\(n\)-dimensional projective varieties with the action of an abelian group of rank \(n-1\) (Q2822722)

Let \(X\) be a normal projective variety of dimension \(n\). In 2004 Dinh and Sibony have proved that every commutative subgroup of \(\mathrm{Aut}(X)\) whose non-trivial elements all have positive entropy has rank at most \(n-1\); this paper deals with the extremal case of rank \(n-1\).NEWLINENEWLINEThe main result of this paper is the following. Let \(X\) be a normal projective variety of dimension \(n\geq 3\), and assume that \(G=\mathbb{Z}^{n-1}\) acts on \(X\) in such a way that every non-trivial element has positive entropy. Assume furthermore that (i) \(X\) is not rationally connected, or that (ii) \(X\) admits an effective \(G\)-invariant divisor \(D\) such that the pair \((X,D)\) has at worst \(\mathbb{Q}\)-factorial klt singularities and \(K_X+D\) is pseudo-effective, or that (iii) \(X\) has at worst klt singularities and has no proper \(G\)-invariant subvarieties of positive dimension. Then \(X\) is \(G\)-equivariantly birational to the quotient of an abelian variety.