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arXiv · 1910.03856

Int-amplified endomorphisms of compact K\"ahler spaces

Guolei Zhong
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Abstract

Let $X$ be a normal compact K\"ahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*\xi-\xi=\eta$ for some K\"ahler classes $\xi$ and $\eta$. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact K\"ahler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a positive power, we can run the minimal model program (MMP) $f$-equivariantly for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of Picard number one.

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Guolei Zhong. 2019-10-09. Int-amplified endomorphisms of compact K\"ahler spaces. https://doi.org/10.4310/ajm.2021.v25.n3.a3

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