Inverse images of positive closed currents under holomorphic endomorphisms of compact K\"ahler manifolds

Taeyong Ahn

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arXiv · 2405.00607

Inverse images of positive closed currents under holomorphic endomorphisms of compact K\"ahler manifolds

Taeyong Ahn

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Abstract

We prove that for a surjective holomorphic endomorphism $f$ of a compact K\"ahler 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 $d_p^{-n}(f^n)^*(S-\alpha_S)$ 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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Taeyong Ahn. 2024-05-01. Inverse images of positive closed currents under holomorphic endomorphisms of compact K\"ahler manifolds. https://arxiv.org/abs/2405.00607

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