Integrable compatible complex structures In reading Gauduchon's notes (cannot link them, anyway it is standard material) I ran into the following construction.
Let $(M, \omega_0)$ be a compact symplectic manifold which is fixed.
An almost complex structure $J$ on $M$ is said to be compatible with $\omega_0$ if $\omega_0( \cdot, J \cdot)$ is a Riemannian metric.
Let $\mathcal J$ be the space of such $J$'s and let $\mathcal J_\text{int}$ be the subspace of integrable ones.
Then a Riemannian metric and a complex structure are constructed on the infinite dimensional manifold $\mathcal J$ which turns out to be Kaehler and it is proved that $\mathcal J_\text{int}$ is a Kaehler submanifold, that is its tangent space at $J$, which is constructed by deriving the equation $N_J=0$, is invariant under the complex structure on $\mathcal J$.
My question is: can it happen that $\mathcal J_\text{int}$ is made by isolated points? I can see no way to say that its tangent space has positive dimension.
Thank you
David
 A: The space of integrable $J$ is acted on by the diffeomorphism group, making it infinite dimensional. Symplectomorphisms will preserve the compatibility, so the orbit of an integrable $J$ under the symplectomorphism group is still infinite dimensional. The quotient by the symplectomorphism group is not known to be even an orbifold, as far as I know.
A: Yes! It can happen and is quite important.
The Teichmüller space of a Riemann surface provides a counterexample to your discreteness assertion. See, eg. Tromba's book, Teichmüller Theory in Riemannian Geometry, p.10.  This space encodes the set of inequivalent complex structures on a fixed Riemannian manifold and positive dimension as soon as the genus is greater than 1.  
Even simpler, for genus 1, the two-torus, we know that  the set of moduli of complex structures is identified with the modular surface (upper half-plane)/$Sl(2,{\mathbb Z})$ which has complex dimension 1. To be  explicit, think of the standard torus as ${\mathbb R}^2/{\mathbb Z}^2$ endowed with the symplectic structure $\omega_0 = dx \wedge dy$ where $x, y$ are standard coordinates for ${\mathbb R}^2$.  Consider the usual description of the moduli space of elliptic curves, eg. from Ahlfor's text on complex analysis.  The Teichmuller space is the upper half plane 
and a point $\tau = \tau_0 + i \tau_1$ in this plane (so $\tau_1 > 0$)
gets sent to the elliptic curve  ${\mathbb C}/{\mathbb Z}1 \oplus {\mathbb Z} \tau$. The complex structure on this curve is induced by projection from the standard complex structure on ${\mathbb C}$.  Let $A$ be the real linear invertible shear which is the identity on the x axis and maps the unit y vector ${\partial}_x$ to $(\tau_0, \tau_1)$.  Then $A$ defines a diffeo.from our standard torus   onto this $\tau$-torus.  Pulling back the $\tau$-complex structure 
using $A^{-1}$ we get a new complex structure $J_{\tau}$ on our standard torus given by $J_{\tau}(\partial_x) = e_{\tau}$ and $J_{\tau}e_{\tau} = -\partial_x$
where $e_{\tau}= (-\tau_0/\tau_1)\partial_x + (1/\tau_1) \partial_y$.  The $J_{\tau}$ form a continuum of inequivalent complex structures on the standard torus.  (What about the metric condition? I compute the $\omega_0 ( \cdot, J_{\tau} \cdot)$  metric to be $(1/\tau_1) dx^2 + (\tau_0/\tau_1) dy^2$,  a little wierd. I would've expected a quadratic form which is positive definite iff   $\tau_1 > 0$, not  iff both $\tau_0 , \tau_1 > 0$... . Maybe someone else can shed light on this (?))  
To understand the  higher dimensional version of this construction, look up the buzzword is Abelian varieties.  
