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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.

edited Aug 4 2010 at 21:42

asked Aug 4 2010 at 2:02

algori
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The standard observation to prove this is that the tangent space to the automorphism scheme at its identity point is the space of global vector fields (consider how to describe a $\mathbf{C}[\epsilon]$-automorphism of $X_ {\mathbf{C}[\epsilon]}$ which lifts the identity modulo $\epsilon$), and a non-etale locally finite type group scheme has a non-zero tangent space at the identity point. If you don't assume completeness for $X$ then the automorphism functor is generally not represented by a scheme (nor is it even an algebraic space); consider the automorphism functor of an affine space. – BCnrd Aug 4 2010 at 2:09

BCnrd: re the automorphisms of the affine space: true, but the affine space does have nonzero vector fields (and the automorphism functor has representable subfunctors). – algori Aug 4 2010 at 2:37

algori: in the absence of representability, how do you propose to replace the hypothesis that the automorphism scheme has positive dimension? Could demand that the automorphism functor contains a representable subfunctor locally of finite type with positive dimension, and then use the same argument as above, but is that satisfactory to you? Curiously, in the affine case the aut functor is always a 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 2010 at 2:46

BCnrd -- I'm not sure what the "right" generalization is. The problem is how to translate the notion of discrete. (But I guess the version you mention would suffice for most practical purposes). Re the observations on the automorphism schemes: is there a reference for that? – algori Aug 4 2010 at 3:22

algori: the interpretation of the tangent space to the identity of the automorphism functor is something I think I first learned from either examples near the end of Schlessinger's first paper on versal deformations (from his thesis) or perhaps from Grothendieck's Seminaire Cartan expose on Hilbert schemes (in which he discusses Aut schemes near the end, as an application). – BCnrd Aug 4 2010 at 3:51

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Automorphism groups and vector fields

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