TITLE: Examples of non-Kähler compact complex manifolds which satisfy the Dolbeault isomorphism
QUESTION [6 upvotes]: Let $X$ be a complex manifold and let $$f_X^n : \bigoplus_{p + q = n} H^{p, q}(X) \to H^n_{\mathrm{DR}}(X, \, \mathbb{C})$$ be the natural map from the Dolbeault cohomology to the de Rham cohomology with complex coefficients. 

Question. Is there an example of a compact complex manifold $X$ which is not Kähler such that $f_X^n$ is an isomorphism for all $n$?

REPLY [4 votes]: I am slightly confused by Francesco's comment and answer, because the Frolicher spectral sequence does degenerate for compact complex surfaces. So why isn't a non-Kahler surface an example? Perhaps there is some subtlety in defining the groups $H^{p,q}(X)$ that has been swept under the rug. In any case, because the Frolicher spectral sequence degenerates at the $E_1$-page, Hodge decomposition (without Hodge symmetry!!) should hold, even for non-Kahler surfaces. An example would be the quotient of $\mathbb{C}^2\backslash 0$ by multiplication by $2$. This compact complex manifold is diffeomorphic to $S^1\times S^3$ so is non-Kahler (the first Betti number is odd). But it is the case that $$H^k(X,\mathbb{C})=\bigoplus_{p+q=k} H^{p,q}(X).$$ Also, there is a subtlety to phrasing your question, as in general, there is no natural map from Dolbeault to de Rham cohomology, only a spectral sequence from one to the other.