TITLE: Birational automorphisms of canonical models
QUESTION [11 upvotes]: Let $X$ be a variety with canonical singularities such that $K_X$ is ample.
Do you have a reference of the fact that every birational map from $X$ to itself is biregular?
Thank you

REPLY [2 votes]: Let $R$ be the canonical ring of $X$. We know that $R$ is finitely generated. Furthermore 
$$X\cong Proj(R) = X_{can}$$
since $K_{X}$ is ample and $X$ has at most canonical singularities. Now, any birational automorphism $f:X\dashrightarrow X$ induces an automorphism of the canonical ring $R$ which in turns induces a biregular automorphism of $X$. So $f$ itself is biregular.
More generally, if $X,Y$ are projective varieties with $K_{X},K_{Y} $ ample and at most canonical singularities, then any birational map $f:X\dashrightarrow Y$ is indeed biregular.