TITLE: Non-Kahler Complex manifolds
QUESTION [11 upvotes]: For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define cohomology classes $H^{p,q}_{\bar\partial}(M)$.
In general is there any relation between $h^r(M)$ and $h^{p,q}(M)$?

REPLY [15 votes]: Let $X$ be a compact complex manifold of complex dimension $n$. The Hodge-Frölicher spectral sequence starts with
$$
E_1^{p,q}=H^{p,q}(X,\mathbb C)
$$ 
and the limit term $E^{p,q}_\infty$ is the graded module associated  to a filtration of the de Rham cohomology group $H^{p+q}_{\text{dR}}(X,\mathbb C)$. In particular
$$
b_k=\sum_{p+q=k}\dim E_\infty^{p,q}\le\sum_{p+q=k}\dim E_1^{p,q}=\sum_{p+q=k}h^{p,q}.
$$ 
The equality is equivalent to the degeneration of the spectral sequence at the $E_1^\bullet$ level.
On the other hand, an elementary lemma on bounded complexes of finite dimensional vector spaces applied to $E^\bullet_r$, tells you that you always have equality for the (topological) Euler characteristic:
$$
\chi_{\text{top}}(X)=\sum_{k=0}^{2n}(-1)^kb_k=\sum_{p,q=0}^n(-1)^{p+q}h^{p,q}.
$$