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Non-density of Points of Small Arithmetic Degrees

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Abstract

Given a surjective endomorphism \(f: X \rightarrow X\) on a projective variety over a number field, one can define the arithmetic degree \(\alpha _f(x)\) of f at a point x in X. The Kawaguchi–Silverman Conjecture (KSC) predicts that any forward f-orbit of a point x in X at which the arithmetic degree \(\alpha _f(x)\) is strictly smaller than the first dynamical degree \(\delta _f\) of f is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus \(Z_f(d)\) of all points of small arithmetic degree is not Zariski dense and verify this sAND Conjecture for endomorphisms on projective varieties including surfaces, HyperKähler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fibre spaces. We show the equivalence of the sAND Conjecture and another conjecture on the periodic subvarieties of small dynamical degree; we also show the close relations between the sAND Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on endomorphisms of projective spaces and another long-standing conjecture on Uniform Boundedness of torsion points in abelian varieties.

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Acknowledgements

The first, third, and last authors are supported by a JSPS Overseas Research Fellowship, a Research Fellowship of NUS, and an ARF of NUS, respectively. The second author is supported in part by Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014) and by a Research Fellowship of KIAS (MG075501). The authors would like to thank Fei Hu for many helpful suggestions, Shu Kawaguchi for kindly pointing out Remark 1.10 (1), and colleagues for pointing out that our proof of Theorem 1.11 actually used an assumption weaker than (initially stated) Conjecture 1.9. The authors would also like to thank the referee for valuable suggestions to improve this paper.

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Authors and Affiliations

  1. Department of Mathematics, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 5588585, Japan

    Yohsuke Matsuzawa

  2. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, People’s Republic of China

    Sheng Meng

  3. Korea Institute For Advanced Study, Seoul, 02455, Republic of Korea

    Sheng Meng

  4. National University of Singapore, Singapore, 119076, Republic of Singapore

    Takahiro Shibata & De-Qi Zhang

  5. National Fisheries University, 2-7-1 Nagata-Honmachi, Shimonoseki, 759-6595, Japan

    Takahiro Shibata

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  1. Yohsuke Matsuzawa
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  2. Sheng Meng
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  3. Takahiro Shibata
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  4. De-Qi Zhang
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Correspondence to Sheng Meng.

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Matsuzawa, Y., Meng, S., Shibata, T. et al. Non-density of Points of Small Arithmetic Degrees. J Geom Anal 33, 112 (2023). https://doi.org/10.1007/s12220-022-01156-y

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