New explicit formulas for Faltings' delta-invariant

TL;DR

This paper derives new explicit formulas for Faltings' delta-invariant using theta functions, providing bounds and extending invariants to the moduli space of abelian varieties.

Contribution

It introduces explicit formulas for the delta-invariant and extends key invariants to a broader moduli space, enhancing computational and theoretical understanding.

Findings

New explicit formulas for Faltings' delta-invariant

Explicit bounds for the delta-invariant and Arakelov-Green function

Canonical extension of invariants to moduli space of abelian varieties

Abstract

In this paper we give new explicit formulas for Faltings' $\delta$-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for $\delta$ only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of $\delta$. Furthermore, we give a canonical extension of $\delta$ and the Zhang-Kawazumi invariant $\varphi$ to the moduli space of indecomposable principally polarised complex abelian varieties.