More pseudoholomorphic curves than complex valued functions

Question

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.

Answer 1

I think that, instead of complex-valued functions $f:M\to\mathbb{C}$ you mean $J$-holomorphic functions, i.e., complex-valued functions $f:M\to\mathbb{C}$ that satisfy $f'(x)(Jv) = i\,f'(x)(v)$ for all $v\in T_xM$. The complex-valued functions don't have anything to do with the almost-complex structure $J$, while the $J$-holomorphic functions clearly do.

The reason these latter aren't usually interesting on an almost-complex 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 almost-complex 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.