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arxiv: 2606.12188 · v1 · pith:Q6IY4HW4new · submitted 2026-06-10 · 🧮 math.OA · math.KT

Schubert Calculus and uniform property Gamma

Andrew S. Toms This is my paper

Pith reviewed 2026-06-27 07:29 UTC · model grok-4.3

classification 🧮 math.OA math.KT
keywords C*-algebrasuniform property GammaSchubert calculusThom-Porteous classesdeterminantal Schur classesinductive limitsnuclear C*-algebrastrace comparison
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The pith

A nuclear C*-algebra lacks uniform property Γ because a quadratic Schubert calculus obstruction from determinantal Schur classes persists through its inductive construction and blocks trace comparison of projections.

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

The paper constructs a simple separable unital nuclear C*-algebra without uniform property Γ by using determinantal Schur classes to force every map between certain equal-rank vector bundles to vanish at some point. A quadratic Schubert calculus computation shows these Thom-Porteous classes remain nonzero in the inductive system of homogeneous C*-algebras. The obstruction then prevents projections from being compared by traces in the uniform tracial completion. This approach identifies quadratic dimension growth as the threshold separating algebras that do and do not have the property.

Core claim

The author constructs a nuclear C*-algebra in which every bundle map between certain equal-rank vector bundles vanishes somewhere, using Thom-Porteous theory of degeneracy loci. Quadratic Schubert calculus shows that the associated determinantal Schur classes remain nonzero through the inductive system of homogeneous C*-algebras. This prevents projections from being compared by traces in the uniform tracial completion, yielding an algebra without uniform property Γ. The classes live in degree proportional to the square of the forced rank loss, forcing quadratic dimension growth in the constituent algebras.

What carries the argument

The non-vanishing of determinantal Schur classes (Thom-Porteous classes) across an inductive system of homogeneous C*-algebras, which enforces vanishing of bundle maps and obstructs trace comparison of projections.

If this is right

  • The resulting algebra is simple, separable, unital and nuclear yet lacks uniform property Γ.
  • Dimension growth in the homogeneous building blocks is quadratic in the forced rank loss.
  • The obstruction arises from forcing bundle maps to vanish rather than from the absence of large trivial subbundles.
  • Uniform property Γ is equivalent to the ability to compare projections by traces in the uniform tracial completion for this class of algebras.

Where Pith is reading between the lines

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

  • The same persistence argument might apply to other regularity properties such as strict comparison when dimension growth is controlled.
  • One could test whether every nuclear C*-algebra with linear dimension growth must have uniform property Γ by attempting to build a counterexample with slower growth.
  • The method supplies a concrete geometric criterion that could decide the presence of uniform property Γ for other inductive systems of homogeneous algebras.

Load-bearing premise

The Thom-Porteous classes arising from the determinantal Schur classes remain non-zero after passage through the inductive system of homogeneous C*-algebras.

What would settle it

A computation showing that the relevant Thom-Porteous class becomes zero in the inductive limit would show the obstruction does not persist and the algebra might satisfy uniform property Γ.

Figures

Figures reproduced from arXiv: 2606.12188 by Andrew S. Toms.

Figure 1
Figure 1. Figure 1: A coarse taxonomy of geometric dimension-growth regimes. Every region of the diagram above is populated by simple separable unital nuclear C∗ -algebras: • Slow: These are precisely the AH algebras satisfying the Toms-Winter regularity con￾jecture, and are classified by the Elliott invariant (this is the work of many, but a good modern reference is [CGS+23]). • Linear: This region contains Villadsen’s algeb… view at source ↗
read the original abstract

We construct a simple, separable, unital, nuclear C$^*$-algebra without uniform property $\Gamma$. The construction is based on a new topological obstruction arising from the Thom-Porteous theory of degeneracy loci. Constructions of pathological nuclear C$^*$-algebras over the past 30 years have used Chern class calculations introduced by Villadsen to obstruct the existence of large trivial subbundles. Here, by contrast, we use determinantal Schur classes to force every bundle map between certain equal-rank vector bundles to vanish somewhere on the base space. A quadratic Schubert calculus computation shows that this obstruction can persist across an inductive system and ultimately obstructs the comparison of projections by traces in the uniform tracial completion. The relevant Thom-Porteous classes live in degree proportional to the square of the forced rank loss, which in turn forces dimension growth of the same order in the constituent homogeneous C$^*$-algebras of our example. This identifies a new geometric threshold in the structure theory of nuclear C$^*$-algebras, linking the presence or absence of uniform property $\Gamma$ to quadratic dimension growth.

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

2 major / 1 minor

Summary. The manuscript constructs a simple, separable, unital, nuclear C*-algebra without uniform property Γ. The construction relies on a new topological obstruction from Thom-Porteous theory of degeneracy loci, using determinantal Schur classes to force every bundle map between equal-rank vector bundles to vanish somewhere. A quadratic Schubert calculus computation is invoked to show that this obstruction persists across an inductive system of homogeneous C*-algebras, ultimately obstructing comparison of projections by traces in the uniform tracial completion and forcing quadratic dimension growth in the constituent algebras.

Significance. If the central claims hold, the work supplies a new obstruction mechanism for nuclear C*-algebras that links absence of uniform property Γ to quadratic dimension growth, extending earlier Chern-class techniques. The explicit use of Schubert calculus to verify persistence of Thom-Porteous classes offers a concrete computational bridge between topology and C*-algebra structure theory.

major comments (2)
  1. [Abstract] Abstract (paragraph on persistence of the obstruction): the claim that the Thom-Porteous classes remain nonzero after passage through the inductive system is load-bearing for the main theorem, yet the quadratic Schubert calculus computation is asserted without explicit equations, cycle representatives, or verification that the classes survive each bonding map of the system.
  2. [Construction] Construction of the inductive system: no explicit verification is supplied that the direct limit remains nuclear and simple once the dimension growth is imposed by the degree of the Thom-Porteous classes; this check is required to confirm the algebra meets the stated hypotheses.
minor comments (1)
  1. [Abstract] The abstract states that the classes live in degree proportional to the square of the forced rank loss; a brief reference to the precise degree formula or Schubert calculus identity used would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the work's significance and for the constructive major comments. We address each point below and will revise the manuscript accordingly to provide the requested explicit details.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on persistence of the obstruction): the claim that the Thom-Porteous classes remain nonzero after passage through the inductive system is load-bearing for the main theorem, yet the quadratic Schubert calculus computation is asserted without explicit equations, cycle representatives, or verification that the classes survive each bonding map of the system.

    Authors: We agree that the persistence claim is central and that the current presentation of the quadratic Schubert calculus is condensed. In the revised manuscript we will expand the relevant section (or add an appendix) to include the explicit determinantal Schur class equations, the cycle representatives on the appropriate flag varieties, and a direct verification that the classes remain nonzero after each bonding map. These additions will make the computation fully traceable without altering the underlying argument. revision: yes

  2. Referee: [Construction] Construction of the inductive system: no explicit verification is supplied that the direct limit remains nuclear and simple once the dimension growth is imposed by the degree of the Thom-Porteous classes; this check is required to confirm the algebra meets the stated hypotheses.

    Authors: We concur that explicit verification is needed. Nuclearity of the direct limit follows at once from the nuclearity of each homogeneous stage and the stability of nuclearity under inductive limits; we will insert a short paragraph stating this standard fact. For simplicity we will add a verification paragraph showing that the chosen bonding maps ensure the limit is simple, appealing to the density of the images of the finite-stage unitaries and the fact that the dimension growth does not introduce nontrivial ideals. These clarifications will be placed in the construction section. revision: yes

Circularity Check

0 steps flagged

No circularity; obstruction derived from independent Schubert calculus computation

full rationale

The paper's derivation uses determinantal Schur classes and a quadratic Schubert calculus computation to establish persistence of the Thom-Porteous obstruction through the inductive system of homogeneous C*-algebras. This step is presented as an external topological verification (standard in algebraic geometry) rather than a reduction to fitted parameters, self-citations, or definitional equivalence within the paper. No load-bearing claim reduces by construction to the paper's own inputs; the non-vanishing result is computed independently of the C*-algebra construction itself. The overall argument therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the ledger entries below are inferred from the abstract's references to standard tools.

axioms (1)
  • standard math Standard results of Thom-Porteous theory and Schubert calculus apply to the degeneracy loci of the bundle maps under consideration.
    Invoked to produce the vanishing obstruction.

pith-pipeline@v0.9.1-grok · 5715 in / 1179 out tokens · 19450 ms · 2026-06-27T07:29:05.178677+00:00 · methodology

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