Here is a question the answer to which I've been trying to locate for some time.
Let $X$ be a smooth projective complete intersection over an algebraically closed field $k$; assume that $X$ is not contained in a hyperplane. Is it true that $X$ admits no nonzero tangent vector fields, unless $X$ is a quadratic hypersurface or a genus 1 curve?
Here are some remarks.
In the case $char(k)=0$ the positive answer to this question is stated as Proposition 2.11 of "Derivations, automorphisms and deformations of quasi-homogeneous singularities" by J. Wahl (Proceedings of symposia in pure mathematics, vol. 40, part 2, 613-625). However, no proof is given there, it is only mentioned that this can be obtained by the same method as theorem 2.8 (whose proof is only briefly sketched). I was wondering if there is a more complete account.
When $X$ is a hypersurface, the answer is positive; two characteristic free proofs of that are given in Katz-Sarnak, Random matrices, Frobenius eigenvalues and monodromy, 11.6 and 11.7. Either proof may in principle generalize to complete intersections, but I don't see a straightforward way to do that.
It is easy to check that when $X$ is a curve, the answer is positive as well.