On Mumford's construction of degenerating abelian varieties

TL;DR

This paper proves the canonical compactification of certain degenerating abelian varieties, introduces SQAVs with a Cartier theta divisor, and explores their geometric and cohomological properties.

Contribution

It provides a canonical compactification method for degenerating abelian varieties and introduces SQAVs with a unique Cartier theta divisor, along with their combinatorial and geometric analysis.

Findings

Canonical compactification of degenerating abelian varieties achieved.

Introduction of SQAVs with a canonical Cartier theta divisor.

Computed cohomologies of line bundles on SQAVs.

Abstract

We prove that a 1-dimnl family of abelian varieties with an ample sheaf defining principal polarization can be canonically compactified (after a finite base change) to a projective family with an ample sheaf. We show that the central fiber (P,L), which we call an SQAV, has a canonical Cartier theta divisor. We give a combinatorial definition for SQAVs and describe their geometrical properties, in particular compute cohomologies of L^n, n\ge0.