In Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius' book Compact complex surfaces. Second edition p.142, there is a Lefschetz theorem on (1,1) classes for compact surface:
Theorem Let $X$ be a compact complex surface. Then the image of $Pic(X)$ in $H^2(X,\mathbb C)$ is $H^{1,1}(X)\cap i^*H^2(X,\mathbb Z)$. In other words: an element of $H^2(X,\mathbb C)$ is in the image of $Pic(X)$ if and only if it is "integral" and can be represented by a real closed (1,1)-form.
First, I think the definition of $H^{1,1}(X)$ is not clear, obviously, it can't be $H^{1,1}_{\bar\partial}(X)$ whose representatives are not necessarilly $d$-closed while the image of $Pic(X)$ in $H^2(X,\mathbb C)$ is closed.
But if we interpret it as a subgroup of $H^2(X,\mathbb C)$ which can be represented by $d$-closed (1,1) form, then we will have a stronger statement.
According to Chern's book Complex manifolds without potential theory. Second edition p.51-52.
For any complex manifold $X$, the image of $H^1(X,\mathcal O^*)\to H^2(X,\mathbb Z)$ is a subgroup of $H^2(X,\mathbb Z)$ which can be represented by a closed $(1,1)$ form.
My question is:
Since we already have a theorem holds for a complex manifold $X$ without any further assumptions, what's the role played by the properties of compactness, of dimension 2, Frölicher spectral sequence degenerating at $E_1$ as in BHPV's book?
