# Vector fields on complete intersections

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.

1. 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.

2. 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.

3. It is easy to check that when $X$ is a curve, the answer is positive as well.

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I haven't checked the indexing completely but I think Thm 1.1 and Prop. 1.3 of SGA 7 II:Exp II does what you want (with possibly a small number of exceptions if I've got the indexing a little bit wrong). You have to use that $T^1_X=\Omega_X^{d-1}\bigotimes \omega_X^{-1}$, $d=\dim X$, and the formula for $\omega_X$ in terms of the degrees of the defining hyperplanes.
Dear Torsten -- thanks! Unless I'm mistaken, Proposition 1.3 is applicable as is when $n+1-\sum d_i-d_{min}<0$ where $d_i$'s are the degrees of the hypersurfaces, $d_{min}$ is the minimal degree and $n$ is the dimension of the ambient projective space. Maybe one can modify the argument so as to get the general statement. – algori Oct 11 '10 at 23:29
Dear Krampusz -- I've realized my last comment may have been ambiguous: $d_{min}$ is included in the sum, but then one has to subtract it once more to get the expression which one would like to be positive. – algori Oct 12 '10 at 5:55