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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 $h$?

Let $h$ be a polynomial. Then results of several authors (including 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 $h$?

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 $h$?

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There are only finitely many varieties up to deformation

Let $h$ be a polynomial. Then results of several authors (including 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 $h$?