Mazur‘s Incidence Structure for Projective Varieties (I)

Abstract

Let X be an m dimensional smooth projective variety with a Kähler metric. We construct a metrized line bundle $\cL$ with a rational section s over the product $\cC$p(X)× $\cC $q(X) of Chow varieties $\cC$p(X), $\cC$q(X) such that $\[{1\over (m-1)!}\log|s(A,B)|^2=\langle A, B\rangle \]$ for disjoint A, B. That gives an answer to a part of Barry Mazur‘s proposal in a private communication to Bruno Horris about the Archimedean height pairing 〈 A, B〉 on a smooth projective variety X.