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arxiv: 2604.04134 · v2 · pith:L2BDBZRJnew · submitted 2026-04-05 · 🧮 math.DG

On the Classification of Vaisman Manifolds with Vanishing First Basic Chern Class and Large First Betti Number

Lucas H. S. Gomes This is my paper

Pith reviewed 2026-05-22 11:25 UTC · model grok-4.3

classification 🧮 math.DG
keywords Vaisman manifoldbasic Chern classfirst Betti numberKodaira-Thurston manifoldlocally conformal KählerAlbanese mapcharacteristic foliationLCK rank
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The pith

Vaisman manifolds with large first Betti number and vanishing first basic Chern class are diffeomorphic to Kodaira-Thurston manifolds.

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

The paper classifies Vaisman manifolds under the twin conditions of large first Betti number and vanishing first basic Chern class. It proves that any such manifold must be diffeomorphic to a Kodaira-Thurston manifold. The complex structure is necessarily left-invariant, the characteristic foliation is regular, and the fibration is realized by the Albanese map. When the LCK rank equals one the Vaisman structure itself is left-invariant, and when all basic harmonic one-forms have constant length the complex structure is the standard one on the Kodaira-Thurston manifold.

Core claim

Every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold. Its complex structure is left-invariant, the characteristic foliation is regular, and the associated fibration is given by the Albanese map. Under the additional assumption that the LCK rank is 1, the Vaisman structure is also left-invariant. If all basic harmonic 1-forms have constant length, then the Vaisman manifold with large first Betti number is diffeomorphic to a Kodaira-Thurston manifold and its complex structure is the standard complex structure.

What carries the argument

The vanishing of the first basic Chern class together with the largeness condition on the first Betti number, which together force diffeomorphism to a Kodaira-Thurston manifold via the Albanese map and left-invariance of the complex structure.

If this is right

  • The complex structure is left-invariant.
  • The characteristic foliation is regular.
  • The associated fibration is realized by the Albanese map.
  • When the LCK rank equals one the Vaisman structure is itself left-invariant.
  • When all basic harmonic one-forms have constant length the complex structure is the standard one.

Where Pith is reading between the lines

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

  • The same topological constraints may produce similar rigidity for locally conformal Kähler structures that are not Vaisman.
  • Regularity of the foliation could simplify calculations of transverse invariants on these manifolds.
  • The link to the Albanese map suggests the classification may be rephrased in terms of the fundamental group.

Load-bearing premise

The manifold carries a Vaisman structure whose first basic Chern class vanishes and whose first Betti number meets the largeness threshold used in the classification.

What would settle it

A single Vaisman manifold with large first Betti number and vanishing first basic Chern class that is not diffeomorphic to any Kodaira-Thurston manifold would disprove the classification.

read the original abstract

We show that every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold. Furthermore, its complex structure is left-invariant, the characteristic foliation is regular, and the associated fibration is given by the Albanese map. Under the additional assumption that the LCK rank is $1$, the Vaisman structure is also left-invariant. We further prove that if all basic harmonic $1$-forms have constant length, then the Vaisman manifold with large first Betti number is diffeomorphic to a Kodaira-Thurston manifold and its complex structure is the standard complex structure. Finally, we discuss the relationship of this condition with transverse geometric formality in this setting.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves that every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold, with left-invariant complex structure, regular characteristic foliation, and fibration given by the Albanese map. Under the additional assumption of LCK rank 1, the Vaisman structure is left-invariant. It further shows that if all basic harmonic 1-forms have constant length, then the manifold is diffeomorphic to a Kodaira-Thurston manifold with the standard complex structure, and discusses the relation of this condition to transverse geometric formality.

Significance. If the central classification holds, the result provides a concrete diffeomorphism-type identification of Vaisman manifolds under natural topological and basic-cohomological hypotheses, linking them to the well-studied Kodaira-Thurston family. The explicit statements about left-invariance, regularity of the foliation, and the role of the Albanese map supply falsifiable geometric predictions that can be checked on known examples. The additional results on constant-length basic harmonic forms and transverse formality broaden the applicability within the theory of locally conformal Kähler structures.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction state the largeness condition on b1 and the vanishing of the first basic Chern class as hypotheses, but a brief reminder of the precise numerical threshold used for “large” (e.g., b1 ≥ 3 or b1 ≥ 4) would help readers locate the result within the existing literature on Vaisman manifolds.
  2. [Proof of the main theorem] When the paper invokes properties of the Albanese map and basic harmonic forms, a short sentence recalling the relevant Hodge-theoretic fact (e.g., that basic harmonic 1-forms are closed and co-closed with respect to the transverse metric) would improve readability for non-specialists.
  3. [Final section] The discussion of transverse geometric formality at the end would benefit from an explicit comparison with the known formality results for Kodaira-Thurston manifolds themselves.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, including the recommendation for minor revision. The summary accurately captures the main results on the diffeomorphism type, left-invariance under LCK rank 1, the role of the Albanese map, and the additional results on constant-length basic harmonic forms and transverse formality. Since the report lists no specific major comments, we have no individual points to address below.

Circularity Check

0 steps flagged

No significant circularity; classification uses standard Vaisman and Kodaira-Thurston properties

full rationale

The paper's main result is a classification theorem stating that Vaisman manifolds with large first Betti number and vanishing first basic Chern class are diffeomorphic to Kodaira-Thurston manifolds, with additional conclusions on left-invariance and the Albanese map under the given hypotheses. These hypotheses are stated explicitly as assumptions in the theorem and enter the proof as external conditions rather than being derived from or equivalent to the conclusions by construction. The derivation relies on established facts about Vaisman structures, basic cohomology, and known diffeomorphism types of Kodaira-Thurston manifolds, without self-definitional loops, fitted inputs relabeled as predictions, or load-bearing self-citations that reduce the central claim to unverified prior work by the same author. The additional results on constant-length basic harmonic forms and transverse formality are likewise presented as independent extensions under the same hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of Vaisman manifolds as LCK manifolds with parallel Lee form, properties of the basic Chern class in the transverse sense, and the largeness condition on the first Betti number; no free parameters or invented entities are evident from the abstract.

axioms (2)
  • domain assumption Vaisman manifold admits an LCK metric whose Lee form is parallel with respect to the Levi-Civita connection.
    This is the defining property invoked throughout the classification statements.
  • standard math The first basic Chern class is a well-defined transverse cohomology class that can vanish independently.
    Used as a hypothesis in the main theorem.

pith-pipeline@v0.9.0 · 5658 in / 1467 out tokens · 28651 ms · 2026-05-22T11:25:15.301578+00:00 · methodology

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Works this paper leans on

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