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Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms?

N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O_X) =0$ for all $i>0$, $K_X$ is numerically trivial and $\mathrm{H}^1(X,T_X) =0$ (or equivalently $h^{2,1}(X) = 0$).

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  • $\begingroup$ Your definition is wrong. For a Calabi-Yau variety of dimension $n$, $h^n(X,\mathcal{O}_X)$ should be nonzero. $\endgroup$
    – Jason Starr
    Commented May 25, 2015 at 17:17
  • $\begingroup$ I assume you mean CY threefolds. Why do you think they would only have a finite automorphism group? $\endgroup$
    – Lev Borisov
    Commented May 25, 2015 at 17:21
  • $\begingroup$ @JasonStarr Sorry about that. I hope it's clear now. $\endgroup$
    – Khedir
    Commented May 25, 2015 at 17:23
  • $\begingroup$ @LevBorisov My apologies to you as well. I indeed meant threefolds. My problem is that I don't know what to expect to be honest. All I know is that the set of automorphisms fixing an ample line bundle is finite. But I guess this leaves open the possibility of automorphisms not fixing an ample line bundle, and I simply do not know how to construct a rigid CY threefold with such automorphisms. $\endgroup$
    – Khedir
    Commented May 25, 2015 at 17:25
  • $\begingroup$ You may want to ask Noriko Yui if anybody looked at infinite order automorphisms of rigid CY threefolds. She probably knows as many of these rigid CYs as anyone. $\endgroup$
    – Lev Borisov
    Commented May 25, 2015 at 18:12

1 Answer 1

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Yes (to the question in the title; no to the question in the first line). You can find an example in this paper by Oguiso and Truong. The variety ``$X$'' should do what you want.

Briefly, let $\omega = (1+\sqrt{3}i)/2$ and let $E$ be the elliptic curve $\mathbb C / (\mathbb Z + \omega \mathbb Z)$. Then $E$ has an automorphism $\tau$ of order $3$ given by multiplication by $\omega$. Let $X$ be the crepant resolution of $(E \times E \times E)/\tau$ with the diagonal action. Then $X$ is a Calabi-Yau threefold, with many automorphisms, induced by the action of $\textrm{SL}_3(\mathbb Z[\omega])$ on $E \times E \times E$. This $X$ is apparently rigid; there are refs in the paper.

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