Automorphism groups of Calabi-Yau manifolds of Picard number 2 (Q2922506)

The object of the paper under review are Calabi-Yau manifolds with Picard number 2. Let \(X\) be such a manifold and let \((x^n)_X\) be the top intersection form of \(X\), where \(n\) is the dimension of \(X\). The paper is devoted to study the finiteness of the automoprhism group \(\mathrm{Aut}(X)\); the main theorem shows that for an \(n\)-dimensional Calabi-Yau manifold \(X\) with Picard number 2, if \(n\) is odd then \(\mathrm{Aut}(X)\) is finite and if \(n\) is even \(\mathrm{Aut}(X)\) is finite provided that there is no real number \(c\) and no quadratic form \(q_X(x)\) on NS\((X)\) taking values in \(\mathbb R\) such that \((x^n)_X=c(q_X(x))^{n/2}\).NEWLINENEWLINESince the definition of Calabi-Yau manifold given in the paper includes projective hyperkähler manifolds, then the authors deals with hyperkähler manifolds of Picard number 2 and shows the relation between the automorphism group and the rationality of the boundary rays of the nef cone of \(X\).NEWLINENEWLINEA third part of the paper is devoted to Calabi-Yau threefolds. Let \(X\) be a Calabi-Yau threefold in the strict sense, i.e., a smooth simply connected projective manifold such that \(\mathcal O_X(K_X)\simeq \mathcal O_X\) and \(h^0(\Omega^k_X)=0\) for \(0<k<3\). Assuming that the cone conjecture proposed in [\textit{Y. Kawamata}, Int. J. Math. 8, No. 5, 665--687 (1997; Zbl 0931.14022)] and [\textit{D. R. Morrison}, Journées de géométrie algébrique d'Orsay, France, 1992. Paris: Société Mathématique de France, Astérisque. 218, 243--271 (1993; Zbl 0824.14007)] holds and that the Picard number of \(X\) is 2, the author shows that \(X\) contains a rational curve.