If the projective automorphism group of a smooth complex projective algebraic variety $X$ has positive dimension, then $X$ admits a nonzero holomorphic vector field i.e. $H^0(X,T_X)\neq 0$. This is easy to prove using analytic methods: every subgroup of $PGL_{n}(\mathbf{C})$ of positive dimension contains a $Ga$ or a $Gm$; both give analytic vector fields on $\mathbf{P}^{n-1}(\mathbf{C})$ tangent to $X$.

There is probably an algebraic proof along the same lines which would work over any algebraically closed field (of any characteristic). I'd like to ask: is there a reference for that? Is there a generalization of this for smooth varieties which are not necessarily complete and/or are not necessarily embedded in any projective space?

upd: settled by BCnrd and Francesco Polizzi in the comments.

alwaysa direct limit of representable affine finite type subfunctors; this follows by the same method used to prove Lemma A.8.13 in the book "Pseudo-reductive groups". – BCnrd Aug 4 '10 at 2:46