Modular interpretation of the Weil-Petersson metric asymptotics for abelian varieties

TL;DR

This paper explores the asymptotic behavior of the Weil-Petersson metric on the moduli space of abelian varieties, connecting it with collapsing limits of flat tori to better understand geometric degenerations.

Contribution

It provides a modular interpretation of the Weil-Petersson metric asymptotics specifically for abelian varieties, serving as a step towards understanding more complex Calabi-Yau cases.

Findings

Link between Weil-Petersson asymptotics and collapsing limits of flat tori

Explicit classification of degenerations via Odaka's work

Foundation for refined asymptotic analysis of moduli spaces

Abstract

As a first step towards a refined description of the asymptotic of the Weil-Petersson metric on the moduli space of polarized Calabi-Yau manifolds we investigate the concrete case of abelian varieties by linking such asymptotic with the multi-scale collapsing limits of the parametrized flat tori, as explicitly classified by Odaka.