Some dynamical properties of pseudo-automorphisms in dimension 3

Abstract

Let X X be a compact Kähler manifold of dimension 3 3 and let f : X → X f:X\rightarrow X be a pseudo-automorphism. Under the mild condition that λ 1 ( f ) 2 > λ 2 ( f ) \lambda _1(f)^2>\lambda _2(f) , we prove the existence of invariant positive closed ( 1 , 1 ) (1,1) and ( 2 , 2 ) (2,2) currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equi-distribution result for Green ( 1 , 1 ) (1,1) currents of meromorphic selfmaps, not necessarily algebraic 1 1 -stable, of a compact Kähler manifold of arbitrary dimension and discuss how a stronger equidistribution result may be proved for pseudo-automorphisms in dimension 3 3 . As a byproduct, we show that the intersection of some dynamically related currents is well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.