The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some explicit examples of compact complex manifolds which satisfy these restrictions, but are not Kähler (i.e. do not admit a Kähler metric).


In dimension 1, every compact complex manifold is Kähler ($d\omega$ is automatically zero). 


In dimension 2, a compact complex surface $X$ is Kähler if and only if the first Betti number $b\_1(X)$ is even. Therefore, if $h^{1,0}(X) = h^{0,1}(X)$, then $b\_1(X) = h^{1,0}(X) + h^{0,1}(X) = 2h^{1,0}(X)$ so by the aforementioned result, $X$ is Kähler.


It follows that any examples must be of dimension at least three. 


The only thing that has come to mind so far is Hironaka's example which is a deformation of Kähler manifolds such that the limiting fibre is not Kähler. As diffeomorphism type is preserved under deformations, I know that the limiting fibre has the same Betti numbers as a Kähler manifold (i.e. the odd Betti numbers are even), but I don't know if its Hodge numbers satisfy the desired symmetry.


**Added Later:** 


1. After consulting the book *Cohomological Aspects in Complex Non-Kähler Geometry* by Angella, I realised that the central fibre of Hironaka's deformation, which I will now call $X\_0$, is indeed an example. We just need to show that $h^{p,q}(X\_0) = h^{q,p}(X\_0)$.


First note that $X\_0$ is a Moishezon manifold and is therefore a proper modification of a projective manifold. As projective manifolds satisfy the $\partial\bar{\partial}$-lemma, so do Moishezon manifolds; see Theorem 5.22 of *Real Homotopy Theory of Kähler Manifolds* by Deligne, Griffiths, Morgan, and Sullivan. In the same paper, they show that if a compact complex manifold satisfies the $\partial\bar{\partial}$-lemma, the Hodge-Frölicher spectral sequence degenerates at the first step, see Theorem 5.21. This in turn equivalent to the equality (for each $k$) $$b\_k(X\_0) = \displaystyle\sum\_{p+q = k}h^{p,q}(X\_0),$$ so $$b\_k(X\_a) = \displaystyle\sum\_{p+q = k}h^{p,q}(X\_a)$$ for all $a$ in the base of the Hironaka deformation. The map $a \mapsto h^{p,q}(X\_a)$ is upper semicontinuous in general (see Corollary 9.19 of Voisin's *Hodge Theory and Complex Algebraic Geometry I* for example), but the map $a \mapsto b\_k(X\_a)$ is constant by Ehresmann's result. Due to the above equality, we see that $a \mapsto h^{p,q}(X\_a)$ is actually constant for all $p$ and $q$. Therefore, $$h^{p,q}(X\_0) = h^{p,q}(X\_a) = h^{q,p}(X\_a) = h^{q,p}(X\_0)$$ where $a \neq 0$ and we use the fact that $X\_a$ is projective and hence Kähler.


The above result can be summarised as follows:



> 
> Let $\mathcal{X} \to B$ be a deformation of compact complex manifolds with $0 \in B$. If $a \mapsto h^{p,q}(X\_a)$ is constant on $B\setminus\{0\}$, and the Hodge-Frölicher spectral sequence of $X\_0$ degenerates at the first step, then $a \mapsto h^{p,q}(X\_a)$ is constant on $B$.
> 
> 
> 


In particular, given a deformation $\mathcal{X} \to B$ such that $h^{p,q}(X\_a) = h^{q,p}(X\_a)$ for all $a \in B\setminus\{0\}$, and the Hodge-Frölicher spectral sequence of $X\_0$ degenerates at the first step, then $h^{p,q}(X\_0) = h^{q,p}(X\_0)$. If $X\_0$ is not Kähler (as in Hironaka's example), $X\_0$ is then an example of the type I was searching for.
2. If $h^{p,q} = h^{q,p}$ for all $p$ and $q$, then $b\_k$ is even for $k$ odd. So one may instead try to solve the easier problem of finding a non-Kähler manifold $X$ such that $b\_k(X)$ is even for $k$ odd. 


As above, in the one or two dimensional case, if $b\_k$ is even for $k$ odd, then $X$ is Kähler. In dimension three however, there is a simple example given by $S^3\times S^3$. A reference for the fact that $S^3\times S^3$ can be equipped with a complex structure is *Complex Structures on $S^3\times S^3$* by Fujiki. 


The Kunneth formula gives $b\_0 = 1$, $b\_1 = 0$, $b\_2 = 0$, $b\_3 = 2$, $b\_4 = 0$, $b\_5 = 0$, and $b\_6 = 1$. All odd Betti numbers are even, but $S^3\times S^3$ is not Kähler as $b\_2 = 0$. In fact, all even Betti numbers of a Kähler manifold are positive ($\omega^k$ is not exact), so one might try to find an example of a non-Kähler manifold with $b\_{2k} > 0$ and $b\_{2k+1}$ even. By the discussion in point one above, an example of such a manifold is the central fibre of the Hironaka deformation.



Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$
Every compact complex manifold satisfying the $\partial\overline{\partial}$-Lemma has such a property. Particular examples are given by - as you already said - Hironaka example (and, more in general, Moishezon manifolds and manifolds in class C of Fujiki), or some deformations of twistor spaces (see LeBrun, Poon, Twistors, Kähler manifolds, and bimeromorphic geometry. II, *J. Amer. Math. Soc.* 5 (1992), no. 2, 317–325).


On the other side: in Ceballos, Otal, Ugarte, Villacampa, [arXiv:1111.5873](http://arxiv.org/pdf/1111.5873.pdf), Proposition 4.3, you find a concrete example of a compact complex manifold with the symmetry of Hodge diamond you require. This example does not satisfy $\partial\overline{\partial}$-Lemma.


