Finding topological obstructions for a complex manifold to be Kaehler - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:06:28Z http://mathoverflow.net/feeds/question/93556 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93556/finding-topological-obstructions-for-a-complex-manifold-to-be-kaehler Finding topological obstructions for a complex manifold to be Kaehler Zhang Xiao 2012-04-09T09:24:26Z 2012-04-09T14:02:57Z <p>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.</p> <p>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:</p> <p>$b_2 \ne 0$ for the symplectic structure and many more;</p> <p>$b_{2k+1}$ are even and $b_{k-2} \leq b_k$ for $k \leq n$ for the Hodge structure, which date back to Lefschetz;</p> <p>Various integral results on Chern numbers by Hirzebruch-Riemann-Roch on Kaehler manifolds;</p> <p>and etc.</p> <p>To make the question precise, I would like to ask:</p> <p>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. </p> <p>Any comments are welcomed and thanks a lot!</p> http://mathoverflow.net/questions/93556/finding-topological-obstructions-for-a-complex-manifold-to-be-kaehler/93563#93563 Answer by Gunnar Magnusson for Finding topological obstructions for a complex manifold to be Kaehler Gunnar Magnusson 2012-04-09T11:19:37Z 2012-04-09T11:19:37Z <p>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.</p> <p>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 <a href="http://arxiv.org/abs/math/0307260" rel="nofollow">http://arxiv.org/abs/math/0307260</a> for the first version of this question).</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/93556/finding-topological-obstructions-for-a-complex-manifold-to-be-kaehler/93569#93569 Answer by Igor Rivin for Finding topological obstructions for a complex manifold to be Kaehler Igor Rivin 2012-04-09T14:02:57Z 2012-04-09T14:02:57Z <p>You might want to consult the really excellent book by Amoros,Burger, Corlete, Kotschick, Toledo called "Fundamental groups of compact Kahler manifolds".</p>