Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an explicit example of such a pair $(X, f)$ such that:

1) The automorphism $f$ lifts to some non trivial first order deformation $\mathfrak{X}$ of the scheme $X$, that is some nonzero element of $\mathrm{H}^1(X, \mathrm{T}X)$ is fixed by the action of $f$.

2) Given any nontrivial deformation $\widetilde{\mathfrak{X}}$ of $X$ over a small extension $A$ of $\mathbf{k}[t]/t^2$ such that the pullback of $\widetilde{\mathfrak{X}}$ is $\mathfrak{X}$, then $f$ doesn't lift to $\widetilde{\mathfrak{X}}$.

edit: by nontrivial, I mean that $\widetilde{\mathfrak{X}}$ is not obtained by pull back from $\mathfrak{X}$ via a section of the map $A \rightarrow \mathbf{k}[t]/t^2$.


Choose a homogeneous polynomial $f(x_0,x_1, \ldots, x_n) = x_0^d+ f_1(x_1,\ldots, x_n)$ such that $V(f)$ is smooth. Then multiplication by a $d$th root of unity on $x_0$ gives an automorphism $\sigma$ of $X= V(f)$. By choosing $d$ large, we can assume that there are no vector fields. Now choose a second polynomial $g= x_0^d + g_1(x_1,\ldots)$ satisfying the same conditions. Let $h(x_0,\ldots)$ be a generic homogeneous degree $d$ polynomial. Then $$ Proj\, k[t]/(t^3)[x_0,\ldots, x_n]/(f + tg+ t^2h)$$ provides an example where the automorphism $\sigma$ extends to first order but not to second. I realize, after I answered it, that I missed the "any" in your condition 2. Your question is either very hard or trivially false (the automorphism always extends to $$\widetilde{\mathfrak{X}}= \mathfrak{X}\times_{Spec k[t]/t^2} Spec A$$ whenever $A\to k[t]/(t^2)$ has a section, but I suspect you meant to disallow this sort of family).

  • $\begingroup$ Yes, I forgot to say that $\widetilde{\mathfrak{X}}$ has to be nontrivial, I've edited the question. $\endgroup$ – Julien Grivaux Feb 5 '16 at 23:52

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