TITLE: Do varieties with ample canonical bundle have finite automorphism group in small characteristic?
QUESTION [12 upvotes]: Suppose $X$ is a smooth projective variety over a field $k$, with ample canonical bundle.  If $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>\dim(X)$ and $X$ lifts to $W_2(k)$ (thanks Olivier Benoist!), it's not hard to see that $X$ has finite automorphism group.
Proof.  Let $X\hookrightarrow \mathbb{P}\Gamma(X, \omega_X^{\otimes n})^\vee$ be a closed embedding.  Then the induced map $\operatorname{Aut}(X)\hookrightarrow PGL(\Gamma(X, \omega_X^{\otimes n})^\vee)$ is a closed embedding, so $\operatorname{Aut}(X)$ is finite type.  But the tangent space to $\operatorname{Aut}(X)$ at the identity is $\Gamma(X, T_X)$, which is trivial by Kodaira vanishing.  So $\operatorname{Aut}(X)$ is a $0$-dimensional finite type group scheme, hence finite. $\blacksquare$  
(An essentially identical argument avoiding the representability of $\operatorname{Aut}(X)$ may be found here.)
This proof breaks down for $\operatorname{char}(k)$ small:  $\Gamma(X, T_X)$ need not be trivial.  But I don't know of an example of a canonically polarized variety with infinite (thus necessarily positive-dimensional) automorphism group in small characteristic.

Is the automorphism group of a smooth variety with ample canonical bundle always finite, even if $\operatorname{char}(k)$ is small?

REPLY [16 votes]: A much more general result is proven in Martin-Deschamps, Lewin-Menegaux : Applications rationnelles séparables dominantes sur une variété de type général. Théorème 2 : if $X$ and $Y$ are smooth and proper, and if $Y$ is of general type, the set of dominant separable rational maps from $X$ to $Y$ is finite !
  What you want follows immediately: it is their Corollaire 4, and they attribute it to Matsumura.