Automorphisms and Dynamics: A List of Open Problems

Abstract

We survey a few results concerning groups of regular or birational transformations of projective varieties, with an emphasis on open questions concerning these groups and their dynamical properties.

Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings

“…Together with the log concavity of dynamical degrees d i (f ) (which follows from Khovanskii-Teissier's inequality), this implies that h top (f ) > 0 if and only if d i (f ) > 1 for some (and hence all) i ∈ {1, • • • , d − 1}. Thus the equivalence of the first four assertions follows from Kronecker's theorem, and also (5) implies these assertions.…”

“…Among the complex dynamical study of (X, f ), there has been a growing interest in compact Kähler manifolds with slow dynamics (see e.g. [5,6,12,9]), and typically, one studies the dynamics of automorphisms having zero entropy. In terms of cohomological data, f has zero entropy if and only if the first dynamical degree of f satisfies…”

“…To see that (2) implies (5), recall that in order to compute Plov(f ), by Lemma 3.4 we can replace f by some iteration of it, so that f * : H 1,1 (X) is unipotent. Hence Plov(f ) < ∞ is a consequence of Lemma 3.12 below.…”

Subgroups of elliptic elements of the Cremona group

“…These results leave open the question whether every linear algebraic group is the automorphism group of a normal projective variety. Further open questions are discussed in the recent survey [Can19], in the setting of smooth complex projective varieties; most of them address dynamical aspects of automorphisms, which are not considered in the present notes but play an important rôle in many developments.…”

Ricci-flat metrics and dynamics on K3 surfaces

“…The theory of holomorphic dynamics in 1 complex variable (on the Riemann sphere) is of course an enormous research area, and when one passes to 2 complex variables, it turns out that the only dynamically interesting automorphisms exist on K3 and rational surfaces (see [10] for the precise statement), and interesting K3 automorphism are relatively easy to construct. The dynamical study of such automorphisms was initiated by Cantat [11], and we refer the reader to the survey articles [12,13,14] and lecture notes [21] for a broader overview.…”