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user97068
user97068

Finite generation Finiteness aspects of Deligne cohomology?

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user97068
user97068

Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.

For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{X},\mathbf{Z}(d))$ of $\mathcal{X}$, an extension of a finitely generated/a abelian group by a compact group in some (any, I don't know) suitable sense?

Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.

For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{X},\mathbf{Z}(d))$ of $\mathcal{X}$, finitely generated/a compact group in some (any, I don't know) suitable sense?

Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.

For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{X},\mathbf{Z}(d))$ of $\mathcal{X}$, an extension of a finitely generated abelian group by a compact group?

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user97068
user97068

Finite generation of Deligne cohomology?

Say $X$ is a smooth projective variety over $\mathbf{C}$, and $\mathcal{X} = X^{\rm an}$ its $\mathbf{C}$-analytic space.

For what integers $i,d$ is the Deligne cohomology $H^i_{\mathcal{D}}(\mathcal{X},\mathbf{Z}(d))$ of $\mathcal{X}$, finitely generated/a compact group in some (any, I don't know) suitable sense?