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

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.