Geometry of A_g and Its Compactifications

TL;DR

This survey reviews recent advances in understanding the geometry of moduli spaces of principally polarized abelian varieties, focusing on compactifications, birational geometry, and intersection theory, and discusses open problems and future directions.

Contribution

It provides an updated comprehensive overview of the progress in the geometric study of A_g moduli spaces over the past decade.

Findings

Analysis of compactifications and their properties

Insights into nef and effective cones of A_g

Discussion of open problems and future research directions

Abstract

In this survey we give a brief introduction to, and review the progress made in the last decade in understanding the geometry of the moduli spaces A_g of principally polarized abelian varieties and its compactifications. Topics surveyed include: compactifications; birational geometry: nef and effective cones, canonical models; homology, Chow rings and intersection theory; and subvarieties of moduli spaces. We also discuss some open problems and possible further directions. This is an expanded and updated version of the talk given at the 2005 Summer Institute for Algebraic Geometry