Singularities of the biextension metric for families of abelian varieties

TL;DR

This paper investigates the singularities of the invariant metric on the Poincaré bundle over families of abelian varieties and proves the effectiveness of height jump divisors, confirming a conjecture by Hain.

Contribution

It provides a detailed analysis of metric singularities and establishes the effectiveness of height jump divisors in a broad setting, advancing understanding in algebraic geometry.

Findings

Singularities of the invariant metric are characterized over arbitrary base dimensions.

Proves the effectiveness of height jump divisors for families of pointed abelian varieties.

Confirms Hain's conjecture in the context of variations of polarized Hodge structures.

Abstract

In this paper we study the singularities of the invariant metric of the Poincar\'e bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$.