Positivity of heights of codimension 2 cycles over function field of characteristic 0

TL;DR

This paper demonstrates how the classical Hodge index theorem implies Beilinson's Hodge index conjecture for height pairings of homologically trivial codimension two cycles over function fields of characteristic zero, linking classical and modern conjectures.

Contribution

It establishes a connection between the classical Hodge index theorem and Beilinson's conjecture, providing a new approach to understanding height pairings in algebraic geometry.

Findings

Hodge index theorem implies Beilinson's conjecture in this context

Application to Gross-Schoen cycles and the Bogomolov conjecture

Lower bounds for Faltings height derived from graph conjectures

Abstract

In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir.