An application of $p$-adic integration to the dynamics of a birational transformation preserving a fibration

TL;DR

This paper applies $p$-adic integration techniques to study the dynamics of birational transformations on projective manifolds with non-negative Kodaira dimension, revealing conditions under which certain automorphisms have finite order and characterizing dense orbits.

Contribution

It introduces a novel application of $p$-adic integration to analyze the dynamics of birational transformations preserving fibrations, establishing finiteness results and orbit characterizations.

Findings

Pseudo-automorphisms preserving a big line bundle have finite order.

The first dynamical degree characterizes transformations with Zariski-dense orbits on certain symplectic manifolds.

Application of $p$-adic integration provides new insights into birational dynamics.

Abstract

Let $f\colon X \dashrightarrow X$ be a birational transformation of a projective manifold $X$ whose Kodaira dimension $\kappa(X)$ is non-negative. We show that, if there exist a meromorphic fibration $\pi \colon X\dashrightarrow B$ and a pseudo-automorphism $f_B\colon B\dashrightarrow B$ which preserves a big line bundle $L\in Pic(B)$ and such that $f_B\circ \pi=\pi\circ f$, then $f_B$ has finite order. As a corollary we show that, for projective irreducible symplectic manifolds of type $K3^{[n]}$ or generalized Kummer, the first dynamical degree characterizes the birational transformations admitting a Zariski-dense orbit.