularity. The solvable half (Theorem 5.4) is self-contained via Lemma 5.2, and the reductive classification (Theorem 5.15) is proved inside the paper using Lemma 5.14. No equation reduces a claimed prediction to its input, so the principal results are not circular.

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

The paper introduces no free parameters and no new postulated entities. Its central claims rest on a chain of prior structural results about Lie algebras, compact quotients, and positivity of forms, several of which come from the authors' own earlier work. The load-bearing assumptions are standard in the field but are not re-derived here.

assumptions (6)

domain assumption Wang's theorem: compact holomorphically parallelizable manifolds are quotients of simply connected complex Lie groups by lattices.
Used at the start of Section 5 to reduce to biinvariant complex structures on Lie algebras; cited as [33].
domain assumption Symmetrization: existence of a p-Kähler structure on a compact quotient Γ\G implies existence on the Lie algebra g (Lemma 2.9).
Allows all subsequent work to be done at Lie-algebra level; cited as [12,14].
domain assumption Nilpotent complex structures admit J-adapted bases and satisfy the obstruction of Proposition 2.11.
Central to Theorem 4.10 and Lemma 4.9; cited as [8,30].
domain assumption Quasi-nilpotent reduction: a p-Kähler nilpotent Lie algebra with quasi-nilpotent complex structure is an a-extension of a (p−1)-Kähler nilpotent Lie algebra.
Used to reduce dimension in Theorem 4.10 and Proposition 5.8; cited as [14]. The paper does not explicitly prove that nilpotent complex structures are quasi-nilpotent.
standard math Standard root-system structure theory for semisimple Lie algebras, including Chevalley bases and root-string computations.
Used in Lemma 5.14 to prove that exact decomposable 2-forms on rank≥2 semisimple factors vanish; cited as [19].
standard math H^2 of a semisimple Lie algebra is zero and the Bott-Chern/∂∂-exactness facts used in Proposition 5.16.
Used to show the constructed (n−2)-Kähler form has vanishing Bott-Chern class; cited as [7] and [22].

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Pith. "Pith review of $p$-K\"ahler structures on nilmanifolds and holomorphically parallelizable manifolds." pith.science (2026). https://pith.science/paper/Y3OCTDL4

@misc{pith2026260729506,
  author       = {Pith},
  title        = {Pith review of: $p$-K\"ahler structures on nilmanifolds and holomorphically parallelizable manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3OCTDL4}},
  note         = {Machine review of arXiv:2607.29506}
}

abstract

We prove the Alessandrini-Bassanelli conjecture on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds. As a consequence, we classify holomorphically parallelizable solvmanifolds admitting $p$-K\"ahler structures, for lower values of $p$. We further provide new examples of $(n-2)$-K\"ahler manifolds, in the setting of compact holomorphically parallelizable manifolds with reductive universal cover.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [14]

A note on p-K¨ ahler structures on compact quotients of Lie groups

Anna Fino and Asia Mainenti. A note on p-K¨ ahler structures on compact quotients of Lie groups. Ann. Mat. Pura Appl. (4) , 203:2111–2124, 2024

2024

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p-K¨ ahler structures on compact complex manifolds

Ettore Lo Giudice. p-K¨ ahler structures on compact complex manifolds. Math. Z. , 312(2):Paper No. 66, 24, 2026

2026

[3] [1]

Closed transverse (p,p)-forms on compact complex manifolds

Lucia Alessandrini and Marco Andreatta. Closed transverse (p,p)-forms on compact complex manifolds. Compos. Math., 61:181–200, 1987

1987

[4] [2]

Compact p-K¨ ahler manifolds.Geom

Lucia Alessandrini and Giovanni Bassanelli. Compact p-K¨ ahler manifolds.Geom. Dedicata , 38(2):199–210, 1991

1991

[5] [3]

Positive ∂∂-closed currents and non-K¨ ahler geometry

Lucia Alessandrini and Giovanni Bassanelli. Positive ∂∂-closed currents and non-K¨ ahler geometry. J. Geom. Anal. , 2(4):291–316, 1992

1992

[6] [4]

Plurisigned Hermitian metrics

Daniele Angella, Vincent Guedj, and Hoang-Chinh Lu. Plurisigned Hermitian metrics. Trans. Am. Math. Soc. , 376(7):4631–4659, 2023. 23

2023

[7] [5]

Bott-Chern cohomology of solvmanifolds

Daniele Angella and Hisashi Kasuya. Bott-Chern cohomology of solvmanifolds. Ann. Global Anal. Geom., 52(4):363–411, 2017

2017

[8] [6]

Squares of positive (p, p)-forms

Zbigniew B locki and Szymon Pli´ s. Squares of positive (p, p)-forms. C. R. Math. Acad. Sci. Paris , 351(1-2):27–32, 2013

2013