TITLE: Poincaré duality for Deligne (co)homology
QUESTION [5 upvotes]: My question is about the papers

Hélène Esnault and Eckart Viehweg, Deligne-Beı̆linson cohomology
Uwe Jannsen, Deligne homology, Hodge-D-conjecture, and motives

(both from the Beı̆linson's conjectures volume).
Let $X$ be a smooth complex variety. We may define Deligne-Beı̆linson cohomology groups
$$H^\bullet_\mathcal{D} (X, A (n)),$$
where $\mathbb{Z} \subseteq A \subseteq \mathbb{R}$, and $n \in \mathbb{Z}$ is some twist. (See the paper by Esnault and Viehweg.) There is also Deligne homology defined by Jannsen:
$$H_\bullet^\mathcal{D} (X, A (n)).$$
Jannsen establishes the "twisted Poincaré duality"
$$\tag{*} H^i_\mathcal{D} (X, A (n)) \cong H^\mathcal{D}_{2d-i} (X, A (d-n)),$$
where $d = \dim_\mathbb{C} X$.
Does it give some classic duality result as a particular case?

My guess is that it's supposed to be some generalization of the duality between singular cohomology and Borel-Moore homology. According to Lemma 1.11 from the same paper of Jannsen,
$$H^\mathcal{D}_\bullet (X, \mathbb{Z} (0)) \cong H^\mathrm{BM}_\bullet (X, \mathbb{Z}).$$
So that the above duality gives us as a particular case
$$H^i_\mathcal{D} (X, \mathbb{Z} (d)) \cong H^\mathrm{BM}_{2d-i} (X, \mathbb{Z}).$$
But what is the left hand side?
Edit: The above is false, I misread the statement of Lemma 1.11.

I hope someone can explain me the motivation behind (*).
Thank you.

REPLY [4 votes]: Deligne complex $A(p)$ is the complex 
$$A((2\pi i)^p) \rightarrow \mathcal O \rightarrow \Omega^1 \rightarrow {}...{} \rightarrow \Omega^{p-1}.$$
You can work with similar bigraded complexes, taking into account not only holomorphic, but also antiholomorphic forms. Let us define $A(p,q)$ as the complex of sheaves
$$A((2\pi i)^{p+q}) \rightarrow \mathcal O \oplus \overline{\mathcal O} \rightarrow \Omega^1\oplus \overline{\Omega^1} \rightarrow {}...{} \rightarrow \Omega^{p-1} \oplus \overline{\Omega^{p-1}} \rightarrow \overline{\Omega^p} \rightarrow  ... \rightarrow \overline{\Omega^{q-1}}.$$
(Here it's supposed that $q>p$). In this situation, we may take $A$ to be a subring of $\Bbb C$, not necessarily of $\Bbb R$. Suppose that $A = \Bbb C$.
Then we can write a resolution of this complex of sheaves by differential forms, and it's possible to see that $A(p,q)$ is quasiisomorphic to the complex
$$ ... \stackrel{d}{\rightarrow} \mathcal{A}^{p-2, q-1} \oplus \mathcal{A}^{p-1, q-2} \stackrel{d}{\rightarrow}
\mathcal{A}^{p-1, q-1} \stackrel{\partial\bar\partial}{\rightarrow}
\mathcal{A}^{p, q} \stackrel{d}{\rightarrow} \mathcal{A}^{p, q-1} \oplus \mathcal{A}^{p-1, q} \stackrel{d}{\rightarrow}{} ...$$
where the term $\mathcal{A}^{p-1, q-1}$ has the grading $p+q-2$.
(The $\partial\bar\partial$ operator arises because of local $\partial\bar\partial$-lemma. You can find the proof that these complexes are quasiisomorphic in M. Schweitzer's text https://arxiv.org/abs/0709.3528).
In particular, $\Bbb H^{p+q-2}(X,{~}A(p,q))$ is known as Aeppli cohomology 
$H_{Ae}^{p-1,q-1}$, and $\Bbb H^{p+q-1}(X,{~}A(p,q))$ is Bott-Chern cohomology $H_{BC}^{p,q}$. If your $X$ satisfies global $\partial\bar\partial$-lemma (for example, if it is compact Kaehler), then these are
just isomorphic to the corresponding Dolbeaut cohomology; on a general complex manifold, they are different.
The Deligne complex with real coefficients $\Bbb R(p)$ is isomorphic to the fixed points of complex conjugation on $\Bbb R(p,p)$, so your can take real forms in the Aeppli-Bott-Chern resolution and obtain the resolution for the real Deligne cohomology.
Now, from the looks of the resolution and a bit of harmonic theory one can conclude that there's a non-degenerate pairing between 
$\Bbb H^{k}(X,{~}\Bbb C(p,q))$ and $\Bbb H^{2n+1-k}(X,{~}\Bbb C(n+1-p,n+1-q))$ (at least when $X$ is compact, otherwise you need to take cohomology with compact support on one side). This is complex version of the duality on Deligne cohomology, and when $k=p+q$ it specializes to the well-known duality between Bott-Chern and Aeppli cohomology (and to Poincare duality when $X$ satisfies $\partial\bar\partial$-lemma).
I don't know what happens if you work with Deligne-Beilinson complex instead of Deligne complex (with logarithmic forms instead of regular), but I'd suppose that everything works the same and gives you the duality between Bott-Chern cohomology and Aeppli cohomology with compact support on an open part of your variety as a particular case.