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Prescribing three Chern classes successively

PDF p. 17: Theorem B surjectivity construction

Statement

Assume the Axiom of Choice and the site and universe size bounds required by the cited motivic filtration and Chern-class constructions. Let $k$ be an algebraically closed field and $X$ a normal affine finite-type $k$ -scheme of Krull dimension three. Given classes $a_{i} \in H_{m o t}^{2 i, i} (X, Z)$ for $i = 1, 2, 3$ , there is a rank-three vector bundle $E$ with $c_{i} (E) = a_{i}$ for all three indices.

More precisely, choose a line bundle $L$ with $c_{1} (L) = a_{1}$ , set $α_{1} = [L] - 1$ , choose $α_{2} \in F_{m o t}^{2} K_{0} (X)$ with $c_{2} (α_{2}) = a_{2}$ , put $b_{3} = c_{3} (α_{1} + α_{2})$ , and choose $α_{3} \in F_{m o t}^{3} K_{0} (X)$ with $c_{3} (α_{3}) = a_{3} - b_{3}$ . Then $α_{1} + α_{2} + α_{3} = [E] - 3$ for a rank-three bundle $E$ , and its first three Chern classes are the prescribed classes. The sum lies in $F_{m o t}^{1} K_{0} (X)$ .

Here a vector bundle is a finite locally free $O_{X}$ -module, its rank is its local free rank, and a line bundle is a rank-one vector bundle. Normal means that all local rings are integrally closed domains; affine finite type means $X = Spec A$ for a finitely generated $k$ -algebra $A$ . Krull dimension is the supremum of lengths of strict chains of irreducible closed subsets. An algebraically closed field is one in which every nonconstant one-variable polynomial has a root. A perfect complex is a complex locally quasi-isomorphic to a bounded complex of vector bundles; $K_{0} (X)$ is the zeroth homotopy group of its nonconnective K-theory spectrum. A bundle gives the class of its degree-zero complex, $1 = [O_{X}]$ , and an integer denotes that multiple of $1$ . The subgroup $F_{m o t}^{j} K_{0} (X)$ is the image in $K_{0} (X)$ of the $j$ th stage of the supplied motivic filtration. The group $H_{m o t}^{2 i, i} (X, Z)$ is degree- $2 i$ cohomology of the weight- $i$ motivic complex; $c_{i}$ is the actual motivic Chern operation, agreeing on bundle classes with the bundle Chern class. Choice asserts that every family of nonempty sets has a choice function; the size bounds fix the sites and ambient universes used by the suppliers.

Facts & Assumptions

Given: The field, scheme, choice and size hypotheses above and the three target classes.

Goal: Carry out the stated successive choices and obtain a rank-three bundle with the specified classes.

[F1]

The first Chern class gives an isomorphism from the Picard group of line bundles to $H_{m o t}^{2, 1} (X, Z)$ . Line bundles and first motivic Chern class

[F2]

Under the given normal finite-type hypotheses, $c_{2}$ is surjective from $F_{m o t}^{2} K_{0} (X)$ to $H_{m o t}^{4, 2} (X, Z)$ . Second Chern surjectivity in dimensions two and three

[F3]

For the given normal affine finite-type threefold, $c_{3}$ is an isomorphism from $F_{m o t}^{3} K_{0} (X)$ to $H_{m o t}^{6, 3} (X, Z)$ . Third Chern isomorphism

[F4]

Every class in $F^{2}$ has pointwise rank zero and first Chern class zero. Every class in $F^{3}$ has rank zero and first and second Chern classes zero. Rank and determinant on the second piece The first two perfect Chern classes vanish on the third image piece in full scope

[F5]

On this nonempty affine finite-type threefold, each pointwise rank-zero perfect $K_{0}$ class is $[V] - 3$ for a rank-three bundle $V$ . The affine bridge identifies bundle and perfect $K_{0}$ classes compatibly. Critical-rank representatives of rank-zero classes Affine perfect K zero and bundle K zero

[F6]

The bundle total Chern series is multiplicative under direct sums, a line has no Chern classes above degree one, and the virtual series of a difference is the quotient of the bundle series. A trivial bundle has total class $1$ . Whitney identities and virtual Chern arithmetic

[F7]

The image filtration is decreasing, and the kernel of the zeroth stage projection is $F^{1}$ . On this normal affine finite-type scheme the zeroth projection is pointwise rank. Motivic filtration and image groups Stage projections, kernels and image groups Zeroth projection is rank

[F8]

The Axiom of Choice and the indicated site and universe bounds are the assumptions used in the cited constructions. The Axiom of Choice

Proof

technique · Choose the classes in successive filtration pieces, represent the higher pieces by rank-three bundles, and calculate using Whitney multiplication

1.1F1F2F3F7F8givenchoose

By [F1], choose $L$ with $c_{1} (L) = a_{1}$ and put $α_{1} = [L] - 1$ . Its pointwise rank is zero, so [F7] places it in $F^{1}$ . By [F2], choose $α_{2} \in F^{2}$ with $c_{2} (α_{2}) = a_{2}$ . Put $b_{3} = c_{3} (α_{1} + α_{2})$ . By [F3], choose $α_{3} \in F^{3}$ with $c_{3} (α_{3}) = a_{3} - b_{3}$ . These choices use the hypotheses and bounds in [F8].

2.1F4F5F6step 1.1

By [F4], $α_{2}$ and $α_{3}$ have pointwise rank zero. The dimension assumption makes $X$ nonempty, so [F5] gives rank-three bundles $E_{2}, E_{3}$ with $α_{i} = [E_{i}] - 3$ for $i = 2, 3$ . Since the trivial bundle has total Chern class $1$ , [F6] identifies $c_{j} (E_{i})$ with $c_{j} (α_{i})$ for these virtual classes. In particular $c_{1} (E_{2}) = 0$ , $c_{2} (E_{2}) = a_{2}$ , $c_{1} (E_{3}) = c_{2} (E_{3}) = 0$ , and $c_{3} (E_{3}) = a_{3} - b_{3}$ .

3.1F5F6step 1.1step 2.1algebra

Let $V = E_{3} \oplus E_{2} \oplus L$ , a rank-seven bundle. In $K_{0} (X)$ , $α_{1} + α_{2} + α_{3} = [V] - 7$ . By [F6] and step 2.1, the total class of $E_{3}$ through degree three is $1 + c_{3} (E_{3}) t^{3}$ , and that of $E_{2} \oplus L$ begins $1 + a_{1} t + a_{2} t^{2} + b_{3} t^{3}$ . The latter equality follows because $α_{1} + α_{2} = [E_{2} \oplus L] - 4$ and trivial summands do not change total Chern classes. Multiplying these series through degree three gives $c_{1} (V) = a_{1}$ , $c_{2} (V) = a_{2}$ , and $c_{3} (V) = b_{3} + c_{3} (E_{3}) = a_{3}$ .

4.1F5F6F7step 1.1step 3.1∎

The class $[V] - 7$ has pointwise rank zero. Applying [F5] gives a rank-three bundle $E$ with $[E] - 3 = [V] - 7 = α_{1} + α_{2} + α_{3}$ . The affine bridge and the representative-independent virtual Chern series in [F5] and [F6] imply $c_{j} (E) = c_{j} (V) = a_{j}$ for $j = 1, 2, 3$ , since trivial bundles have total class $1$ . Finally, $α_{1} \in F^{1}$ by step 1.1 and $F^{3} \subseteq F^{2} \subseteq F^{1}$ by [F7]; hence their sum lies in $F^{1}$ . This proves the construction.

Source notation

The article first identifies the sum with $[E_{3} \oplus E_{2} \oplus L] - 7$ . Its final two occurrences use $E_{1} \oplus E_{2} \oplus L$ , although the preceding rank-three bundle is called $E_{3}$ . The calculation above retains the defined $E_{3}$ .

Depends on

Used by · 1

Motivic Chern classification on normal affine threefolds

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. 17, proof of Theorem B, surjectivity

The three motivic Chern classes of a rank-three bundle can be prescribed successively.

Hypotheses: A normal affine threefold of finite type over an algebraically closed field.

Reading limits: The indicated proof was read in the supplied extracted article text. The named completed suppliers are used on their stated contracts; their proofs are not independently reconstructed here.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 17, proof of Theorem B, surjectivity