Motivic Chern classes

PDF p. 2: Displayed motivic Chern classes; PDF p. 8: Section 3, first display

Assume the Axiom of Choice and the suitable universes used by the supplied constructions. Let $k$ be a field, $X$ a quasi-compact quasi-separated $k$-scheme, and $E$ a vector bundle of constant rank $n\ge 0$ on $X$. A vector bundle means a finite locally free ${\mathcal{O}}_X$-module. Write $H^{a,b}(X)$ for the supplied integral motivic cohomology $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a,b}(X,\mathbb{Z})$, with its natural bigraded ring structure.

If $n>0$, let $\pi :{\mathbb{P}}_X(E)\to X$ be the projective bundle of rank-one quotients of $E$, let $\mathcal{O}(1)$ be its tautological quotient line bundle, and put $\xi =c_1(\mathcal{O}(1))\in H^{2,1}({\mathbb{P}}_X(E))$, using the supplied orientation on line bundles. Define the motivic Chern classes $c_i(E)\in H^{2i,i}(X)$ for $1\le i\le n$ as the unique coefficients satisfying ${\xi }^n+{\sum }_{i=1}^n(-1{)}^i{\pi }^{\ast }{c}_i(E){\xi }^{n-i}=0.$ Set $c_0(E)=1\in H^{0,0}(X)$ and $c_i(E)=0$ for $i>n$. For the rank-zero bundle use the same conventions $c_0(E)=1$ and $c_i(E)=0$ for $i>0$, without using a projective bundle. The total Chern series of an actual bundle is $c_t(E)={\sum }_{i\ge 0}{c}_i(E)t^i$, a polynomial with constant term one.

For every $i\ge 0$ this construction defines a map $c_i:{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n(X)\to H^{2i,i}(X)$, where ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n(X)$ is the set of isomorphism classes of rank-$n$ bundles. It is natural under pullback of $k$-schemes. These definitions apply to singular as well as smooth schemes. The article also uses total Chern classes of virtual bundle classes; their extension and its Whitney identity are supplied separately and are not part of this definition.

1.1 The hypotheses of [F1] and [F2] hold for $X$ and $E$. When $n>0$, [F2] therefore supplies exactly one coefficient $c_i(E)$ in each indicated bidegree satisfying the displayed relation. The unit and zero classes used for $c_0$ and for indices above the rank exist in the bigraded ring; the separate rank-zero prescription is consequently defined without a projective-bundle assertion. These choices give a polynomial $c_t(E)$ with constant coefficient one. [A1, F1, F2, given]

2.1 An isomorphism $E\cong E^{\mathrm{\prime }}$ induces an isomorphism of quotient projective bundles over $X$ carrying the tautological quotient line to the corresponding quotient line. By naturality of its first Chern class, the two defining relations correspond. Their coefficients agree by uniqueness in [F2]. The prescribed zero and unit coefficients also agree, so each $c_i$ is well-defined on the set of isomorphism classes in [F3]. The same conclusion holds in rank zero by its explicit convention. [A1, F1, F2, F3, step 1.1]

3.1 For a morphism $f:Y\to X$ of qcqs $k$-schemes, the pullback bundle has rank $n$. For $1\le i\le n$, [F2] gives $f^{\ast }{c}_i(E)=c_i(f^{\ast }E)$; for $i=0$ or $i>n$ this follows from preservation of the unit and zero. The rank-zero case follows from the same conventions. Thus the maps are natural, and no smoothness hypothesis has been used. This completes the justification. [A1, F1, F2, step 1.1, step 2.1] ∎