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Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties

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Abstract

We consider an arbitrary int-amplified surjective endomorphism f of a normal projective variety X over \(\mathbb {C}\) and its \(f^{-1}\)-stable prime divisors. We extend the early result in Zhang (Adv Math 252:185–203, 2014, Theorem 1.3) for the case of polarized endomorphisms to the case of int-amplified endomorphisms. Assume further that X has at worst Kawamata log terminal singularities. We prove that the total number of \(f^{-1}\)-stable prime divisors has an optimal upper bound \(\dim X+\rho (X)\), where \(\rho (X)\) is the Picard number. Also, we give a sufficient condition for X to be rationally connected and simply connected. Finally, by running the minimal model program (MMP), we prove that, under some extra conditions, the end product of the MMP can only be an elliptic curve or a single point.

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Acknowledgements

The author would like to deeply thank Professor De-Qi Zhang for many inspiring ideas and discussions. Also, he would like to thank Doctor Sheng Meng for the result of running MMP [22] on int-amplified endomorphisms, and the referees for many constructive suggestions to improve the paper.

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  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Republic of Singapore

    Guolei Zhong

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Zhong, G. Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties. J Geom Anal 31, 2568–2593 (2021). https://doi.org/10.1007/s12220-020-00366-6

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