On the cohomological classification of vector bundles on smooth real affine surfaces and threefolds
Pith reviewed 2026-05-22 03:08 UTC · model grok-4.3
The pith
Under suitable cohomological assumptions on the real locus, vector bundles on smooth real affine surfaces and threefolds are classified exactly as over algebraically closed fields.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank 3 bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine R-algebra of dimension 3 with trivial Chern classes which is not stably free.
What carries the argument
Cohomological classification of vector bundles, which uses data on the real locus to replicate the Mohan Kumar-Murthy and Asok-Fasel results from algebraically closed fields.
Load-bearing premise
The real locus of the surface or threefold must satisfy suitable cohomological assumptions for the classification to match the algebraically closed case.
What would settle it
Exhibit a smooth real affine surface or threefold obeying the cohomological assumptions on its real locus for which some vector bundle fails to be determined by the expected Chern classes or other invariants.
read the original abstract
We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and in a coming joint paper with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents results on the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. It proves that under suitable cohomological assumptions on the real locus, this classification is similar to the one for algebraically closed fields due to Mohan Kumar and Murthy and Asok and Fasel. It also gives an efficient proof of a theorem by Kucharz on Chern classes of rank 3 bundles on smooth real affine threefolds and provides the first example of a projective module over a smooth affine R-algebra of dimension 3 with trivial Chern classes that is not stably free.
Significance. This work is significant because it bridges the classification of vector bundles between real and complex settings in algebraic geometry. The construction of the non-stably free module with trivial Chern classes is a key achievement that answers previously open questions. The efficient proof of Kucharz's theorem using Fasel's argument is also noteworthy. These contributions enhance the understanding of projective modules and vector bundles over real affine varieties.
minor comments (2)
- The precise cohomological assumptions on the real locus should be stated explicitly in the main theorem to allow immediate verification of applicability.
- Distinguish more clearly between results from prior joint works (Asok-Fasel, Banerjee-Fasel) and the new contributions in the introduction.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of our manuscript and for recommending minor revision. We are pleased that the referee has recognized the significance of our results, including the bridge between real and complex settings for vector bundle classification, the efficient proof of Kucharz's theorem, and the construction of the first example of a non-stably-free projective module with trivial Chern classes over a smooth affine R-algebra of dimension 3.
Circularity Check
No significant circularity; derivations rely on independent proofs and new constructions
full rationale
The paper explicitly conditions its main classification result on cohomological assumptions on the real locus and states that the mirroring to algebraically closed cases was observed in prior joint work with Asok and Fasel. It separately provides an efficient proof (using an argument due to Fasel) of Kucharz's theorem on Chern classes and constructs a new example of a non-stably-free projective module with trivial Chern classes over a smooth affine R-algebra of dimension 3. These steps are presented as original contributions rather than reductions to fitted parameters or self-definitional loops. No load-bearing claim reduces by construction to the paper's own inputs or unverified self-citations.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
Theorem 1.1 (Kucharz). … the triple (c1,c2,c3) lies in the image of φ3(X) iff a3−a1∪a2−Sq(a2)=0 in H3(X(R),Z/2). … φ3(X) is injective if X(R) has no compact connected components
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IndisputableMonolith/Foundation/AlexanderDuality.leanD3_admits_circle_linking echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
the map [X+,BGL2]A1→[X+,BGL(3)2]A1 is a bijection … H3(X(R),Z/2)=0 implies Euler class bijection and φ2 bijective
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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discussion (0)
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