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
-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.
Keywords
arithmetic dynamics, intersection theory, Arakelov theory
