Ttykuu

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 ($mathbbCP^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.) but 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.

Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.

For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.

In fact, the $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.

Moreover, by standard results on rigidity, any Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.