Criteria for the existence of equivariant fibrations on algebraic surfaces and hyperkähler manifolds and equality of automorphisms up to powers: a dynamical viewpoint

Abstract

Let X be a projective surface or a hyperkähler manifold and G⩽Aut(X). We give a necessary and sufficient condition for the existence of a non‐trivial G‐equivariant fibration on X. We also show that two automorphisms gi of positive entropy and polarized by the same nef divisor are the same up to powers, provided that either X is not an abelian surface or the gi share at least one common periodic point. The surface case is known among experts, but we treat this case together with the hyperkähler case using the same language of hyperbolic lattice and following Ratcliffe or Oguiso.

On the Primitivity of Birational Transformations of Irreducible Holomorphic Symplectic Manifolds

“…In the case of surfaces, an automorphism being of parabolic type has a clear geometric interpretation (see [14], [15], or [9] for the birational case). We could expect the situation to be similar in the irreducible symplectic context; indeed, Hu, Keum and Zhang have proved a partial analogue to Theorem 3.8, see [22]: Theorem 3.9. Let X be a 2n-dimensional projective irreducible symplectic manifold of type K3 [n] or of type generalized Kummer and let f ∈ Bir(X) be a bimeromorphic transformation which is not elliptic; f is parabolic if and only if it admits a rational Lagrangian invariant fibration π : X P n such that the induced transformation on P n is biregular, i.e.…”

On the cohomological action of automorphisms of compact Kähler threefolds

“…Apart from the case of surfaces, the only situation where the answer is known (and affirmative) is that of irreducible holomorphic symplectic (or hyperkähler) manifolds of deformation type K3 [n] or generalized Kummer (see [HKZ15]); the proof uses the hyperkähler version of the abundance conjecture, which was proven in this context by Bayer and Macrí [BM14]. See also [LB17] for a converse statement, which holds for any irreducible holomorphic symplectic manifold.…”

Spectral interpretations of dynamical degrees and applications

“…Let f : X X be a dominant rational self-map of a projective normal variety X of dimension d defined over a field K of characteristic zero. We may define its k-th degree with respect to a given any ample class ω as the intersection product deg k,ω (f ) := (f * ω k • ω d−k ) ∈ (0, +∞) for any 0 ≤ k ≤ d. The growth of the sequence of degrees {deg k,ω (f n )} n is a fundamental invariant of f that governs many of its dynamical features like its entropy [DS04b,Gue05,DDG10,DTV10]; the presence of invariant fibrations [DN11,HKZ15,Ogu18,LB19b,BCK14]; or its periodic points [DNT17,Xie15]. It also controls the behaviour of heights of iterates of points when f is defined over a number field [KS16,MSS18a,MSS18b].…”