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Gap for geometric canonical height functions

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Abstract

We prove the existence of a gap around zero for canonical height functions associated with endomorphisms of projective spaces defined over complex function fields. We also prove that if the rational points of height zero are Zariski dense, then the endomorphism is birationally isotrivial. As a corollary, by a result of S. Cantat and J. Xie, we have a geometric Northcott property on projective plane in the same spirit of results of R. Benedetto, M. Baker and L. Demarco on the projective line.

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Acknowledgements

I would like to thank my advisor T. Gauthier for numerous helpful discussions and all the time he spent with me. I would like to thank N.-B. Dang, C. Favre and G. Vigny for useful discussions. I also thank S. Cantat and the anonymous referee for their useful comments and suggestions.

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  1. Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 307 Rue Michel Magat, 91405, Orsay, France

    Yugang Zhang

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  1. Yugang Zhang
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Zhang, Y. Gap for geometric canonical height functions. Math. Z. 307, 30 (2024). https://doi.org/10.1007/s00209-024-03502-y

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