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arxiv: 2606.16176 · v1 · pith:EUREWPUPnew · submitted 2026-06-15 · 🧮 math.DG · math.AP

Curvature at infinity of scalar-flat ALE four-manifolds

Jiangcheng You This is my paper

Pith reviewed 2026-06-27 03:11 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords scalar-flat ALE four-manifoldsasymptotic expansionWeyl tensor at infinityADM masscrepant resolutionminimal resolutionquotient singularitiesself-dual metrics
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The pith

Scalar-flat ALE four-manifolds admit preferred coordinates at infinity in which the |x|^{-2} metric term splits into an ADM-mass scalar part and a separate algebraic Weyl tensor.

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

The paper examines the refined asymptotics of scalar-flat ALE four-manifolds that are either self-dual or have harmonic curvature, working in the coordinates first obtained by Tian and Viaclovsky. From those coordinates it constructs preferred coordinates that isolate the homogeneous term of degree |x|^{-2} in the metric expansion. This term factors canonically into a scalar piece fixed by the ALE ADM mass and an algebraic Weyl tensor contribution at infinity. As a direct application, scalar-flat Kähler ALE metrics on minimal resolutions of quotient singularities have vanishing leading Weyl tensor precisely when the resolution is crepant.

Core claim

In the Tian-Viaclovsky setting, the construction of preferred coordinates at infinity shows that the homogeneous |x|^{-2} term in the metric expansion decomposes canonically into a scalar component determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. For scalar-flat Kähler ALE metrics on minimal resolutions π:X→C²/Γ of quotient surface singularities, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.

What carries the argument

Preferred coordinates at infinity built from Tian-Viaclovsky ALE coordinates, which isolate the homogeneous |x|^{-2} term and produce its canonical splitting into mass scalar and algebraic Weyl tensor.

If this is right

  • The ADM mass controls the scalar part of the leading asymptotic term independently of the Weyl contribution.
  • The algebraic Weyl tensor at infinity provides an additional invariant that distinguishes different asymptotic behaviors within the self-dual and harmonic-curvature classes.
  • On minimal resolutions of quotient singularities the vanishing of the leading Weyl tensor is equivalent to the resolution being crepant.
  • Non-crepant resolutions necessarily carry a nonzero algebraic Weyl tensor at infinity.

Where Pith is reading between the lines

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

  • The splitting may supply a finer asymptotic invariant that could be used to distinguish moduli components of ALE metrics.
  • The link between crepancy and vanishing Weyl tensor suggests that crepant resolutions admit asymptotically simpler models than non-crepant ones.
  • The same coordinate construction might be tested on other curvature conditions outside the self-dual and harmonic settings.

Load-bearing premise

The manifolds admit the ALE coordinates constructed by Tian-Viaclovsky and lie in the self-dual or harmonic-curvature setting assumed throughout the analysis.

What would settle it

A scalar-flat Kähler ALE metric on a non-crepant minimal resolution of C²/Γ whose leading Weyl tensor at infinity is nevertheless zero, or a crepant resolution whose leading Weyl tensor is nonzero.

Figures

Figures reproduced from arXiv: 2606.16176 by Jiangcheng You.

Figure 1
Figure 1. Figure 1: The Burns metric on Bl0C 2 . The exceptional divisor replaces the blown￾up origin, while the metric is ALE at infinity. This example shows that the leading term in Theorem 1.3 can have both a mass part and a Weyl-type part. Theorems 1.3 and 1.7 associate to a scalar-flat ALE four-manifold a preferred homogeneous |x| −2 coefficient at infinity, as (1). As an application, we explain what this Weyl-type part … view at source ↗
read the original abstract

We study refined asymptotics of scalar-flat ALE four-manifolds in the Tian--Viaclovsky setting, namely for self-dual or anti-self-dual metrics and for metrics with harmonic curvature. Starting from the ALE coordinates obtained by Tian--Viaclovsky, we construct preferred coordinates at infinity and identify the homogeneous $|x|^{-2}$ term in the metric expansion. This term splits canonically into a scalar part determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. As an application, we consider scalar-flat K\"ahler ALE metrics on minimal resolutions $\pi:X\to\mathbb C^2/\Gamma$ of quotient surface singularities. In this case, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.

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 examines refined asymptotics for scalar-flat ALE four-manifolds in the Tian-Viaclovsky setting, covering self-dual/anti-self-dual metrics and those with harmonic curvature. Starting from Tian-Viaclovsky ALE coordinates, it constructs preferred coordinates at infinity and isolates the homogeneous |x|^{-2} term in the metric expansion. This term is shown to split canonically into a scalar component fixed by the ALE ADM mass and an algebraic Weyl tensor contribution at infinity. As an application, for scalar-flat Kähler ALE metrics on minimal resolutions π:X→C²/Γ of quotient surface singularities, the leading Weyl tensor at infinity vanishes if and only if the resolution is crepant.

Significance. If the central claims hold, the work supplies a canonical decomposition of the leading curvature term at infinity for a class of scalar-flat ALE four-manifolds. This decomposition separates the mass contribution from the Weyl part in a manner independent of coordinate choice within the Tian-Viaclovsky charts, and the Kähler application ties the vanishing of the Weyl term directly to the crepant condition on minimal resolutions. The result strengthens the asymptotic toolkit for gravitational instantons and could inform moduli-space questions or gluing constructions in four-dimensional geometry. The manuscript credits the input coordinates explicitly and isolates the new splitting as an independent contribution.

minor comments (3)
  1. [Section introducing preferred coordinates] §2 (or the section introducing the preferred coordinates): the transition from Tian-Viaclovsky charts to the new preferred coordinates is outlined but would benefit from an explicit statement of the error term in the coordinate change, even if it follows from prior work.
  2. [Main theorem statement] The statement of the main splitting result (likely Theorem 1.1 or equivalent) should include a brief remark on the decay assumptions needed to guarantee that the |x|^{-2} term is well-defined and the splitting is unique.
  3. [Application to Kähler ALE metrics] In the Kähler application, the proof that the Weyl tensor vanishes precisely for crepant resolutions relies on the self-dual/harmonic-curvature hypothesis; a short sentence clarifying why this hypothesis is preserved under the minimal resolution would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

Derivation self-contained from external coordinates; no circular steps

full rationale

The paper takes the Tian-Viaclovsky ALE coordinates as given input and performs a coordinate normalization to extract the homogeneous |x|^{-2} term, then algebraically splits it into scalar (ADM-mass) and Weyl parts under the self-dual/harmonic-curvature hypotheses. The vanishing statement for crepant resolutions is a direct consequence of that splitting applied to the Kähler case. No equation reduces to a fitted parameter renamed as prediction, no self-citation chain is load-bearing, and the central claims do not rest on an ansatz or uniqueness theorem imported from the author's own prior work. The derivation is therefore independent of its starting data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or non-standard axioms are visible. The work rests on the standard axioms of Riemannian geometry and the existence of ALE coordinates from prior literature.

axioms (2)
  • domain assumption Existence of ALE coordinates as constructed by Tian–Viaclovsky for scalar-flat metrics in the self-dual or harmonic-curvature setting.
    Invoked at the start of the coordinate refinement step.
  • standard math Standard differential-geometric identities for the Weyl tensor and ADM mass in four dimensions.
    Used to decompose the |x|^{-2} term.

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