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By a result of Mori, we know that there are only finitely many deformation classes of dimension $n$ smooth projective Fano varieties, for any $n$. For curves we only have $\mathbb{P}^1$, while for surfaces we have a total of $10$ deformation classes of del Pezzos. There's also a classification for $n=3$, which IIRC gives around $200$ or so classes.

What are the best known bounds for the number of classes in general? In particular, can we prove that it grows faster than polynomially in $n$?

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    $\begingroup$ The numbers seem to grow pretty fast for toric Fanos (~smooth reflexive polytopes) - perhaps these are enough to answer the last question? $\endgroup$
    – gsvr
    Commented Oct 16, 2015 at 0:53
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    $\begingroup$ Actually I just realized that complete intersections already give a superpolynomial number. Presumably one can get better bounds with a bit more cleverness. $\endgroup$
    – dhy
    Commented Oct 16, 2015 at 1:24

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