For every even integer $n>2$, does there exist a smooth $n$ dimensional manifold $M$ that admits a complex structure but not a symplectic one?
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13$\begingroup$ Well, for $n>2$, yes. For example, when $n>2$ is even, $S^1\times S^{n-1}$ has a complex structure (the Calabi-Eckmann complex structure), but it cannot have a symplectic structure because it has vanishing second deRham cohomology. $\endgroup$– Robert BryantCommented Feb 10, 2015 at 20:20
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$\begingroup$ Then the Gromov result of Open manifolds can not be generalized in general? But on the other hand, Yau's conjecture that every almost complex manifold admits a complex structure. In particular if it is true, symplectic manifolds are a subset of complex ones and these Calabi-Eckmann manifolds asserts that are not equal is it true? $\endgroup$– user66943Commented Feb 10, 2015 at 21:13
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10$\begingroup$ Gromov's convex integration methods and h-principle methods in these cases depend on the manifold not having any compact component, so, no his results for open manifolds do not carry over to the compact case. It is not known whether a compact almost-complex (in particular, a symplectic) manifold of dimension at least 6 always has a complex structure; this is an open problem. Dimension 4 is, of course, different. $\endgroup$– Robert BryantCommented Feb 11, 2015 at 0:04
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