Faltings delta-invariant and semistable degeneration

TL;DR

This paper analyzes the asymptotic behavior of key Arakelov invariants, including the delta-invariant, for degenerating families of complex curves, linking these to combinatorial structures on graphs.

Contribution

It provides a detailed asymptotic analysis of Arakelov invariants during semistable degeneration, connecting complex geometry with graph theory.

Findings

Asymptotic formulas for Arakelov metric and Green's function

Leading terms interpreted via admissible Green's functions on graphs

Results applicable to arbitrary one-parameter degenerations

Abstract

We determine the asymptotic behavior of the Arakelov metric, the Arakelov-Green's function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang's theory of admissible Green's functions on polarized metrized graphs.