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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?

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Any complex submanifold of a Kähler manifold is Kähler, sure. Are you allowed to change the complex structure on $H\times\mathbb C^k$ to look for a Kähler structure? For example $S^3\times S^1$ is compact and not Kähler (no dimension-2 homology so no symplectic form), but you can embed it as the standard spheres of $\mathbb C^2\times\mathbb C$, and cutting out origins you get an alternative Kähler structure on $S^3\times S^1\times\mathbb C$. –  Elizabeth S. Q. Goodman Feb 20 '12 at 7:11
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I haven't given this much thought, but would the Hopf surface with a point deleted work? –  David Speyer Feb 20 '12 at 7:29
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2 Answers

up vote 19 down vote accepted

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 $b_2(Y)\not=0$.

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Very very nice! –  diverietti Feb 20 '12 at 16:53
    
Oh right--so, no $\omega$-positive curves in an exact symplectic manifold, in particular the 2nd-homology obstruction still is useful for closed curves in an open Kähler manifold. Nice. –  Elizabeth S. Q. Goodman Feb 20 '12 at 17:34
    
the point is that often a non-kähler, and hence non-projective, manifold does not have any closed submanifold at all... so the hopf surface is a quite particular case... –  diverietti Feb 20 '12 at 17:44
    
@diverietti: How sure are we about that "often"? The main examples of non-Kahler manifolds - Hopf manifolds and the Iwasawa manifold - are torus fibrations and thus contain submanifolds. The only examples I know of manifolds that contain no submanifolds are general tori and certain hyperkahler manifolds, and both are Kahler. –  Gunnar Magnusson Feb 20 '12 at 21:59
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@Gunnar: you might also know about Inoue surfaces, which are compact complex non-Kahler surfaces of class VII with $b_2=0$, and which have no complex curves. Higher-dimensional versions of Inoue surfaces are the Oeljeklaus-Toma manifolds, which are non-Kahler and also have no closed complex subvarieties other than points, see arXiv.org/abs/1009.1101 –  YangMills Feb 21 '12 at 0:39
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From any compact non-Kähler surface $X$ remove a point $p$. You're left with a non-Kahler, non-compact surface. For the proof, see Théorème 2.3 in A. Lamari - Courants kählériens et surfaces compactes. First, by a theorem of Shiffman, a Kähler form on $X\setminus {p}$ extends as a closed positive current to all of $X$. Then, locally around $p$, the singularity at $p$ can be fixed by using convolutions to obtain a smooth Kähler form on $X$.

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Could you please give a reference of Shiffman's theorem? And spend some more words on the regularization part? It would be interesting! –  diverietti Feb 21 '12 at 7:52
    
Shiffman's extension theorem appeared in "Extension of positive line bundles and meromorphic maps", Invent. Math., 15(1972), 332-347. It is the Main Lemma on page 333. I think the regularization part is due to Miyaoka "Extension theorems for Kähler metrics", Proc. Japan Acad., 50(1974), 407-410. You might want to check this paper for details on the regularization. It also appears in Lamari's paper cited above. –  user20497 Feb 21 '12 at 9:21
    
Thanks! I'll take a look! –  diverietti Feb 21 '12 at 23:50
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