Examples of compact complex manifolds for which the $dd^c$ lemma does not hold

Question

The well known $dd^{c}$ lemma in complex geometry claimed that

Let $X$ be a compact Kähler manifold. Let $p,q\ge 1$. Let $\eta$ be a $(p,q)$ form on $X$ and assume $\eta$ is $d$-exact. Then there exists a $(p-1,q-1)$ form $\beta$ such that $$ \eta=dd^{c}\beta $$ If $p=q$ and $\eta$ is real, then we may take $\beta$ to be real.

(Source, Griffths and Harris, page 149, sign of $\beta$ corrected). The traditional proof breaks down for non-Kähler complex manifolds because in this case $\Delta_{\overline{\partial}}=\Delta_{\partial}=\frac{1}{2}\Delta_{d}$ may no longer holds. And then the lemma seems too good to be true in general.

I am wondering if we have examples of compact complex manifolds for which we know the $dd^{c}$ lemma does not hold. Here is a counter-example for an open domain. The Hodge decomposition still holds for any elliptic pseudodifferential operator, and we can still take its complex conjugate. But I am kind of lost how to construct a specific counter-example. The topic is classical, so I am not sure if this is common sense among expert circle already.

If the statement is true for certain non-Kähler complex manifolds, then I suspect we need to show that $$ (\eta, (\partial\overline{\partial}^{*}+\overline{\partial}^{*}\partial)\eta)\le 0, \forall \eta\in \mathcal{A}^{p,q}(M) $$ in these cases. I am wondering what does this imply for $M$ from an analysis perspective.

Answer 1

Gauduchon proved that a compact complex manifold satisfies the $dd^c$ lemma for $(1, 1)$-forms if and only if $b_1 = 2h^{0,1}$. As a compact complex surface is Kähler if and only if $b_1$ is even, a compact complex non-Kähler surface does not satisfy the $dd^c$ lemma.

The reference for the above result of Gauduchon is the paper

Gauduchon, P. La classe de Chern pluriharmonique d’un fibré en droites, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 9, Aii, A479–A482

a link to which can be found in the comments below. Better still, an elementary proof of this result is given in the exercises from a course taught by Valentino Tosatti. A link to the exercises, complete with solutions, also appears in the comments. Thanks to YangMills for both links.

Answer 2

A known consequence of the $dd^c$-lemma is the vanishing of Massey products, which are certain secondary cohomology operations, see Deligne, Griffiths, Morgan, Sullivan, Real homotopy theory of Kähler manifolds, Inventiones 1975. However, the Iwasawa manifold (cf Griffiths-Harris) is a compact complex manifold with nontrivial Massey products.

Answer 3

Below are just some more examples illustrating Donu's answer.

In the paper A new geometric construction of compact complex manifolds in any dimension Laurent Meersseman constructed a large family of complex non-Kahler manifolds. Each of this manifolds is (a deformation of) a principle torus bundle over toric variety.

The same manifolds are known in the toric topology under the name moment-angle-manifolds (see, e.g., Real quadrics in ℂn, complex manifolds and convex polytopes and Complex-analytic structures on moment-angle manifolds) and it is has been demonstrated that many of them have non-trivial triple Massey products.