Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert polynomial $h$ is of finite type (even quasi-projective).
This means, in particular, that the moduli space has only finitely many connected components, and therefore, that there are only finitely many polarized varieties with Hilbert polynomial $h$ up to deformation.
Proving that the moduli space is of finite type is formally more difficult than the latter statement, I believe.
Question. Is there an "easy" proof for the finiteness of the number of deformation classes of polarized varieties with Hilbert polynomial $h$?