Cycle class map in smooth quasi-projective varieties

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ be a closed subvariety of codimension $k$.

Q1. How to define a cycle class $[Z]\in H^k(X,\Omega_X^{k})$ ?

Q2. More general, which are necessary conditions to have a "good" cycle class map of this type? The "good" means it is coincide with the usual cycle class map when $X$ is smooth projectve complex variety. I mean that if $X$ can be define over arbitrary field or in the case $X$ is a variety but maybe not smooth quasi-projective.

• This looks rather strange; one usually defines cycle classes only for those cohomology theories that satisfy some sort of the homotopy invariance property (that fails for the cohomology you consider unless $X$ is also proper). May 19, 2015 at 20:39
• I find this in subsection 1.1 of Variations de structure de Hodge et zéro-cycles sur les surfaces générales, where the author give a cycle class in $H^2(X,\Omega_X^2)$ May 20, 2015 at 1:13
• In that paper $X$ is proper, so this just the standard definition.
– abx
May 20, 2015 at 5:13
• There does indeed exist a cycle class map in Hodge cohomology for any smooth variety. See, for example, the article by El Zein "Complexe dualisant et applications à la classe fondamentale d'un cycle".
– naf
May 20, 2015 at 9:08

2 Answers

Here's one way to obtain a cycle class, at least over a field $$k$$ of characteristic 0 where resolutions of singularities are available:

Let $$Z \subset X$$ be a subvariety of codimension $$c$$ (possibly singular), and let $$\pi: \tilde{Z} \to Z$$ be a resolution of singularities. This means $$\pi$$ is proper and birational, and $$\tilde{Z}$$ is smooth. Let $$f = \iota \circ \pi : \tilde{Z} \to X$$ where $$\iota: Z \to X$$ is the inclusion.

The (dual of the) differential $$d f^\vee: f^*\Omega_X^* \to \Omega_{\tilde{Z}}^*$$ induces homomorphisms of Hodge cohomology groups $$f^*: H^q(X, \Omega_X^p) \to H^q(\tilde{Z}, \Omega_{\tilde{Z}}^p) \text{ for all } p, q$$ Now, $$H^{\dim \tilde{Z}}(\tilde{Z}, \Omega_\tilde{Z}^{\dim \tilde{Z}}) \xrightarrow{\mathrm{Tr}, \simeq} k$$ via the trace map, and the composition $$H^{\dim \tilde{Z}}(X, \Omega_X^{\dim \tilde{Z}}) \xrightarrow{f^*} H^{\dim \tilde{Z}}(\tilde{Z}, \Omega_{\tilde{Z}}^{\dim \tilde{Z}}) \xrightarrow{\mathrm{Tr}, \simeq} k$$ is an element of $$H^{\dim \tilde{Z}}(X, \Omega_X^{\dim \tilde{Z}})^{\vee} \simeq H^{\dim X -\dim \tilde{Z}}(X, \Omega_X^{\dim X -\dim \tilde{Z}})$$ This isomorphism comes from Poincare duality -- since $$\Omega_X^{\dim \tilde{Z}} \otimes \Omega_X^{\dim X - \dim \tilde{Z}} \xrightarrow{\wedge} \omega_X$$ is a perfect pairing it induces an isomorphism $$\Omega_X^{\dim X - \dim \tilde{Z} } \simeq \mathrm{Hom}(\Omega_X^{\dim \tilde{Z}}, \omega_X)$$. Hence we've assigned a cohomology class, say $$\eta(Z) \in H^c(X, \Omega_X^c)$$ to $$Z$$. Well, strictly speaking we've assigned it to the morphism $$f : \tilde{Z} \to X$$; it'd take more work to prove $$\eta(Z)$$ is idependent of the resolution $$\pi : \tilde{Z} \to Z$$ (idea: any 2 resolutions of $$Z$$ can be dominated by a third).

An alternative approach without using resolution of singularities is given in Lipman's "Blue Book":

Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque No. 117 (1984)

See Chap. 3, Remark (ii) on page 39.

In short, for a $$d$$-dimensional variety $$V$$ there is a canonical map $$c_V \colon \Omega^n_V \to \omega_V$$ that provides an element in $$H^{N-d}_V(X,\Omega^{N-d}_X)$$ where $$V \hookrightarrow X$$ is the embedding of $$V$$ into a regular variety $$X$$ of dimension $$N$$. Take $$Z := V$$ in your notation. Of course there is map from cohomology with supports to usual cohomology.