The question [Are automorphism groups of hypersurfaces reduced ?](https://mathoverflow.net/questions/10743/are-automorphism-groups-of-hypersurfaces-reduced) reminded me of the following related question that I have not seen discussed.


If $X$ is a smooth projective variety over an algebraically closed field $k$ of characteristic $p$ such that the canonical bundle $K\_X$ is ample, then is the automorphism group of $X$ reduced?


Some observations: 


1. It is equivalent to ask if $H^0(X,T\_X) = 0$.
2. When $X$ is a hypersurface in $\mathbb{P}^n$, this is exactly the question answered in the link above.
3. If $d := \mathrm{dim}(X) < p$ and $X$ lifts flatly to $W\_2(k)$, then the answer is "yes" as $H^0(X,T\_X) = (H^d(X, \Omega^1\_X \otimes K\_X))^\vee = 0$ by Kodaira vanishing (which holds because of Deligne-Illusie-Raynaud under these hypotheses).


Arguments or references or counterexamples would be very appreciated!



Automorphism groups of general type varieties
William Lang produced examples of surfaces of general type in positive characteristic with non-zero vector fields. Since surfaces of general type must have finite automorphism groups, this gives examples. See William Lang, "Examples of surfaces of general type with vector fields", Arithmetic and geometry, Vol. II, Progr. Math., 36, Birkhäuser, pp. 167–173


