Vol. 309, No. 1, 2020

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The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields

Alexander Carney

Vol. 309 (2020), No. 1, 71–102
DOI: 10.2140/pjm.2020.309.71

In one of the fundamental results of Arakelov’s arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge index theorem for arithmetic surfaces by relating the intersection pairing to the negative of the Néron–Tate height pairing. More recently, this has been generalized to higher dimensions by Moriwaki and by Yuan and Zhang. We extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with nonconstant but numerically trivial line bundles coming from the constant field via Chow’s Kk-trace functor. As an application of the Hodge index theorem, we also prove a rigidity theorem for the set of canonical height zero points of polarized algebraic dynamical systems over function fields. For function fields over finite fields, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, of Baker and DeMarco, and of Yuan and Zhang.

arithmetic dynamics, intersection theory, Arakelov theory
Mathematical Subject Classification
Primary: 11G50, 14G40, 37P30
Received: 1 December 2019
Revised: 24 July 2020
Accepted: 8 August 2020
Published: 26 December 2020
Alexander Carney
Department of Mathematics
University of Rochester
Rochester, NY
United States