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arxiv: 2604.22642 · v1 · submitted 2026-04-24 · 🧮 math.DG · math.AG

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B-complex manifolds with generalized corners. I. Newlander-Nirenberg Theorems

H\"ulya Arg\"uz , Dominic Joyce
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Pith reviewed 2026-05-08 09:38 UTC · model grok-4.3

classification 🧮 math.DG math.AG
keywords b-complex manifoldsgeneralized cornersNewlander-Nirenberg theoremb-tangent bundlelog geometryKato-Nakayama spacecomplex structures on corners
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The pith

B-complex structures on manifolds with corners agree with standard models to infinite order along each stratum.

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

This paper generalizes complex manifolds to manifolds with corners and generalized corners by defining complex structures on the b-tangent bundle. It proves a formal Newlander-Nirenberg theorem establishing that these structures match a standard local model to infinite order along every corner stratum. A sympathetic reader would care because the result supplies the analytic foundation needed to relate such objects to log smooth schemes and their Kato-Nakayama spaces in later work.

Core claim

We generalize complex manifolds to manifolds with corners X, and to manifolds with generalized corners in the sense of the second author, using complex structures on the b-tangent bundle bTX. We prove a formal Newlander-Nirenberg type theorem showing that along each corner stratum of X, the b-complex structure agrees with a standard model to infinite order.

What carries the argument

The formal Newlander-Nirenberg theorem for b-complex structures on the b-tangent bundle, which verifies infinite-order agreement along corner strata of manifolds with generalized corners.

If this is right

  • The Kato-Nakayama space of any log smooth log C-scheme carries the structure of a b-complex manifold with generalized corners.
  • Necessary and sufficient conditions can be stated for a b-complex manifold with generalized corners to arise as a Kato-Nakayama space.
  • Complex geometry on spaces with corners becomes compatible with the logarithmic methods used in algebraic and analytic geometry.

Where Pith is reading between the lines

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

  • The same formal agreement might extend the construction to more singular log spaces beyond the log smooth case.
  • Explicit coordinate computations on simple log schemes could serve as a direct test of the infinite-order matching.
  • The framework suggests a route to treat deformations of complex structures on cornered spaces within logarithmic geometry.

Load-bearing premise

The b-complex structure satisfies the required integrability conditions along strata so that the formal theorem applies using the prior definition of generalized corners and the local standard model.

What would settle it

A concrete b-complex structure on a manifold with generalized corners that satisfies integrability yet fails to match the standard model to infinite order along at least one corner stratum would disprove the theorem.

Figures

Figures reproduced from arXiv: 2604.22642 by Dominic Joyce, H\"ulya Arg\"uz.

Figure 2.1
Figure 2.1. Figure 2.1: 3-manifold with g-corners XQ in (2.9) We sketch XQ in view at source ↗
read the original abstract

We generalize complex manifolds to manifolds with corners $X$, and to manifolds with generalized corners (g-corners) in the sense of the second author arXiv:1501.00401, using complex structures on the b-tangent bundle (log tangent bundle) ${}^bTX$. We prove a formal Newlander-Nirenberg type theorem showing that along each corner stratum of $X$, the b-complex structure agrees with a standard model to infinite order. In the sequel we show that if $S$ is a log smooth log $\mathbb C$-scheme, or log smooth log complex analytic space, then the Kato-Nakayama space $S^{\rm KN}$ has the structure of a b-complex manifold with g-corners. Using our Newlander-Nirenberg theorem we give necessary and sufficient conditions for a b-complex manifold with g-corners to be a Kato-Nakayama space.

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 / 2 minor

Summary. The paper generalizes complex manifolds to manifolds with corners and generalized corners (g-corners) by defining b-complex structures via integrable complex structures on the b-tangent bundle (log tangent bundle). The central result is a formal Newlander-Nirenberg theorem asserting that, along each corner stratum of such an X, the b-complex structure agrees with a standard local model to infinite order. The work also sketches applications to Kato-Nakayama spaces of log smooth log C-schemes or log complex analytic spaces, with necessary and sufficient conditions for a b-complex manifold with g-corners to arise as one.

Significance. If the formal theorem holds, the manuscript supplies a differential-geometric framework that extends the classical Newlander-Nirenberg theorem to singular settings with corners, directly linking to log geometry. The formal (infinite-order) character of the result circumvents analytic convergence questions and provides a clean foundation for subsequent work on integrability and deformation theory in the presence of logarithmic singularities. Explicit ties to Kato-Nakayama spaces constitute a concrete application with potential impact on log complex analytic geometry.

major comments (1)
  1. §3 (proof of the main formal NN theorem): the argument that vanishing of the b-Nijenhuis tensor on the b-tangent bundle implies infinite-order agreement with the standard model along each stratum is only sketched at the level of the abstract; the manuscript must supply the precise inductive step or jet-space argument that lifts the integrability condition stratum-by-stratum, as this step is load-bearing for the central claim.
minor comments (2)
  1. Introduction, paragraph 2: the transition from ordinary complex structures to b-complex structures on the b-tangent bundle would benefit from a short local coordinate example before the general definition.
  2. Notation section: the symbol for the b-tangent bundle is introduced as {}^bTX but occasionally appears without the superscript in later statements; uniform usage would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for the constructive suggestion regarding the presentation of the central theorem. We address the major comment below.

read point-by-point responses

  1. Referee: §3 (proof of the main formal NN theorem): the argument that vanishing of the b-Nijenhuis tensor on the b-tangent bundle implies infinite-order agreement with the standard model along each stratum is only sketched at the level of the abstract; the manuscript must supply the precise inductive step or jet-space argument that lifts the integrability condition stratum-by-stratum, as this step is load-bearing for the central claim.

    Authors: We agree that the inductive lifting argument in §3 is presented concisely and would be strengthened by greater explicitness. In the revised manuscript we will expand the proof to include a detailed jet-space argument: we will describe the induction on the order of jets along each corner stratum, showing step-by-step how the vanishing of the b-Nijenhuis tensor on the b-tangent bundle forces the b-complex structure to agree with the standard local model to all finite orders. This addition will render the stratum-by-stratum lifting fully precise without changing the formal character of the result. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for foundational g-corners definition; central NN theorem derivation remains independent

full rationale

The paper generalizes complex structures via the b-tangent bundle on manifolds with g-corners (defined via self-citation to arXiv:1501.00401) and proves a formal Newlander-Nirenberg theorem asserting infinite-order agreement with a standard local model along corner strata. This self-citation supplies only the ambient category and standard model, not the integrability proof or the formal power-series argument itself. The central claim rests on the classical Newlander-Nirenberg technique adapted to the b-Nijenhuis tensor vanishing condition, which is an independent differential-geometric statement. No equation reduces to a prior result by construction, no parameter is fitted and relabeled as a prediction, and no uniqueness theorem is imported from the authors' own prior work to force the conclusion. The derivation chain is therefore self-contained apart from one non-load-bearing definitional citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the classical Newlander-Nirenberg theorem as background, the definition of generalized corners from prior work, and the new b-complex structure concept. No free parameters or data fitting involved as this is a pure existence/integrability theorem.

axioms (2)
  • standard math The classical Newlander-Nirenberg theorem holds for standard almost complex structures on smooth manifolds.
    Invoked as the base case being generalized to the b-setting along strata.
  • domain assumption Manifolds with generalized corners are as defined in arXiv:1501.00401 by the second author.
    The b-tangent bundle and corner strata are taken from this prior definition.
invented entities (1)
  • b-complex structure no independent evidence
    purpose: To equip the b-tangent bundle of a manifold with corners or g-corners with an almost complex structure.
    New concept introduced to generalize complex manifolds; no independent evidence outside this framework provided in abstract.

pith-pipeline@v0.9.0 · 5454 in / 1537 out tokens · 82219 ms · 2026-05-08T09:38:15.094830+00:00 · methodology

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Reference graph

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