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Rank-two bundles with prescribed first two classes

PDF p. 1: Abstract; PDF p. 3: Corollary D; PDF p. 17: Corollary 6.10

Statement

Assume the Axiom of Choice and the site and universe size bounds required by the cited motivic filtration, support and Chern-class constructions. Let $k$ be an algebraically closed field of arbitrary characteristic and let $X$ be a normal affine finite-type $k$ -scheme of Krull dimension three. Then the map $(c_{1}, c_{2}) : {V e c t}_{2} (X) ⟶ H_{m o t}^{2, 1} (X, Z) \times H_{m o t}^{4, 2} (X, Z)$ is surjective.

Here an algebraically closed field is a field in which every nonconstant polynomial in one variable has a root. Affine and finite type mean $X = Spec A$ for a finitely generated $k$ -algebra $A$ ; normal means every local ring is an integrally closed domain; and Krull dimension is the supremum of lengths of strict chains of irreducible closed subsets. A vector bundle is a finite locally free $O_{X}$ -module, and ${V e c t}_{2} (X)$ is the set of isomorphism classes of bundles locally isomorphic to $O_{X}^{2}$ . The group $H_{m o t}^{2 i, i} (X, Z)$ is degree- $2 i$ cohomology of the weight- $i$ integral motivic complex, and $c_{i}$ is the corresponding motivic Chern-class operation on bundles. Choice says every family of nonempty sets has a choice function; the size bounds fix the sites and universes used by the suppliers.

Facts & Assumptions

Given: $k$ and $X$ as in the statement, the Axiom of Choice, the stated size bounds, and a pair $(a_{1}, a_{2})$ in the displayed target.

Goal: Produce a rank-two bundle $E^{'}$ with $c_{i} (E^{'}) = a_{i}$ for $i = 1, 2$ .

[F1]

Under these field, scheme, Choice and size hypotheses, every triple in $\prod_{i = 1}^{3} H_{m o t}^{2 i, i} (X, Z)$ is the first three Chern classes of a rank-three bundle. Motivic Chern classification on normal affine threefolds The Axiom of Choice

[F2]

Under the same hypotheses, a rank-three bundle with third Chern class zero is isomorphic to $E^{'} \oplus O_{X}$ for some rank-two bundle $E^{'}$ . Third Chern vanishing and a trivial summand

[F3]

Chern classes are invariant under bundle isomorphism, vanish above the rank, and have $c_{0} = 1$ . The total Chern polynomial satisfies $c_{t} (V \oplus W) = c_{t} (V) c_{t} (W)$ , and first Chern classes of line bundles are additive under tensor product. Motivic Chern classes Whitney identities and virtual Chern arithmetic

Proof

technique · Prescribe a zero third class, split off a trivial line, and compare Chern polynomials

1.1F1given

Apply [F1] to $(a_{1}, a_{2}, 0)$ . There is a rank-three bundle $E$ with $c_{1} (E) = a_{1}$ , $c_{2} (E) = a_{2}$ , and $c_{3} (E) = 0$ . The field, dimension, normality, finite-type, Choice and size hypotheses required by [F1] are precisely those given.

2.1F2givenstep 1.1

Apply [F2] to $E$ . Its hypotheses hold by the statement and step 1.1, so $E ≅ E^{'} \oplus O_{X}$ for a rank-two bundle $E^{'}$ .

3.1F3algebrastep 2.1

By the line-bundle tensor identity in [F3], $c_{1} (O_{X}) = 2 c_{1} (O_{X})$ because $O_{X} \otimes O_{X} ≅ O_{X}$ . Cancellation in the cohomology group gives $c_{1} (O_{X}) = 0$ . Its higher Chern classes vanish by rank, and $c_{0} = 1$ ; hence $c_{t} (O_{X}) = 1$ .

4.1F3step 1.1step 2.1step 3.1∎

Using the isomorphism from step 2.1 and the Whitney identity, $c_{t} (E) = c_{t} (E^{'}) c_{t} (O_{X}) = c_{t} (E^{'})$ . Comparing coefficients of $t$ and $t^{2}$ and using step 1.1 gives $c_{1} (E^{'}) = a_{1}$ and $c_{2} (E^{'}) = a_{2}$ . Thus every target pair has a preimage, proving surjectivity.

Depends on

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

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 1, abstract; p. 3, Corollary D; p. 17, Corollary 6.10 and its proof

The first two motivic Chern classes of rank-two bundles give a surjection on a normal affine threefold over an algebraically closed field.

Hypotheses: Normal affine threefold over an algebraically closed field, under the article’s conventions.

Reading limits: The supplied extracted article text at Corollary D and Corollary 6.10 and its proof was read. The classification and splitting arguments are used through the named completed suppliers, not independently reconstructed here.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 1, abstract; p. 3, Corollary D; p. 17, Corollary 6.10 and its proof