Not every complex manifold is a Kähler manifold (i.e. a manifold which can be equipped with a Kähler metric). All Riemann surfaces are Kähler, but in dimension two and above, at least for compact manifolds, there is a necessary topological condition (i.e. the odd Betti numbers are even). This condition is also sufficient in dimension two, but not in higher dimensions. Therefore the task of finding examples of compact complex manifolds which are not Kähler is reduced to topological considerations.
In the non-compact setting, we can also find such manifolds. For example, let $H$ be a Hopf surface, which is a compact complex surface which is not Kähler. Then for $k > 0$, $M_{k+2} = H\times\mathbb{C}^k$ is a non-compact complex manifold which is not Kähler – any submanifold of a Kähler manifold is Kähler, and $H$ is a submanifold of $M_{k+2}$. This generates examples in dimensions three and above. So I ask the following question:
Does anyone know of some (easy) explicit examples of non-compact complex surfaces which are not Kähler?
Best Answer
Following David Speyer's suggestion, let $X=\mathbb{C}^2-\{0\}/\lbrace(x,y)\mapsto (2x,2y)\rbrace$ be the standard Hopf surface. The image, $E$, of the $x$-axis is an elliptic curve. Remove a point of $X-E$ to get $Y$. The second Betti number $b_2(Y)=0$ because it is homeomorphic to $S^3\times S^1-pt$. If $Y$ were Kähler then $\int_E\omega\not=0$, where $\omega$ is the Kähler form, and this would imply that $b_2(Y)\not=0$.