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Mohan Kumar existence of a nontrivial stably trivial bundle on a complex affine fourfold

Statement

There exist a nonempty smooth affine complex algebraic variety $X$ of dimension four and a rank-two vector bundle $E$ with $E ≇ O_{X}^{2}$ but $E \oplus O_{X}^{N} ≅ O_{X}^{2 + N}$ for some integer $N \geq 1$ . The variety can be taken to be the complement of a union of three hypersurfaces in $P_{C}^{3} \times A_{C}^{1}$ , as in Mohan Kumar's construction. No particular defining equations are selected in this input. On writing $X = Spec A$ , there is therefore a right-invertible matrix $q : A^{2 + N} \to A^{N}$ whose associated kernel bundle is $E$ and which cannot be completed to an invertible $(2 + N) \times (2 + N)$ matrix.

External input

The deep construction and the assertion of algebraic nontriviality are explicitly imported from the precise authoritative subsequent statement. The local numbered steps expand the definition and translate nontriviality into a matrix-completion obstruction. No equations for the hypersurfaces or entries of a matrix are invented, and no original Kumar proof or independent Hermitian-K-theory invariant computation is claimed. The five preceding fully authored prerequisites give the construction/application interface and the actual Chern-class consequences.

Facts & Assumptions

Given: The data and hypotheses in the statement, AC and the article's fixed universe bounds where its K-theory is used.

Goal: Establish the stated conclusion.

[F1]

The exact established existence statement and the geometric description are imported from Brazelton–Opie–Syed Example 6.6, which attributes them to N. Mohan Kumar MK84/MK85.

[F2]

Stable triviality means an actual finite free stabilization. Stable triviality and stable bundle equivalence

[F3]

The local matrix-kernel construction and completion criterion are Stable triviality as the kernel of a right-invertible matrix and Matrix completion detects the actual nontrivial bundle.

Proof

1.1F1F2

Apply the imported existence statement over the explicit base field $C$ . It supplies the smooth affine fourfold X and the rank-two nontrivial stably trivial bundle E, with the stated geometric description. Expanding stable triviality via [F2] supplies a finite N and an actual isomorphism. Since E is nontrivial, N cannot be zero.

2.1F1F3step 1.1∎

Choose one such stable isomorphism. By [F3] it yields a right-invertible matrix q whose kernel sheaf is E. If q were completable, the completion criterion in [F3] would make E trivial, contradicting [F1]. Thus q is not completable. This matrix consequence is local algebra deduced from the imported bundle existence; it is not a claim that the source gives this matrix explicitly.

Depends on

Used by · 1

A nontrivial rank-two bundle with trivial Chern classes on a smooth complex affine fourfold

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Thomas Brazelton, Morgan Opie and Tariq Syed, On Algebraic Vector Bundles of Rank 2 over Smooth Affine Fourfolds · Example 6.6, printed/PDF pp. 33–34, opening existence assertion and p=2 geometric description; bibliography MK84/MK85, p. 36

There exists a smooth complex affine fourfold carrying a nontrivial but stably trivial algebraic vector bundle of rank two. The p=2 Mohan Kumar variety is described as the complement of a union of three hypersurfaces in P^3 x A^1.

Hypotheses: Base field C, algebraic vector bundles, smooth affine variety of complex algebraic dimension four, bundle of constant rank two. The general construction is stated for a prime p over an algebraically closed field; only the explicit p=2 complex case is imported here. C satisfies the section-6 standing characteristic restriction.

Reading limits: Freshly downloaded the author-hosted 37-page PDF to tmp/fourfolds-source.pdf in this run; SHA256 e7e83f41926c3b85502f5ca4e3b8f41e15f6945cd3d5746eece28afbd58f32ee. Read the title/author identification, the section-6 base-field convention, Example 6.6 on printed/PDF pp. 33–34, and the MK84/MK85 bibliography entries on p. 36. Rendered p. 33 was also inspected. The exact existence assertion and its p=2 geometric description are used as an authoritative external input. The nontriviality claim itself is imported, not established from an independently computed invariant here. Original Mohan Kumar papers MK84 and MK85, their defining equations, and original nontriviality proofs were not read. The Hermitian-K-theory discussion continuing on p. 34 was read only as context and is not used as a supplier; its cited Sye24 and the present paper's classification theorem proofs were not reconstructed.

Sources

Thomas Brazelton, Morgan Opie and Tariq Syed, On Algebraic Vector Bundles of Rank 2 over Smooth Affine Fourfolds · Example 6.6, printed/PDF pp. 33–34, opening existence assertion and p=2 geometric description; bibliography MK84/MK85, p. 36