Positivity of the height jump divisor

TL;DR

This paper proves that height jump divisors are always effective under certain conditions, confirming a conjecture of Hain and impacting the understanding of line bundles in algebraic geometry.

Contribution

It demonstrates the positivity of height jump divisors in a broad setting, including applications to Hodge theory and moduli spaces, and confirms Hain's conjecture for the Ceresa cycle.

Findings

Height jump divisors are always effective under specified conditions.

The result confirms Hain's conjecture for the Ceresa cycle.

The Moriwaki divisor has non-negative degree on all relevant curves.

Abstract

We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain, we show that under these assumptions the so-called height jump divisors are always effective. This result is of particular interest in the context of biextension line bundles on Griffiths intermediate jacobian fibrations of polarized variations of Hodge structure of weight -1, pulled back along normal function sections. In the case of the normal function on M_g associated to the Ceresa cycle, our result proves a conjecture of Hain. As an application of our result we obtain that the Moriwaki divisor on \bar M_g has non-negative degree on all complete curves in \bar M_g not entirely contained in the locus of irreducible singular curves.