Ample canonical heights for endomorphisms on projective varieties

Takahiro Shibata

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arXiv · 1710.05278

Ample canonical heights for endomorphisms on projective varieties

Takahiro Shibata

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Abstract

We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell--Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety.

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BibTeX RIS

Takahiro Shibata. 2017-10-15. Ample canonical heights for endomorphisms on projective varieties. https://arxiv.org/abs/1710.05278

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