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Fix a smooth projective variety $X$ over the complex numbers.

We write $H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch's higher Chow groups.

Notation

For a field $k$, recall $\Delta^n_{k} := \text{Spec}k[t_0,t_1,\ldots,t_n]/(t_0+\ldots+t_n-1)$, and that Bloch defines $z^d(X, 2d-*)$ to be the complex of abelian groups whose $n$th $\mathbf{Z}$-module is defined to be the subgroup of the free abelian group of cycles on $X\times\Delta^n_k$ that, in addition, are "in good position" (see [Bl]). It turns out the assignment $T\mapsto z^d(T, 2d-*)$ is a sheaf of complexes of abelian groups on $X_{\rm \acute{e}t}$. Define $H^n_L(X,\mathbf{Z}(d))$ to be its $n$th étale hypercohomology.

There is a cycle class map to Deligne cohomology $$c_{n,d}: H^n_L(X,\mathbf{Z}(d)) \to H^n_{\mathcal{D}}(X,\mathbf{Z}(d)).$$

One can endow the Deligne cohomology groups with the structure of complex analytic spaces $\mathcal{H}^n_{\mathcal{D}}(X,d)_{/\mathbf{C}}$ in a canonical way.

Let $k\subset\mathbf{C}$ be the field of definition of $X$, a finitely generated extension of $\mathbf{Q}$, ie. $X = (X_0)_{\mathbf{C}}$ for a smooth projective variety $X_0$ over $k$.

We have a natural map $H^n_L(X_0,\mathbf{Z}(d))\to H^n_L(X,\mathbf{Z}(d))$.

Questions.

(1) Does there exist a scheme $\mathcal{H}^n_L(X_0,d)$ such that $\mathcal{H}^n_L(X_0,d)(k) = H^n_L(X_0,\mathbf{Z}(d))$, $\mathcal{H}^n_L(X_0,d)_{\mathbf{C}}(\mathbf{C})=H^n(X,\mathbf{Z}(d))$, the natural map above is induced by $k\to\mathbf{C}$ by functoriality, and such that there is a morphism (an isomorphism?) of complex analytic spaces:

$$\mathcal{H}^n_L(X,d)^{\rm an} \to\mathcal{H}^n_D(X,d) ?$$

I'd expect the answer to be no, given the case $(n,d) = (3,1)$ (ie. the Brauer group functor is not representable).

(2) Do there exist a higher(?) (group) schemes $\mathcal{Z}_L^*(X_0,d), \mathcal{Z}_L^*(X,d)$ such that the $n$th homotopy of $\mathcal{Z}_L^*(X_0,d)$ is $H^n_L(X_0,\mathbf{Z}(d))$, the $n$th homotopy of $\mathcal{Z}_L^*(X,d)_{\mathbf{C}}$ is $H^n(X,\mathbf{Z}(d))$, the analytification $\mathcal{Z}_L^*(X,d)^{\rm an}$ is a higher analytic (group) space with a morphism (an isomorphism?) of higher analytic spaces:

$$\mathcal{Z}^*_L(X,d)^{\rm an}\to\mathcal{Z}^*_{\mathcal{D}}(X,d)$$ with the right side being a higher analytic space whose $n$th homotopy is $H^n_{\mathcal{D}}(X,\mathbf{Z}(d))$.

$\mathcal{Z}^*_{\mathcal{D}}(X,d)$ does exist, as suggested in the notes [DLitt].

Example.

When $n = 2d$, let $\text{Chow}^d(X_0)$ be the Chow variety of codimension $d$ cycles on $X_0$. This almost plays the role of $\mathcal{H}_L^{2d}(X_0,d)$ (up to replacing étale hypercohomology with Zariski hypercohomology at the beginning of this post).

I will benefit from any references.

References.

[Bl] Bloch, S. Algebraic cycles and higher K-theory

[DLitt] https://static1.squarespace.com/static/57bf2a6de3df281593b7f57d/t/57bf63e33e00be08153d8c0a/1472160739280/homotopicalcycleclasses.pdf

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    $\begingroup$ So for $n=2d$ (the classical case, corresponding to the usual Chow group), $c_{n,d}$ would be zero??? The Chow variety parametrizes positive cycles. $\endgroup$ – abx Jan 4 '18 at 5:27
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    $\begingroup$ I fail to see any possible relation between the bigrading of the Deligne (or motivic) cohomology and the degree of the cycles involved. $\endgroup$ – abx Jan 4 '18 at 8:36
  • $\begingroup$ May I suggest to change the title into "homotopical enhancements of cycle class maps"? Seems appropriate. Maybe the author of the notes thought about this and made it work? @Daniel Litt $\endgroup$ – user95222 Jan 4 '18 at 9:46

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