# Kähler metric on a Zariski open subset of a non-Kähler manifold

I may be missing an obvious example, but here goes...

Let $X$ denote a complex manifold of dimension $n$. If $X$ is Kähler, then the induced metric on any complex submanifold is also Kähler. If instead $X$ is non-Kähler, we can at least say that any coordinate neighborhood $(U,\varphi)$ inherits a Kähler metric from the induced Euclidean metric on an open subset of $\mathbb{C}^n$. If $X$ is already equipped with a metric, note that this metric on $U$ may not be compatible with it. However, I am unsure of what can be said about the metric structure of $U$ if instead we allow something a little more exotic. For example, what if we take $U$ to be the complement of an analytic set? This is the situation in which I am most interested.

My question is:

Is there an example of a non-Kähler manifold, $X$, and a Zariski open subset $U\subset X$, such that $U$ admits a Kähler metric?

Added: Francesco has provided a class of examples in dimensions $\geq 3$ and BS has provided an example in dimension 2 as a comment to Benoît's answer. I am going to write up BS's example, with a few minor additions, as an answer. In case he decides to write his own answer, I'll delete mine and encourage everyone to vote his up.

One can consider the following example.

A Moishezon manifold $M$ is a compact connected complex manifold such that the field of meromorphic functions on $M$ has transcendence degree equal to the complex dimension of $M$. Complex algebraic varieties have this property, but the converse is not true if the dimension is at least $3$.

In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kaehler metric. Now take a non-projective (hence non-Kaehler) Moishezon manifold $M$. One can prove that such a manifold admits a bimeromorphic modification $\pi \colon \widetilde{M} \to M$ such that $\widetilde{M}$ is projective.

This implies that there exist an analytic subset $S \subset M$ such that $U:=M - S$ is biholomorphic to an open subset of a projective variety (one can take as $S$ a $1$-codimensional analytic subset).

Hence $U$ admits a Kaehler metric.

For further details see the paper by Shanyu Ji Currents, metrics and Moishezon manifolds, Pacific J. Math. Volume 158, Number 2 (1993), 335-351.

• Any chance you'd know where to look for an example in dimension 2? It seems the above construction breaks down there because all Moishezon surfaces are Kahler, by the Chow-Kodaira Theorem. Jul 15, 2012 at 1:38
• You are welcome. Right, this construction cannot be applied in dimension $2$; one needs another idea. I should think about it. Jul 15, 2012 at 8:05
• More concretely, any smooth non-projective toric manifold is such an example. Use Chow's lemma you can find an open which is projective, so Kahler.
– temp
Aug 26, 2012 at 18:58

This example is due to BS -- see the edit to the original question.

For an example in dimension two, consider the Hopf surface $X = \left(\mathbb{C}^2\setminus \{0\}\right)/\mathbb{Z}$, where $\mathbb{Z}$ acts as $n\cdot(x,y) = (2^nx,2^ny)$. One can show that $X$ is diffeomorphic to $S^3\times S^1$. It follows that $X$ is non-Kähler because $H^2(X,\mathbb{Z}) = 0$. The $\mathbb{Z}$ action restricts to an action on $\mathbb{C}^*$, which is identified with $\mathbb{C}^*\times \{0\}$, and the smooth curve $E = \mathbb{C}^*/\mathbb{Z}$ of genus 1 includes into $X$ as a submanifold. $Y = X\setminus E$ is then biholomorphic to $E\times \mathbb{C}$ via the map $[(x,y)]\rightarrow \left([y],x/y\right)$. Thus $Y$ admits a Kähler structure. In fact, $Y$ is a quasiprojective variety, as it is isomorphic to an open subset of $E\times \mathbb{P}^1$.

More generally, the Hopf manifolds $X_n = \left(\mathbb{C}^n\setminus \{0\}\right)/\mathbb{Z},\ \ n\geq 2$, are all non-Kähler, as $X_n\simeq S^{2n-1}\times S^1$. The identification $\mathbb{C}^{n-1}\simeq \mathbb{C}^{n-1}\times \{0\}\subset \mathbb{C}^n$ gives rise to an embedding $X_{n-1}\subset X_n$ and, by an argument identical to the one above, we get an isomorphism $Y_n = X_n\setminus X_{n-1} \simeq E\times \mathbb{C}^{n-1}$. Again, $Y_n\subset E\times \mathbb{P}^{n-1}$ is quasiprojective. This gives an example in every dimension $n\geq 2$.

• This is only a slight nitpick: "... and the smooth curve $E$ of genus 1 includes canonically into $X$". In fact $X$ is the total space of a family of elliptic curves $E$ (all isomorphic between them) over the projective line, so the inclusion of $E$ into $X$ is not very canonical (unless we can agree on one point of $\mathbb P^1$---a homogeneous manifold---being more canonical than all the others!). Jul 19, 2012 at 4:46
• Good point. I'll tone down the wording a bit. Jul 19, 2012 at 5:38

You might be interested in the theory of complex manifolds of class C, due to Fujiki: a complex manifold is of class C if it is bimeromorphic to a compact Kaehler manifold. The concept was introduced in:

Fujiki, A., Closedness of the Douady spaces of compact Kahler spaces, Publ. RIMS, Kyoto Univ., 14 (1978), 1-52.

Class C manifolds have Kaehler metrics on Zariski dense open sets, by definition, but those metrics also compactify to compact manifolds, so class C is stronger than what you are looking for, apparently. Poon constructed self-dual metrics on compact 4-manifolds for which the twistor space is in Fujiki's class C, so there are interesting and important examples. to a compact Kahler manifold), but not Kahler.

Y. S. Poon, Compact self-dual manifolds with positive scalar curvature, J. Differential Geometry 24 (1986) 97-132.