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arxiv: 2606.25110 · v1 · pith:Y3BDJWCXnew · submitted 2026-06-23 · 🧮 math.DG · math.AG

Point Singularities and Bubbling in Degenerations of Rank-Two Bundles on Threefolds

Xuemiao Chen This is my paper

Pith reviewed 2026-06-25 21:49 UTC · model grok-4.3

classification 🧮 math.DG math.AG
keywords rank-two bundlespoint singularitiesbubblingdegenerationsthreefoldsHermitian-Yang-Millsreflexive hullExt-length
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The pith

In one-parameter degenerations of rank-two bundles on threefolds the algebraic bubbling multiplicity equals half the Ext-length of the singularity in the reflexive hull.

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

The paper examines one-parameter degenerations of rank-two vector bundles on complex threefolds whose central fiber develops an isolated point singularity, becoming a torsion-free sheaf. It proves a rigidity identity that equates the algebraic bubbling multiplicity of this central fiber to one half the Ext-length of the singularity of its reflexive hull. The identity shows bubbling must occur whenever such a singularity appears and supplies obstructions to smoothability together with constructions of sharp local smoothings. The same relation is used to realize a local example as a global degeneration of smooth Hermitian-Yang-Mills connections and to produce smoothings of elementary modifications of projective-cone singularities that serve as local models for point bubbling.

Core claim

In a one-parameter degeneration of rank-two vector bundles on a complex threefold to a rank-two torsion-free sheaf with an isolated point singularity, the algebraic bubbling multiplicity of the central fiber equals one half of the Ext-length of the singularity of its reflexive hull. Bubbling is forced by the appearance of the singularity, and the identity yields smoothability obstructions plus explicit constructions of smoothings, including global realizations of Hermitian-Yang-Mills degenerations and smoothings of elementary modifications.

What carries the argument

The rigidity identity that sets algebraic bubbling multiplicity equal to one half the Ext-length of the singularity in the reflexive hull.

If this is right

  • Bubbling is forced whenever the central fiber develops an isolated point singularity.
  • Smoothability obstructions for the degeneration follow directly from the identity.
  • Sharp local smoothings of the singularities can be constructed explicitly.
  • The local example is realized as a global degeneration of smooth Hermitian-Yang-Mills connections.
  • Rescaling the degeneration produces a smooth non-flat Hermitian-Yang-Mills connection on C^3 with density one at infinity whose tangent cone has a multiplicity-one line.

Where Pith is reading between the lines

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

  • The constructions distinguish point bubbling from bubbling along complex codimension-two loci and supply local models specific to dimension three.
  • The identity may supply a template for relating algebraic and analytic invariants in degenerations of bundles of other ranks.

Load-bearing premise

The family is a one-parameter degeneration of rank-two vector bundles on a complex threefold whose central fiber develops an isolated point singularity whose reflexive hull has a well-defined Ext-length.

What would settle it

A concrete one-parameter degeneration in which the algebraic bubbling multiplicity differs from half the Ext-length of the singularity would falsify the identity.

read the original abstract

We study one-parameter degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity. We prove a rigidity identity: the algebraic bubbling multiplicity of the central fiber equals one half of the Ext-length of the singularity of its reflexive hull. Furthermore, bubbling is forced when a family develops such an isolated point singularity. We use this identity to obtain smoothability obstructions and construct sharp local smoothings. We realize the local example from earlier joint work with Sun as a global degeneration of smooth Hermitian-Yang-Mills connections. Rescaling this degeneration produces a smooth non-flat Hermitian-Yang-Mills connection on $\mathbb C^3$ with density one at infinity, whose tangent cone at infinity has flat connection part and a multiplicity-one line as the blow-up cycle. We also construct smoothings of elementary modifications of projective-cone singularities with explicit algebraic bubbles. These examples give local models for Hermitian-Yang-Mills point bubbling in complex dimension three and distinguish this phenomenon from bubbling along complex codimension-two loci.

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 / 2 minor

Summary. The paper studies one-parameter degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity in the central fiber. It proves a rigidity identity stating that the algebraic bubbling multiplicity of the central fiber equals one half of the Ext-length of the singularity of its reflexive hull. Bubbling is shown to be forced by such singularities. The identity is applied to obtain smoothability obstructions and to construct sharp local smoothings. The work realizes a local example from joint work with Sun as a global degeneration of smooth Hermitian-Yang-Mills connections, rescales it to produce a smooth non-flat HYM connection on C^3 with density one at infinity, and constructs smoothings of elementary modifications of projective-cone singularities, providing local models for point bubbling in complex dimension three that distinguish it from bubbling along codimension-two loci.

Significance. If the identity holds, the result supplies a precise algebraic relation between bubbling multiplicity and Ext-length in these degenerations, bridging algebraic geometry and gauge-theoretic analysis. The explicit constructions, including the global HYM degeneration and the non-flat connection on C^3 whose tangent cone has a multiplicity-one line, furnish concrete local models that could inform moduli problems and bubbling analysis in higher-dimensional Hermitian-Yang-Mills theory. The distinction from codimension-two bubbling adds clarity to the geometric picture.

minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from a brief statement of the precise definition of algebraic bubbling multiplicity used in the identity, to make the central claim immediately accessible without reference to prior literature.
  2. [Section on global realizations] In the discussion of the rescaled non-flat HYM connection on C^3, the precise normalization of the density-one condition at infinity and its relation to the blow-up cycle could be stated more explicitly for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of our results, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation to prior joint work; central rigidity identity proved via independent algebraic techniques

full rationale

The paper establishes the main rigidity identity (algebraic bubbling multiplicity equals half the Ext-length) through algebraic geometry arguments on degenerations of rank-two bundles, supported by local-to-global constructions. The only self-reference is to earlier joint work with Sun, used solely to realize a local example globally and rescale it; this is not load-bearing for the identity itself. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling appear in the derivation chain. The result rests on standard Ext and reflexive hull machinery external to the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

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Reference graph

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