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April, 2019 Ample canonical heights for endomorphisms on projective varieties
Takahiro SHIBATA
J. Math. Soc. Japan 71(2): 599-634 (April, 2019). DOI: 10.2969/jmsj/79157915


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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Takahiro SHIBATA. "Ample canonical heights for endomorphisms on projective varieties." J. Math. Soc. Japan 71 (2) 599 - 634, April, 2019.


Received: 30 October 2017; Revised: 16 December 2017; Published: April, 2019
First available in Project Euclid: 4 March 2019

zbMATH: 07090058
MathSciNet: MR3943453
Digital Object Identifier: 10.2969/jmsj/79157915

Primary: 37P30
Secondary: 14G05

Keywords: canonical height , dynamical degree

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 2 • April, 2019
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