On the primitivity of birational transformations of irreducible symplectic manifolds

TL;DR

This paper proves that certain birational transformations on irreducible symplectic manifolds with eigenvalues greater than one in their induced action are primitive, meaning they do not preserve any non-trivial fibration, leading to Zariski-dense orbits.

Contribution

It establishes a criterion linking eigenvalues of the induced automorphism to the non-existence of invariant fibrations for these transformations.

Findings

Transformations with eigenvalues > 1 have no invariant fibrations.

Such transformations have Zariski-dense generic orbits.

Provides a dynamical criterion for primitivity in symplectic manifolds.

Abstract

Let $f\colon X\dashrightarrow X$ be a bimeromorphic transformation of a complex irreducible symplectic manifold $X$. Some important dynamical properties of $f$ are encoded by the induced linear automorphism $f^*$ of $H^2(X,\mathbb Z)$. Our main result is that a bimeromorphic transformation such that $f^*$ has at least one eigenvalue with modulus $>1$ doesn't admit any invariant fibration (in particular its generic orbit is Zariski-dense).