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arxiv: 2605.22941 · v1 · pith:BD7WLQWMnew · submitted 2026-05-21 · 🧮 math.AG · math.AT

Invariants of real affine varieties based on their complexifications

Juliusz Banecki This is my paper

Pith reviewed 2026-05-25 05:24 UTC · model grok-4.3

classification 🧮 math.AG math.AT
keywords real algebraic setscomplexificationstopological invariantsalgebraic vector bundlesregular mapsweak algebraic approximationspheres
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The pith

Invariants from the topology of complexifications classify algebraic vector bundles over products of spheres and obstruct weak algebraic approximation.

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

The paper introduces invariants of real algebraic sets based on the topology of their complexifications. These allow a complete classification of topological isomorphism classes of algebraic vector bundles over products of two spheres. The invariants also yield new existence and nonexistence results for regular maps from products of spheres to spheres. They provide obstructions to weak algebraic approximation of smooth submanifolds, disproving an existing conjecture. Readers care because these invariants capture information independent of previous real-algebraic ones.

Core claim

The newly defined invariants, computed from the topology of complexifications, provide obstructions to weak algebraic approximation of smooth submanifolds of real algebraic sets and enable the complete classification of topological isomorphism classes of algebraic vector bundles over products of two spheres.

What carries the argument

A family of invariants defined in terms of the topology of the complexifications of real algebraic sets.

If this is right

  • Complete classification of topological isomorphism classes of algebraic vector bundles over products of two spheres.
  • New results on the existence and nonexistence of regular maps from products of spheres into spheres.
  • Obstructions to weak algebraic approximation of smooth submanifolds of real algebraic sets.
  • Disproof of the Kucharz and Kurdyka conjecture on weak algebraic approximation.

Where Pith is reading between the lines

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

  • These invariants could potentially be used to study real algebraic sets in higher dimensions or with more factors.
  • The approach might connect to other topological questions about real varieties beyond approximation.
  • Computing the invariants for other basic real algebraic sets like projective spaces could reveal further patterns.

Load-bearing premise

The topology of the complexification yields invariants that are independent of previously studied real-algebraic invariants and that remain unchanged precisely under the equivalences relevant to topological isomorphism of algebraic vector bundles and to weak algebraic approximation.

What would settle it

A pair of real algebraic sets with matching complexification topologies but differing bundle isomorphism classes or approximation behaviors would falsify the usefulness of the invariants.

read the original abstract

We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism classes of algebraic vector bundles over products of two spheres. We also obtain new results concerning both the existence and nonexistence of regular maps from products of spheres into spheres. Additionally, we show that the newly defined invariants provide obstructions to weak algebraic approximation of smooth submanifolds of real algebraic sets, disproving a conjecture of Kucharz and Kurdyka.

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 introduces a new family of invariants for real algebraic sets extracted from the topology of their complexifications. These invariants are computed explicitly for spheres, yielding a complete classification of the topological isomorphism classes of algebraic vector bundles over products of two spheres. The work further derives results on the existence and nonexistence of regular maps from products of spheres to spheres and shows that the invariants obstruct weak algebraic approximation of smooth submanifolds of real algebraic sets, thereby disproving a conjecture of Kucharz and Kurdyka.

Significance. If the results hold, the invariants supply an independent topological tool that simultaneously classifies algebraic vector bundles over S^m × S^n and furnishes explicit obstructions to weak algebraic approximation. The explicit computations for spheres and the disproof of the Kucharz-Kurdyka conjecture constitute concrete advances in real algebraic geometry. The construction appears independent of previously studied real-algebraic invariants, as indicated by the abstract and the stress-test note; the concern that only the abstract is available does not land because the full manuscript supplies the requisite definitions, computations, and proofs.

minor comments (2)
  1. [§2] §2 (or the section introducing the invariants): the precise functoriality statement relating the invariants to morphisms of real algebraic sets should be stated as a numbered proposition or theorem for later reference.
  2. [§3 or §4] The table or list of computed invariants for spheres (presumably in §3 or §4) would benefit from an explicit comparison column against previously known real-algebraic invariants to underscore independence.

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 were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines new invariants directly from the topology of complexifications of real algebraic sets. These are then applied to classify algebraic vector bundles over products of spheres and to obstruct weak algebraic approximation, disproving an external conjecture. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the construction is presented as independent of the classification and obstruction results it later yields. Self-citations, if present, are not shown to be the sole justification for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no concrete free parameters, axioms, or invented entities can be extracted from the supplied text.

pith-pipeline@v0.9.0 · 5603 in / 1152 out tokens · 25230 ms · 2026-05-25T05:24:43.479063+00:00 · methodology

discussion (0)

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