Consider the problem of trying to identify which $n$-dimensional compact complex manifolds can be endowed with a Kähler metric.
$\underline{n = 1}:$ Any hermitian metric on a Riemann surface is a Kähler metric as the Kähler form is a two-form, and all two-forms are closed. (This is also true of non-compact Riemann surfaces).
$\underline{n = 2}:$ A necessary and sufficient condition for a complex surface to admit a Kähler metric is that its odd Betti numbers are even.
The condition that odd Betti numbers are even is necessary for a complex manifold to admit a Kähler metric, but not sufficient for $n \geq 3$. To see it is not sufficient, consider the example of Hironaka which gives a deformation of three-dimensional Kähler manifolds to a non-Kähler manifold. The Betti numbers are invariant under diffeomorphisms, so the central fibre of the deformation has odd Betti numbers even (as it is diffeomorphic to a Kähler manifold) but it is not itself Kähler.
In general, for $n \geq 3$, it is not easy to determine when a compact complex manifold can be equipped with a Kähler metric. In the simplest of these cases, $n = 3$, are we any closer to solving this problem?
What conditions (necessary or sufficient) are there for a three-dimensional compact complex manifold to admit a Kähler metric? How far are we from a single necessary and sufficient condition (like we have for $n = 1$ and $2$)?
Of course I'm interested in results which apply for all $n$, but I'm guessing that there are some results specifically for $n = 3$.
Best Answer
The main obstruction to existence of Kahler metric (in addition to Lefschetz SL(2)-action and Riemann-Hodge relations in cohomology) is homotopy formality: the cohomology ring of a Kahler manifold is related to its de Rham algebra by a chain of homomorphisms of differential graded algebras inducing isomorphisms on cohomology. This is proven by Deligne-Griffiths-Morgan-Sullivan in 1970-ies.
This is a very strong topological condition; for instance, no nilmanifold (except torus) is homotopy formal. There are symplectic nilmanifolds satisfying hard Lefschetz and the rest of Riemann-Hodge conditions for cohomology.
Another obstruction is existence of a positive, exact current. As shown by Peternell, all non-Kahler Moishezon manifolds admit a positive, exact (n-1,n-1)-current, hence they are not Kahler. However, Moishezon manifolds are homotopy formal ([DGMS]), and often satisfy the Riemann-Hodge. This argument is also used to prove that twistor spaces of compact Riemannian 4-manifolds are not Kahler, except CP^3 and flag space (Hitchin).
The sufficient condition in this direction is obtained by Harvey-Lawson: they proved that a manifold is Kahler if and only if it does not admit an exact (2n-2)-current with positive, non-zero (n-1, n-1)-part.
Finally, Izu Vaisman has shown that any compact locally conformally Kahler manifold (a manifold with Kahler metric taking values in a non-trivial 1-dimensional local system) is non-Kahler.
Also, a complex surface is Kahler if and only if its $b_1$ is even. This was known from Kodaira classification of surfaces, and the direct proof was obtained in late 1990-ies by Buchdahl and Lamari using the Harvey-Lawson criterion.