TITLE: Examples of smooth manifolds admitting inbetween one and a continuum of complex structures
QUESTION [15 upvotes]: For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).
For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($\mathbb{CP}^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.) 

Q. Are there examples admitting only finitely many or a countably infinite number? 

By deformation theory the tangent space to the moduli space of complex structures on $X$ should be given by $H^1(X, TX)$ (at least morally) so it must be necessary for this to vanish for every possible complex structure on $X$ to have any hope.

REPLY [17 votes]: There are countably many complex structures on $S^2 \times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_{2k}$ are the only options. This is the main result of 
Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.
The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.