The Moduli Space of Abelian Varieties and Its Compactifications

Abstract

Abelian varieties are high-dimensional generalizations of elliptic curves. The moduli space A_g of principally polarized abelian varieties of dimension g is the quotient of the Siegel upper half-space by the action of Sp(2g, Z). This moduli space is not compact. The first compactification is due to Satake. In this thesis, we mainly discuss the toroidal compactification of A_g due to Mumford. We provide an introduction to complex abelian varieties, discuss the construction of their moduli spaces, cover the necessary knowledge of toric varieties, outline the general steps of toroidal compactifications, and present the toroidal compactification of A_2.