The Schottky problem

The Riemann-Schottky problem is the problem of determining which complex principally polarized abelian varieties arise as Jacobian varieties of complex curves. The history of the problem is very long, going back to the works of Abel, Jacobi, and Riemann. The first approach, culminating in a complete solution in genus 4 (the first nontrivial case), was developed by Schottky and Jung [Sch88; SJ09]. Since then a variety of different approaches to the problem have been developed, and many geometric properties of abelian varieties in general and Jacobian varieties in particular have been studied extensively. Numerous partial and complete solutions to the Schottky problem have been conjectured, and some were proven. In this survey we will describe many of the ideas and methods that have been applied to or developed for the study of the Schottky problem. We will present some of the results, as well as various open problems and possible connections among various approaches. To keep the length of the text reasonable, the proofs for the most part will be omitted, and references will be given; when possible, we will try to indicate the general idea or philosophy behind the work done. We hope that an interested reader may consider this as an introduction to the ideas and results of the subject, and would be encouraged to explore the field in greater depth by following some of the references. This text is in no way the first (and will certainly not be the last) survey written on the Schottky problem. Many excellent surveys, from various points of view, and emphasizing various aspects of the field, have been written, including those by Dubrovin [Dub81b], Donagi [Don88], Beauville [Bea88], Debarre [Deb95b], Taimanov [Tai97], van Geemen [vG98], Arbarello [Arb99], Buchstaber and Krichever [BK06]. A beautiful introduction is [Mum75], while many relevant results on curves, abelian varieties, and theta functions can be found in [Igu72; MumO7a; MumO7b; MumO7c; ACGH85; BL04].