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arxiv: 2607.01402 · v1 · pith:E3Q2MB2Jnew · submitted 2026-07-01 · 🧮 math.DG · math.AG

Point Singularities and Local Third Chern Classes for Rank-Two Torsion-free Sheaves on Threefolds

Xuemiao Chen This is my paper

Pith reviewed 2026-07-03 18:29 UTC · model grok-4.3

classification 🧮 math.DG math.AG
keywords point singularitiesthird Chern classtorsion-free sheavesthreefoldsK-theoryreflexive sheavesBuchsbaum-Rim multiplicitygauge theory
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The pith

The local third Chern class at isolated point singularities of rank-two torsion-free sheaves on threefolds is defined by finite-length algebraic data and equals a K-theoretic charge.

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

The paper defines an invariant for the local third Chern class contribution of isolated point singularities in rank-two torsion-free sheaves on complex threefolds using finite-length local algebraic data. It shows this invariant is deformation invariant because it can be computed from the total family. Its parity recovers a topological invariant of the smooth rank-two vector bundle on the boundary sphere. The class equals the local K-theoretic charge from a self-dual complex associated to the sheaf, and for reflexive sheaves it matches the Fitting scheme and Buchsbaum-Rim multiplicity. This provides a way to study singularity formation in gauge theory.

Core claim

In the local rank-two setting considered here, the invariant is defined in terms of finite-length local algebraic data at the singular point. We prove that it can be computed from data on the total family; in particular, it is deformation invariant. We also prove that its parity recovers a topological invariant of the underlying smooth complex rank-two vector bundle on the boundary sphere. We then give a relative K-theoretic interpretation: a self-dual complex naturally associated with the sheaf defines a local K-theoretic charge, and this charge is equal to the local third Chern class. For rank-two reflexive sheaves, we relate the same invariant to several classical algebraic quantities, in

What carries the argument

finite-length local algebraic data at the singular point defining the local third Chern class

If this is right

  • It is deformation invariant as it computes from the total family data.
  • Its parity recovers a topological invariant of the underlying smooth rank-two vector bundle on the boundary sphere.
  • It equals the local K-theoretic charge from an associated self-dual complex.
  • For reflexive sheaves it equals the Fitting scheme and Buchsbaum-Rim multiplicity.
  • It applies to analyzing the boundary of moduli spaces of Hermitian-Yang-Mills connections.

Where Pith is reading between the lines

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

  • This could facilitate the computation of global invariants in singular moduli spaces by adding local contributions.
  • The link to Buchsbaum-Rim multiplicity suggests connections to algebraic multiplicity theory for counting purposes in geometry.
  • Similar methods might apply to other Chern classes or sheaf ranks if appropriate algebraic data can be identified.

Load-bearing premise

Singularities are isolated points and sheaves are rank-two torsion-free on complex threefolds, so finite-length algebraic data captures the local contribution and allows K-theoretic and topological comparisons.

What would settle it

A concrete example of a rank-two torsion-free sheaf on a threefold with an isolated point singularity where the computed local third Chern class from algebraic data does not match the K-theoretic charge or the expected parity of the boundary bundle.

read the original abstract

In this paper, motivated by singularity formation in gauge theory, we study the local third Chern class contribution carried by isolated point singularities of rank-two torsion-free sheaves on complex threefolds. In the local rank-two setting considered here, the invariant is defined in terms of finite-length local algebraic data at the singular point. We prove that it can be computed from data on the total family; in particular, it is deformation invariant. We also prove that its parity recovers a topological invariant of the underlying smooth complex rank-two vector bundle on the boundary sphere. We then give a relative K-theoretic interpretation: a self-dual complex naturally associated with the sheaf defines a local $K$-theoretic charge, and this charge is equal to the local third Chern class. For rank-two reflexive sheaves, we relate the same invariant to several classical algebraic quantities, including the Fitting scheme and the Buchsbaum-Rim multiplicity. We also discuss applications to the boundary of moduli spaces of Hermitian-Yang-Mills connections.

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 / 1 minor

Summary. The paper defines the local third Chern class contribution carried by isolated point singularities of rank-two torsion-free sheaves on complex threefolds via finite-length local algebraic data at the singular point. It proves that this invariant can be computed from data on the total family (hence deformation invariant), that its parity recovers a topological invariant of the underlying smooth rank-two vector bundle on the boundary sphere, that it equals the local K-theoretic charge from an associated self-dual complex, and that for reflexive sheaves it equals the Fitting scheme and Buchsbaum-Rim multiplicity. Applications to the boundary of moduli spaces of Hermitian-Yang-Mills connections are discussed.

Significance. If the results hold, this provides a precise algebraic framework for local invariants arising from point singularities in rank-two sheaves, with direct applications to gauge theory moduli spaces. The deformation invariance, parity matching, and equalities with K-theoretic charge and classical multiplicities (Fitting scheme, Buchsbaum-Rim) are load-bearing strengths that connect algebraic geometry to topology and gauge theory without ad-hoc parameters or fitting.

minor comments (1)
  1. Notation for the local algebraic data and the self-dual complex should be introduced with explicit references to the relevant local rings or modules in the first section where the definition appears, to aid readers unfamiliar with the Buchsbaum-Rim setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the local third Chern class via finite-length algebraic data at isolated points for rank-two torsion-free sheaves, then proves deformation invariance from the total family, parity recovery of the topological invariant on the link sphere, equality to the K-theoretic charge of an associated self-dual complex, and (for reflexive sheaves) equality to the Fitting scheme and Buchsbaum-Rim multiplicity. These steps consist of explicit algebraic definitions followed by standard comparisons and proofs; no equations reduce by construction to fitted inputs, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness theorem is smuggled in. The results are externally verifiable through the local-algebraic setup and do not rely on renaming known results or self-referential loops.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the definition itself relies on the existence of finite-length modules at isolated points and the standard properties of Chern classes and K-theory on threefolds.

pith-pipeline@v0.9.1-grok · 5706 in / 1250 out tokens · 20835 ms · 2026-07-03T18:29:50.212658+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Michael Atiyah,K-theory, CRC Press, 2018

    work page 2018

  2. [2]

    David A Buchsbaum and Dock S Rim,A generalized Koszul complex. ii. depth and multiplicity, Transactions of the American Mathematical Society111(1964), no. 2, 197–224

    work page 1964

  3. [3]

    Xuemiao Chen,Bubbling of rank two bundles over surfaces, arXiv preprint arXiv:2506.09614 (2025)

    work page arXiv 2025

  4. [4]

    Xuemiao Chen,Point singularities and bubbling in degenerations of rank-two bundles on threefolds, preprint (2026)

    work page 2026

  5. [5]

    4, 2061–2108

    Xuemiao Chen and Song Sun,Analytic tangent cones of admissible Hermitian-Yang- Mills connections, Geometry & Topology25(2021), no. 4, 2061–2108

    work page 2021

  6. [6]

    Math.225(2021), no

    ,Reflexive sheaves, Hermitian-Yang-Mills connections, and tangent cones, In- vent. Math.225(2021), no. 1, 73–129. MR 4270664

    work page 2021

  7. [7]

    Volume XVI

    Simon Donaldson and Ed Segal,Gauge theory in higher dimensions, II, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, Surv. Differ. Geom., vol. 16, Int. Press, Somerville, MA, 2011, pp. 1–41. MR 2893675

    work page 2011

  8. [8]

    Press, Oxford, 1998, pp

    Simon Donaldson and Richard Thomas,Gauge theory in higher dimensions, The geometric universe (Oxford, 1996), Oxford Univ. Press, Oxford, 1998, pp. 31–47. MR 1634503

    work page 1996

  9. [9]

    David Eisenbud,Commutative algebra: with a view toward algebraic geometry, Springer Science & Business Media, 2013

    work page 2013

  10. [10]

    2, Springer-Verlag, Berlin, 1984

    William Fulton,Intersection theory, Ergebnisse der Mathematik und ihrer Grenzge- biete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620

    work page 1984

  11. [11]

    2, 121–176

    Robin Hartshorne,Stable reflexive sheaves, Mathematische annalen254(1980), no. 2, 121–176

    work page 1980

  12. [12]

    1, 155–169

    Birger Iversen,Local Chern classes, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure9(1976), no. 1, 155–169

    work page 1976

  13. [13]

    C Okonek, C Schneider, and H Spindler,Vector bundles on complex projective space (corrected reprint of the 1988 edition), Modern Birkh¨ auser Classics (2011)

    work page 1988

  14. [14]

    POINT SINGULARITIES AND LOCALc 3 35 Department of pure mathematics, University of W aterloo, Ontario, Canada, N2L 3G1 Email address:x67Chen@uwaterloo.ca

    Charles Weibel,The K-book: An Introduction to Algebraic K-theory, Graduate Studies in Mathematics145(2013). POINT SINGULARITIES AND LOCALc 3 35 Department of pure mathematics, University of W aterloo, Ontario, Canada, N2L 3G1 Email address:x67Chen@uwaterloo.ca

    work page 2013