Automorphisms of 3-folds of general type acting trivially on cohomology

DOI10.1007/S10711-021-00619-WarXiv1906.11469MaRDI QIDQ6321137

Hang Zhao

Publication date: 27 June 2019

Abstract: Let $X$ be a minimal projective threefold of general type over $mathbbC$ with only Gorenstein quotient singularities, and let $mathrm{Aut}_{mathbbQ}(X)$ be the subgroup of automorphisms acting trivially on $H^{*}(X,mathbbQ)$. In this paper, we show that if $X$ is of maximal Albanese dimension, then $|mathrm{Aut}_{mathbbQ}(X)|leq6$. Moreover, if $X$ is nonsingular and $K_X$ is ample, then $|mathrm{Aut}_{mathbbQ}(X)|leq5$. Seeking for higher-dimensional examples of varieties with nontrivial $mathrm{Aut}_{mathbbQ}(X)$, we concern $d$-folds $X$ isogenous to an unmixed product of curves. If $d=3$, we show that $mathrm{Aut}_{mathbbQ}(X)$ is a $2$-elementray abelian group whose order is at most $4$ under some conditions on their minimal realizations. Moreover, each of the possible groups can be realized. If $dgeq3$, we give a sufficient condition for $mathrm{Aut}_{mathbbQ}(X)$ being trivial. Curiously, there exist examples of projective threefolds $X$ with terminal singularities and maximal Albanese dimension whose $mathrm{Aut}_{mathbbQ}(X)$ can have an arbitrarily large order.

Mathematics Subject Classification ID

(3)-folds (14J30) Automorphisms of surfaces and higher-dimensional varieties (14J50)

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