Balanced embedding of degenerating Abelian varieties

TL;DR

This paper demonstrates how to embed degenerating Abelian varieties into projective space using theta functions after a base change, and explores their Gromov-Hausdorff limits under flat Kähler metrics.

Contribution

It introduces a method to achieve balanced embeddings of degenerating Abelian varieties and analyzes their metric limits, linking algebraic and differential geometric perspectives.

Findings

Balanced embeddings exist after base change for degenerating Abelian varieties.

The study connects algebraic degenerations with Gromov-Hausdorff limits of Kähler metrics.

Provides a framework for understanding degenerations via theta functions.

Abstract

For a certain maximal unipotent family of Abelian varieties over the punctured disc, we show that after a base change, one can complete the family over a disc such that the whole degeneration can be simultaneously balanced embedded into a projective space by the theta functions. Then we study the relationship between the balanced filling-in and the Gromov-Hausdorff limit of flat K\"{a}hler metrics on the nearby fibers.