Kummer rigidity for hyperkähler automorphisms (Q6592860)

Let \(X\) be a projective hyperkähler manifold, i.e.~simply connected and with a unique (up to scalars) holomorphic symplectic form \(\Omega\). Let \(f \colon X \to X\) be an automorphism of positive topological entropy (= spectral radius of the induced map on cohomology \(\mathrm H^*(X, \mathbb C)\)). Suppose the volume measure (induced by \((\Omega \wedge \overline\Omega)^n\)) is the \(f\)-invariant measure of maximal entropy. Then \((X, f)\) is Kummer in the following sense: there is a complex torus \(T\) and a finite group \(\Gamma\) such that \(X\) is (a blowup of) \(T / \Gamma\), and there is a hyperbolic (= having positive topological entropy) affine-linear map \(A \colon T / \Gamma \to T / \Gamma\) which induces \(f\) in the sense that the natural diagrams involving \(X\), \(T\) and \(T / \Gamma\) commute.\N\NThis generalizes results of \textit{S. Cantat} and \textit{C. Dupont} [J. Eur. Math. Soc. (JEMS) 22, No. 4, 1289--1351 (2020; Zbl 1476.37071)] and \textit{S. Filip} and \textit{V. Tosatti} [Am. J. Math. 143, No. 5, 1431--1462 (2021; Zbl 1480.14025)] for complex surfaces.