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Biholomorphism type of left-invariant complex structures on nilpotent Lie groups

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Pith's one-line read A simply connected nilpotent Lie group of dimension 2n with a left-invariant complex structure is biholomorphic to C^n.

desk verdict This note proves Hasegawa's conjecture by reducing left-invariant complex structures on simply connected nilpotent groups to the Lie algebra level and constructing explicit biholomorphisms to C^n. read the letter →

arxiv 2606.31448 v1 pith:TAFM3ZPR submitted 2026-06-30 math.DG math.AG

Keizo Hasegawa , Sönke Rollenske , Lorenzo Sillari , Adriano Tomassini , Konstantin Wehler This is my paper · ORCID
classification math.DGmath.AG
keywords nilpotentLiegroupsleft-invariantcomplexstructuresbiholomorphismsHasegawaconjecturemanifoldson
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

The paper proves Hasegawa's conjecture that any simply connected nilpotent Lie group of even dimension carrying a left-invariant complex structure is biholomorphic to complex Euclidean space of half the dimension. This holds regardless of whether the underlying Lie group is abelian or non-abelian. A sympathetic reader would care because the result fixes the biholomorphism type completely, showing that left-invariance overrides any potential complexity in the group law when viewed through the complex structure. The proof applies exactly in the setting of left-invariant structures on simply connected nilpotent groups.

What carries the argument

Left-invariant complex structure on a simply connected nilpotent Lie group, which forces the complex manifold to be standard Euclidean space.

What would settle it

An explicit simply connected nilpotent Lie group of dimension 4 with a left-invariant complex structure whose underlying manifold fails to be biholomorphic to C^2 would falsify the claim.

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Extended reading notes

Core claim

In this note we prove a conjecture by Hasegawa stating that a simply connected, nilpotent Lie group of dimension 2n endowed with a left-invariant complex structure is biholomorphic to C^n.

Load-bearing premise

The complex structure must be left-invariant and the Lie group must be simply connected.

Editorial extensions

If this is right

  • All such complex manifolds are holomorphically equivalent to one another.
  • The biholomorphism type does not depend on the specific choice of left-invariant complex structure.
  • The result classifies the complex structure up to biholomorphism for the entire class of simply connected nilpotent groups.

Reading between the lines

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

  • The result suggests left-invariance is rigid enough to eliminate non-standard complex structures even when the group law is non-commutative.
  • Similar conclusions might hold if the left-invariance assumption is weakened to other invariance conditions on nilpotent groups.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

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Summary. The manuscript proves Hasegawa's conjecture: any simply connected nilpotent real Lie group of dimension 2n carrying a left-invariant integrable complex structure is biholomorphic to ℝ^{2n} ≅ ℂ^n. The argument reduces the problem to the Lie algebra via left-invariance, verifies that the given data imply vanishing of the Nijenhuis tensor, and constructs global holomorphic coordinates realizing the biholomorphism.

Significance. If correct, the result supplies a complete resolution of a known conjecture in complex geometry and the theory of nilpotent Lie groups. The proof is direct, reduces cleanly to the Lie-algebra level, and contains no free parameters or ad-hoc constructions; these features constitute a genuine strength of the manuscript.

Simulated Author's Rebuttal

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We thank the referee for their positive report, which accurately summarizes the main result and its significance. We are pleased that the referee recommends acceptance.

Circularity Check

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No significant circularity; direct proof of external conjecture

full rationale

The manuscript states and proves Hasegawa's conjecture under the given hypotheses (simply connected nilpotent real Lie group of dimension 2n with left-invariant integrable complex structure). The derivation reduces the problem to the Lie algebra via left-invariance, verifies the Nijenhuis tensor condition from the given data, and constructs global holomorphic coordinates realizing the biholomorphism to C^n. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the conjecture statement itself is the target being proved rather than an unverified premise imported from overlapping prior work. The argument is self-contained against the stated assumptions.

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

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

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Pith. "Pith review of Biholomorphism type of left-invariant complex structures on nilpotent Lie groups." pith.science (2026). https://pith.science/paper/TAFM3ZPR

@misc{pith2026260631448,
  author       = {Pith},
  title        = {Pith review of: Biholomorphism type of left-invariant complex structures on nilpotent Lie groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAFM3ZPR}},
  note         = {Machine review of arXiv:2606.31448}
}
abstract

In this note we prove a conjecture by Hasegawa stating that a simply connected, nilpotent Lie group of dimension $2n$ endowed with a left-invariant complex structure is biholomorphic to $\mathbb{C}^n$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Invariant forms compute the Dolbeault cohomology of complex nilmanifolds

    math.DG 2026-07 accept novelty 8.0 of 10

    On every compact nilmanifold with left-invariant complex structure, invariant forms induce an isomorphism on Dolbeault cohomology in all bidegrees.

Reference graph

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