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I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$ there exists no irreducible submanifold $Y$ of $X$ of dimension $0 < \dim Y < \dim X$ which contains $x_0$.

Examples of these kinds of manifolds are very general complex tori and quotients thereof, and they're interesting because they give counterexamples to the Hodge conjecture in the analytic category.

I thought I'd take a look at these things, but I can't find any mention of "simple Kahler manifolds" either here or on google. Did I get the name wrong? Do any of you know what I'm talking about and know of some references?

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Non-algebraic Kahler manifolds are not as intensively studied as algebraic one, so one could not exclude that this terminology "simple" is not common. In a certain sense it is a bit contradictory, these manifolds don't seem to be simple at all :) –  Dmitri Jul 15 '10 at 10:03
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@Dmitri: this terminology is analogous to the use of simple in many branches of mathematics. An object is simple if it does not contain non-trivially another object of the same type. Simple groups are far from being simple. No one claims that that terminology is "contradictory". –  Sándor Kovács Feb 23 '11 at 22:55
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For reference: These manifolds have connections to formal logic, or more precisely to model theory. (For a start, see the links on Rahim Moosa's website math.uwaterloo.ca/~rmoosa). Apparently, having that few subspaces means that one can bring methods from formal logic to bear on questions from Kahler geometry. That certainly counts as my unexpected maths discipline connection of the week. –  Gunnar Þór Magnússon Mar 3 '11 at 19:13

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up vote 6 down vote accepted

A generic deformation of a Hilbert scheme of K3 and a generic torus have no subvarieties, hence they are "simple" in the above sense. For a torus it's well known, for a Hilbert scheme of K3 it's in my paper http://arxiv.org/abs/alg-geom/9705004

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This is a very interesting class of manifolds which, to my knowledge, has not been studied in any detail. One should be able to prove interesting structure theorems for such manifolds. For instance, I expect that the abelian category of analytic coherent sheaves on such manifolds is stable under deformation equivalence, i.e. if two such manifolds are deformation equivalent, they should have equivalent categories of coherent sheaves.

The one danger here is that the pool of examples may be very small. I guess, the first thing to look at is to find constructions of more examples. It will be interesting to find an example with a non-abelian infinte fundamental group. One can try to take a quotient of a torus by a freely acting finite group but I suspect that these are never simple.

Also, as Dmitry points out, the terminology is misleading, so if you are seriously thinking of working on these manifolds, now is the time to come up with a better name for them.

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Tony, I am not sure if I understand what is your expectation about these manifolds. What about non-algebraic K3 surfaces that don't have line bundles apart from the trivial one? This can be deformed to an algebraic manifold, so the abelian category is not stable (If you impose that the whole deformation stay in the class of "simple" manifolds, you can multiply this family of K3 by a simple complex torus) –  Dmitri Jul 15 '10 at 13:29
    
@Dmitri: I expect that if we have a smooth complex family (over a connected base) in which two fibers are compact simple Kahler manifolds, then the abelian categories of analytic coherent sheaves on these two manifolds are equivalent. Of course, you can not expect that this holds for all members of the family. For non-algebraic K3s with no curves this is actually known due to a theorem of Verbitsky. I am not sure if it is known for tori. Also, I am not sure if being 'simple' is enough. Perhaps one should require that we have no connected proper analytic subvarieties of positive dimension. –  Tony Pantev Jul 15 '10 at 13:52
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Hmm... I just assumed these manifolds had been studied before. I'm supposed to be doing my PhD so I don't think I'll take a serious look at them for two or three years, but I must admit I'm a bit curious. Most points not belonging to submanifolds seems like such an "oh shit" condition to have, it'd be interesting to see how you get a strong enough hold to prove anything about these manifolds. PS: on the name front, "simple" doesn't do it for me either. Maybe "sparse" is better? –  Gunnar Þór Magnússon Jul 15 '10 at 17:29

I've also never heard simple used in this way, so I expect you would have a hard time finding references. If you are really interested in pursuing this, you could ask Demailly. The condition looks interesting. Manifolds satisfying it are as far from projective algebraic manifolds as you can get. In particular, there are no nonconstant meromorphic functions on them. If you want another example, try a nonalgebraic K3 surface without an elliptic pencil.

One last comment: the Hodge conjecture is vacuously true for very general tori. The counterexamples, due to Voisin and Zucker, are tori but they are more subtle.

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A K3 without an elliptic pencil? I'll have fun verifying that on summer vacation. :) And thanks for correcting me on the Hodge conjecture. –  Gunnar Þór Magnússon Jul 15 '10 at 17:32
    
That was off the top of my head, so I hope I'm not leading on wild goose chase. –  Donu Arapura Jul 15 '10 at 17:45

These manifolds have actually been studied to some extent by Campana and Peternell in their series of papers "Towards a Mori theory on compact Kähler threefolds I, II, III". In those papers they mention a folklore conjecture (which can be found for example here), that says that every simple Kähler manifold of odd dimension $n>1$ must be Kummer (i.e. bimeromorphic to the quotient of a complex torus by a finite group).

In the paper number II Peternell shows that this conjecture in dimension $3$ follows from MMP plus abundance for Kähler threefolds. On the other hand in paper number III he shows that abundance does hold for Kähler threefolds with the possible exception of simple non-Kummer manifolds (which should not exist).

On the other hand more recent developments using model theory seem to suggest that in the even-dimensional case apart from Kummer manifolds the only other simple Kähler manifolds are in generically finite-to-finite correspondence with an irreducible hyperkähler manifold, like in Misha's answer.

Anyway, even after all this work it's unclear whether simple non-Kummer odd-dimensional manifolds exist or not. It's certainly an interesting problem, but probably a very hard one.

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Why doesn't a non-algebraic K3 (without an elliptic fibration) or more generally a deformation of a Hilbert scheme of a K3 (as Misha say in his reply) contradict this folklore conjecture? –  Jim Bryan Jul 24 '12 at 17:44
    
You're right, the conjecture is only for the odd-dimensional case. In the even-dimensional case one should also allow manifolds which are in generically finite-to-finite correspondence with irreducible hyperkahler manifolds. I edited the answer accordingly, thanks a lot! –  YangMills Jul 24 '12 at 20:11

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