# Finding topological obstructions for a complex manifold to be Kaehler

Well, it is of the "straightforward" questions one may ask. I propose it here to see if someone could tell me more on the recent status of this quite long-standing problem.

To initiate, let me give a brief description of the "classical" invariants. Essentially they are provided by either the symplectic structure or the Hodge structure, most of which relating to vanishing theorems and integral theorems. As far as a compact manifold $M$ of complex dimension $n$ is concerned, we have the following:

$b_2 \ne 0$ for the symplectic structure and many more;

$b_{2k+1}$ are even and $b_{k-2} \leq b_k$ for $k \leq n$ for the Hodge structure, which date back to Lefschetz;

Various integral results on Chern numbers by Hirzebruch-Riemann-Roch on Kaehler manifolds;

and etc.

To make the question precise, I would like to ask:

Has there been any "higher" or essentially new invariants discovered so far? Particularly, one may observe that the above invariants are all torsion-free(as Yau did in his problem list), so the torsion invariants would be rather interesting, suppose they do exist.

Any comments are welcomed and thanks a lot!

-
This may not address your question as stated but it is known, by a famous result of Deligne-Griffiths-Morgan-Sullivan, that compact Kahler manifolds are formal. What that entails, for instance, is that all triple Massey products in the cohomology ring vanish. Sometimes, this may be useful in detecting if a complex manifold admits a Kahler structure. –  Somnath Basu Apr 9 '12 at 17:27
Somnath: My knowledge of rational homotopy theory is rather limited, certainly not adequate to understand all the mathematics evolved here. But after all many thanks！ –  Zhang Xiao Apr 10 '12 at 1:10

The cohomology ring of $X$ is probably a good place to look for candidates for such invariants. For example, let $$B = \lbrace \alpha \in H^{1,1}(X,\mathbb R) \mid \text{Vol}(X,\alpha) := \int_X \alpha^n/n! > 0 \rbrace$$ be the big cone of $X$. If $X$ is Kahler, then it contains the Kahler cone $K$ of $X$, but it is in general larger.

The big cone admits a (in general only pseudo-)Riemannian metric $g$, given by the Hessian of the smooth function $- \log \text{Vol}$. Conjecturally, the sectional curvature of this metric is seminegative if $X$ is Kahler (see Wilson's http://arxiv.org/abs/math/0307260 for the first version of this question).

So, suppose we have a compact manifold $X$ and that we know its cohomology ring and its intersection product. Then we can calculate the big cone and the Riemannian metric $g$ and check if its sectional curvature is seminegative. If not, then $X$ should not be Kahler.

Of course this may not be so easy to check in practice, but Wilson's article contains an interesting prototype of an example (Propositon 5.3) where one should obtain the non-existence of a certain type of Kahler structure on a differentiable manifold through these means, that one does not get by the more traditional invariants.

-
Gunnar: I really appreciate your response. Since the criterion mentioned is rather differential-geometric than topological, I am wondering whether the seminegativeness of the sectional curvature of this "moduli space" has some implications on the topology of the original manifold:) –  Zhang Xiao Apr 10 '12 at 1:17
Well, it has some implications, as in Wilson's example, but exactly what those are is very badly understood. One should also be careful to note that manifolds with wildly different topology can have the "same" Kahler cone and thus metric; f.ex. for a manifold with $h^{1,1} = 1$ the Kahler cone is a ray $\mathbb R_+$ and the metric is basically the Poincaré metric on the half-plane, restricted to the half-line $i\mathbb R_+$. Now, the collection of comact manifolds with $h^{1,1} = 1$ is immense, it includes all Riemann surfaces, and all smooth hypersurfaces in $\mathbb P^n$ ($n \geq 4$). –  Gunnar Magnusson Apr 10 '12 at 6:13
I am not familiar with the kaehler cone but all these stuffs sound rather interesting. By the way, personally I regard $h^{1,1}=1$ as a rather strong restriction since they are "immense" only in the cardinality sense but only consist of a few of "types":) –  Zhang Xiao Apr 10 '12 at 23:57

You might want to consult the really excellent book by Amoros,Burger, Corlete, Kotschick, Toledo called "Fundamental groups of compact Kahler manifolds".

-
...as well as Donu Arapura's survey "Fundamental Groups of Smooth Projective Varieties" (which almost entirely deals with Kahler groups). –  Misha Apr 9 '12 at 14:31
@Misha: very true... –  Igor Rivin Apr 9 '12 at 15:58
Nice reference. My thanks to both Igor and Misha. Actually without knowing something has been done on the fundamental group, what I have in mind is the second homotopy group, which by Hurewicz a refinement of $H^2(M)$，the latter being much more important in the Kaehler case than $H^1(M)$.Has any work been done on this direction? –  Zhang Xiao Apr 10 '12 at 1:23
Apart from the rational homotopy theory, I do not think there was anything done in this direction. A good introduction to RHT is "Rational homotopy theory and differential forms" by Griffiths and c Morgan. An interesting question would be to consider $\pi_2(M)\otimes {\mathbb Q}$ as a $\pi_1(M)$-module. The point is that much is known about $\pi_1$ and $\pi_2$, but, I do not think anybody looked (in the Kahler context) at the module. There are probably some interesting restrictions appearing here. Also, every fp group appears as $\pi_1$ of a compact complex manifold (C.Taues), but ... –  Misha Apr 10 '12 at 20:07
...his construction has nontrivial $\pi_2$: The complex 3-fold constructed by Taubes is a twistor bundle over some ASD 4-manifold with huge $\pi_2$. I do not know if there are any restrictions on the above module ($\pi_2$ as a $\pi_1$-module) in the context of compact complex manifolds. –  Misha Apr 10 '12 at 20:11