Inverse images of positive closed currents under holomorphic endomorphisms of compact Kähler manifolds

Abstract

We prove that for a surjective holomorphic endomorphism $f$ of a compact Kähler manifold $X$ of dimension $k\ge 2$ and for some integer $p$ with $1\le p\le k$, there exists a proper invariant analytic subset $E$ for $f$ such that if a positive closed $(p,p)$-current $S$ can be represented by a smooth form in a neighborhood of $E$, the sequence $d_p^{-n}(f^n{)}^{\ast }(S-{\alpha }_S)$ converges to $0$ exponentially fast in the sense of currents, where $d_p$ is the dynamical degree of order $p$ and ${\alpha }_S$ is a smooth closed $(p,p)$-form in the de Rham cohomology class of $S$.

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