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arxiv: 2606.31448 · v1 · pith:TAFM3ZPRnew · submitted 2026-06-30 · 🧮 math.DG · math.AG

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

Keizo Hasegawa , S\"onke Rollenske , Lorenzo Sillari , Adriano Tomassini , Konstantin Wehler This is my paper

Pith reviewed 2026-07-01 04:12 UTC · model grok-4.3

classification 🧮 math.DG math.AG
keywords nilpotent Lie groupsleft-invariant complex structuresbiholomorphismsHasegawa conjecturecomplex manifolds on Lie groups
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The pith

A simply connected nilpotent Lie group of dimension 2n with a left-invariant complex structure is biholomorphic to C^n.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

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.

read the original 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 $\mathbb{C}^n$.

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 / 0 minor

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

0 responses · 0 unresolved

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

0 steps flagged

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.

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 stated or can be extracted.

pith-pipeline@v0.9.1-grok · 5565 in / 1081 out tokens · 67760 ms · 2026-07-01T04:12:57.515032+00:00 · methodology

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Reference graph

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