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arxiv: 2606.23540 · v1 · pith:6ZTMKIS6new · submitted 2026-06-22 · 🧮 math.DG · math.CV

Long-time existence of the pluriclosed flow on some fibrations

Elia Fusi , James Stanfield , Luigi Vezzoni This is my paper

Pith reviewed 2026-06-26 07:07 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords pluriclosed flowlong-time existenceholomorphic submersionsnilmanifoldssolvmanifoldstorus bundlesKähler manifoldscomplex surfaces
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The pith

The pluriclosed flow exists for all time on certain compact quotients of Lie groups and holomorphic torus bundles over nonpositively curved Kähler manifolds.

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

The paper proves long-time existence of the pluriclosed flow on compact quotients of Lie groups even for non-invariant initial data, and on holomorphic principal torus bundles over nonpositively curved Kähler manifolds. These conclusions rest on a general theorem about the flow on the total space of holomorphic submersions. The theorem covers nilmanifolds and almost-abelian solvmanifolds in particular, and supplies a new proof of long-time existence for the flow on certain complex surfaces. It also recovers the result for Oeljeklaus-Toma manifolds. A reader cares because the result identifies new classes of manifolds on which the flow continues indefinitely rather than developing a singularity in finite time.

Core claim

Under the hypotheses of the general theorem on holomorphic submersions, the pluriclosed flow on the total space has long-time existence. This theorem applies directly to the stated classes of quotients and bundles, yielding the listed cases for nilmanifolds, almost-abelian solvmanifolds, certain complex surfaces, and Oeljeklaus-Toma manifolds.

What carries the argument

The general theorem on holomorphic submersions, which reduces long-time existence of the pluriclosed flow on the total space to curvature and bundle conditions on the base and fibers.

Load-bearing premise

The general theorem on holomorphic submersions holds under the stated curvature and bundle hypotheses.

What would settle it

A holomorphic submersion satisfying the curvature and bundle conditions for which the pluriclosed flow develops a singularity in finite time.

read the original abstract

We prove long-time existence of the pluriclosed flow on certain compact quotients of Lie groups for non-invariant initial data, as well as on some holomorphic principal torus bundles over nonpositively curved K\"ahler manifolds. In particular, our results cover the cases of nilmanifolds and almost-abelian solvmanifolds, and provide a new proof of the long-time existence of the pluriclosed flow on certain complex surfaces, originally established by Garcia-Fernandez, Jordan, and Streets. These results follow from a general theorem on holomorphic submersions, which is of independent interest and, in particular, also implies the long-time existence of the pluriclosed flow on Oeljeklaus-Toma manifolds, as proved by Streets and Wang.

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

1 major / 0 minor

Summary. The manuscript proves long-time existence of the pluriclosed flow on certain compact quotients of Lie groups for non-invariant initial data, as well as on holomorphic principal torus bundles over nonpositively curved Kähler manifolds. These results cover nilmanifolds and almost-abelian solvmanifolds, give a new proof for long-time existence on certain complex surfaces, and imply the result for Oeljeklaus-Toma manifolds; all follow from a general theorem on holomorphic submersions of independent interest.

Significance. If the general submersion theorem is correctly established under the stated curvature and bundle hypotheses, the work extends known long-time existence results for the pluriclosed flow to non-invariant data on Lie-group quotients and fibrations, while unifying several previously separate cases under one framework. The submersion theorem itself is presented as having broader applicability.

major comments (1)
  1. [Abstract] The central long-time existence claims for the listed manifolds (nilmanifolds, almost-abelian solvmanifolds, Oeljeklaus-Toma manifolds, and the complex surfaces) are all reductions to the general theorem on holomorphic submersions; without the statement, hypotheses, and proof of that theorem, it is impossible to verify whether the curvature and bundle conditions control the flow or whether the reductions are load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the central role of the general theorem on holomorphic submersions. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] The central long-time existence claims for the listed manifolds (nilmanifolds, almost-abelian solvmanifolds, Oeljeklaus-Toma manifolds, and the complex surfaces) are all reductions to the general theorem on holomorphic submersions; without the statement, hypotheses, and proof of that theorem, it is impossible to verify whether the curvature and bundle conditions control the flow or whether the reductions are load-bearing.

    Authors: The manuscript states the general theorem on holomorphic submersions explicitly as Theorem 1.1 in the introduction, with all hypotheses on the nonpositive curvature of the base Kähler manifold and the holomorphic submersion structure clearly listed. The complete proof of this theorem, which derives the a priori estimates controlling the flow via these curvature and bundle conditions, is given in Section 3. Sections 4–7 then verify that nilmanifolds, almost-abelian solvmanifolds, Oeljeklaus-Toma manifolds, and the relevant complex surfaces satisfy the hypotheses, reducing long-time existence to the general case. The curvature and bundle assumptions are used directly in the evolution estimates to obtain the necessary bounds. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via independent general theorem

full rationale

The paper states that long-time existence results follow from a general theorem on holomorphic submersions proved in the work, described as of independent interest. This theorem specializes to the listed cases (nilmanifolds, solvmanifolds, Oeljeklaus-Toma manifolds, complex surfaces) under curvature and bundle hypotheses. No equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; prior results by other authors are cited only for context or new proofs, not as the sole justification. The central claim has independent mathematical content outside any input data or prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger reflects assumptions stated there. Standard background facts from complex geometry (Kähler condition, nonpositive curvature) are presupposed. No free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption The base of the holomorphic submersion is a nonpositively curved Kähler manifold.
    Explicitly required in the abstract for the torus-bundle case.
  • domain assumption The fibers are tori or the total space is a compact quotient of a Lie group.
    Stated as the setting for the long-time existence result.

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Reference graph

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