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Let $(M^{2n},\omega)$ be a symplectic manifold with an integral symplectic form $\omega$. Due to the work of M.Gromov and D.Tischler (M.Gromov "A topological technique for the construction of solutions of differential equations and inequalities", D.Tischler "Closed 2-forms and an embedding theorem for symplectic manifolds"), there exists a symplectic embedding $$ (M,\omega) \rightarrow (\mathbb{C}P^{2n+1},\omega_{FS}),$$ where $\omega_{FS}$ denote by the Fubini-Study form on the projective space. For example, Kodaira-Thurston manifold is a symplectic submanifold of $\mathbb{C}P^5$.

My questions are as follows :

  1. Is there an example of non-Kaehler symplectic manifold $(M,\omega)$ which can be embedded into $\mathbb{C}P^n$ for some $n \leq 4$? (There is no restriction of the dimension of $M$.)

  2. Is there an example of non-Kaehler symplectic manifold $(M,\omega)$ of dimension $2n$ which can be embedded into $\mathbb{C}P^{n+1}$? (I mean, $M$ is a submanifold of codimension 2)

I really appriciate for your any comments.

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If you are happy with just an opinion, it looks to me that both question might be open. I would think, that the answer to 1 should be positive, i.e., there should be a conter-example, a 4-dimensional symplectic non-Kahler submanifold in $\mathbb CP^4$. The only positive statement for questions of this type that I know, is that it is expected that a symplectic surface in $CP^2$ is symplecticly isotopic to an aglebraic curve. – Dmitri Mar 27 '11 at 21:12

You can embed T^2 into CP^2, and I think by a suitable fibering construction, you should be able to embed the Kodaira-Thurston symplectic non-Kaehler manifold symplectically into a CP^2-bundle over CP^2. I think the bundle can be made trivial, yielding CP^2 \times CP^2 (have not checked the details). Obviously, it's hard to say what the cohomology class of the resulting symplectic form would be.

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