Balanced and pluriclosed metrics on real semisimple Lie groups

Joseph Kwong

arxiv: 2604.25239 · v1 · submitted 2026-04-28 · 🧮 math.DG

Balanced and pluriclosed metrics on real semisimple Lie groups

Joseph Kwong This is my paper

Pith reviewed 2026-05-07 14:27 UTC · model grok-4.3

classification 🧮 math.DG

keywords balanced metricspluriclosed metricsVogan diagramsreal semisimple Lie groupsregular complex structurescompact quotientsnon-Kähler geometry

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The pith

Compact quotients of real semisimple Lie groups with regular complex structures admit balanced metrics or pluriclosed metrics according to their Vogan diagrams, but never both.

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

The paper characterises when balanced metrics and when pluriclosed metrics exist on these compact quotients by reading off conditions from the Vogan diagrams of the regular complex structures. It shows that the two metric types are mutually exclusive on any given such manifold. A reader would care because the result settles the possible coexistence of two distinct non-Kähler metric conditions on a large family of homogeneous complex manifolds. The work also supplies a corrected classification of the regular complex structures themselves on real semisimple Lie groups.

Core claim

The existence of balanced metrics and of pluriclosed metrics on compact quotients of real semisimple Lie groups equipped with regular complex structures is completely determined by the Vogan diagrams of those structures; consequently no such manifold admits both metric types at once. This characterisation is obtained after revisiting and correcting the prior classification of the regular complex structures.

What carries the argument

Vogan diagrams of regular complex structures, which directly encode the algebraic conditions for the existence of each metric.

If this is right

Every such manifold admits at most one of the two metric types.
The Vogan diagram determines which metric, if any, can exist.
The corrected classification of regular complex structures is required for the metric statements to hold.
The mutual exclusion is a direct corollary of the diagram-based conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same diagram method might be used to test metric existence on non-compact quotients or on other homogeneous spaces whose complex structures admit Vogan-type descriptions.
If the diagrams capture the only obstructions, then one could search for examples where neither metric exists by looking for diagrams that satisfy neither set of conditions.

Load-bearing premise

The corrected classification of regular complex structures by Vogan diagrams is complete and those diagrams translate without further hidden constraints into the existence or non-existence of balanced and pluriclosed metrics on the quotients.

What would settle it

An explicit compact quotient whose Vogan diagram predicts the absence of a balanced metric, yet a balanced metric is constructed on it, or the symmetric case for pluriclosed metrics.

read the original abstract

We characterise the existence of balanced and pluriclosed metrics on compact quotients of real semisimple Lie groups equipped with regular complex structures, in terms of Vogan diagrams. Consequently, such complex manifolds cannot simultaneously admit a balanced metric and a pluriclosed metric. Along the way, we revisit and correct the classification of regular complex structures on real semisimple Lie groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a Vogan-diagram test for when balanced or pluriclosed metrics exist on these homogeneous complex manifolds and shows they cannot have both, while correcting an earlier classification.

read the letter

The main contribution is the explicit combinatorial criterion, read off Vogan diagrams, for existence of balanced metrics and of pluriclosed metrics on compact quotients of real semisimple Lie groups carrying regular complex structures. The paper also records that no such manifold admits both types at once and supplies a corrected list of the regular complex structures themselves. This is a concrete step beyond general existence statements because the diagrams make the conditions checkable for specific groups and diagrams. The mutual-exclusion claim follows directly once the two separate criteria are in hand. The correction to the classification is presented as part of the work and uses the same diagram language, so it fits the overall approach. The soft spots are limited. The abstract states the characterization cleanly, but the translation from diagram data to metric existence still needs the proofs to confirm there are no hidden constraints coming from the choice of lattice or from the Hermitian condition itself. The correction to prior work on regular structures is useful only if it is accurate, and that part will require direct comparison with the papers being corrected. This paper is aimed at people already working in non-Kähler Hermitian geometry or homogeneous complex manifolds who know Vogan diagrams. A reader in that niche will be able to apply the criterion to examples without much extra work. It deserves a serious referee because the claims are specific, the method is standard in the area, and the results are falsifiable by checking the diagrams on known groups.

Referee Report

2 major / 2 minor

Summary. The manuscript characterizes the existence of balanced and pluriclosed metrics on compact quotients of real semisimple Lie groups equipped with regular complex structures, expressed in terms of Vogan diagrams. It derives the consequence that no such manifold can admit both a balanced metric and a pluriclosed metric simultaneously. As part of the work, the authors revisit and correct the classification of regular complex structures on real semisimple Lie groups.

Significance. If the characterization and mutual-exclusion result hold, the paper supplies a combinatorial criterion (via Vogan diagrams) for the existence of these special Hermitian metrics on a class of homogeneous complex manifolds. This would be a concrete advance in the study of non-Kähler geometry, providing both existence conditions and obstructions. The correction to the prior classification of regular complex structures is a useful secondary contribution if it is shown to be complete and accurate.

major comments (2)

[classification correction section] The central characterization and the mutual-exclusion claim rest on the corrected list of regular complex structures being complete and on a direct translation from Vogan-diagram data to metric existence conditions. The manuscript should make explicit, in the section presenting the corrected classification, which diagrams are added or removed relative to the literature and why each change preserves the regularity condition without introducing hidden constraints on the quotients.
[mutual-exclusion argument] The proof that balanced and pluriclosed metrics are mutually exclusive appears to follow from the characterization; however, it must be verified that the argument does not rely on any post-hoc choice of lattice or quotient that could allow both metrics in special cases. A concrete check against the Vogan diagrams that permit one metric but not the other would strengthen the claim.

minor comments (2)

[Introduction] The introduction would benefit from a brief reminder of the definitions of balanced (d(ω^{n-1})=0) and pluriclosed (∂∂̄ω=0) metrics in the Hermitian setting, together with a reference to the standard literature on these notions.
[throughout] Notation for the Vogan diagrams and the associated root systems should be standardized throughout; any deviation from the conventions in the cited Lie-theory references should be noted explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

Referee: [classification correction section] The central characterization and the mutual-exclusion claim rest on the corrected list of regular complex structures being complete and on a direct translation from Vogan-diagram data to metric existence conditions. The manuscript should make explicit, in the section presenting the corrected classification, which diagrams are added or removed relative to the literature and why each change preserves the regularity condition without introducing hidden constraints on the quotients.

Authors: We agree that the presentation of the corrected classification would benefit from greater explicitness. In the revised manuscript, we will add a dedicated subsection (or table) within the classification section that directly compares our list of Vogan diagrams to those appearing in the prior literature. For every diagram added or removed, we will state the precise reason for the modification, verify that the regularity condition continues to hold, and confirm that the change introduces no additional constraints on the existence or properties of compact quotients. This will make the completeness of the list and the translation to metric conditions fully transparent. revision: yes
Referee: [mutual-exclusion argument] The proof that balanced and pluriclosed metrics are mutually exclusive appears to follow from the characterization; however, it must be verified that the argument does not rely on any post-hoc choice of lattice or quotient that could allow both metrics in special cases. A concrete check against the Vogan diagrams that permit one metric but not the other would strengthen the claim.

Authors: The mutual-exclusion result follows directly from the combinatorial characterization: each Vogan diagram determines at most one of the two metric types (balanced or pluriclosed) according to explicit conditions on the diagram data. These conditions are intrinsic to the real semisimple Lie algebra and the regular complex structure; they are independent of any particular lattice or quotient. Consequently, if a diagram permits only balanced metrics, no choice of compact quotient can produce a pluriclosed metric (and conversely). To strengthen the exposition as requested, we will add an explicit verification subsection that enumerates the diagrams permitting balanced metrics but not pluriclosed metrics (and vice versa) and confirms that none permit both. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper characterizes existence of balanced and pluriclosed metrics on compact quotients via Vogan diagrams applied to a corrected list of regular complex structures on real semisimple Lie groups. Vogan diagrams constitute an external, standard classification tool from Lie theory (not defined or fitted inside the paper). The correction to the prior classification is presented as an independent contribution rather than a self-referential step. No load-bearing claim reduces by construction to an input, a fitted parameter renamed as prediction, or a self-citation chain. The incompatibility result follows directly from the two separate existence conditions. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5340 in / 1134 out tokens · 52168 ms · 2026-05-07T14:27:02.103253+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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[1] [1]

[Bor63] Armand Borel

Reprint of the 1987 edition. [Bor63] Armand Borel. Compact Clifford-Klein forms of symmetric spaces.Topology, 2:111–122,

work page 1987

[2] [2]

[CH04] Meng-Kiat Chuah and Chu-Chin Hu

Translated from the 1968 French original by Andrew Pressley. [CH04] Meng-Kiat Chuah and Chu-Chin Hu. Equivalence classes of Vogan diagrams.J. Algebra, 279(1):22–37,

work page 1968

[3] [3]

Pluriclosed metrics on compact semisimple lie groups

[LM25] Jorge Lauret and Facundo Montedoro. Pluriclosed metrics on compact semisimple lie groups. arXiv preprint arXiv:2506.21725,

work page arXiv

[4] [4]

[Mic82] M

Corrected reprint of the 1969 original. [Mic82] M. L. Michelsohn. On the existence of special metrics in complex geometry.Acta Math., 149(3- 4):261–295,

work page 1969

[5] [5]

Complex parallisable manifolds.Proc

[Wan54] Hsien-Chung Wang. Complex parallisable manifolds.Proc. Amer. Math. Soc., 5:771–776, 1954

work page 1954