TL;DR
This survey reviews classical and modern approaches to the Schottky problem, which aims to distinguish Jacobians of curves within the broader class of principally polarized abelian varieties, highlighting recent advances and research directions.
Contribution
It compiles recent results and discusses current research directions in the classical and modern methods for the Schottky problem.
Findings
Summary of classical and modern methods
Recent results in characterizing Jacobians
Directions for future research
Abstract
In this survey we discuss some of the classical and modern methods in studying the (Riemann-)Schottky problem, the problem of characterizing Jacobians of curves among principally polarized abelian varieties. We present many of the recent results in this subject, and describe some directions of current research. This paper is based on the talk given at the "Classical algebraic geometry today" workshop at MSRI in January 2009.
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