A lecture I heard had a remark  "There is a rich class of pseudohoplomorphic curves to a symplectic manifold with an almost complex structure (tamed by the symplectic structure). On the other hand, there are very few complex valued functions on a symplectic manifold". Can someone explain why? Or point me to some reference? I always wondered why people do not study complex valued functions that should be equally insightful.
I think that, instead of complexvalued functions $f:M\to\mathbb{C}$ you mean $J$holomorphic functions, i.e., complexvalued functions $f:M\to\mathbb{C}$ that satisfy $f'(x)(Jv) = i\,f'(x)(v)$ for all $v\in T_xM$. The complexvalued functions don't have anything to do with the almostcomplex structure $J$, while the $J$holomorphic functions clearly do.
The reason these latter aren't usually interesting on an almostcomplex manifold is that, while the equations for pseudoholomorphic curves form a determined elliptic system (and so have plenty of local solutions), the equations for $J$holomorphic functions form an overdetermined PDE system when the domain has (real) dimension greater than $2$, and the integrability of the underlying almost complex structure is exactly the condition that this overdetermined system be 'maximally compatible'. In fact, for most almostcomplex manifolds $(M^{2n},J)$ of real dimension $2n>2$, the sheaf of $J$holomorphic functions on $M$ is just the sheaf of constant functions, so there is nothing special to say about it.

$\begingroup$ Indeed, what I had in mind were Jholomorphic functions. Thanks for the answer $\endgroup$ – Divakaran Divakaran Jul 24 '17 at 17:55