The aim of these lecture notes is to update Beauville's beautiful 1987 Seminaire Bourbaki talk on the same subject. The Schottky problem is the problem of finding characterizations of Jacobians among all principally polarized abelian varieties. We review the numerous approaches to this problem. In the “analytical approach”, one tries to find polynomials in the thetaconstants that define the Jacobian locus in the moduli space of principally polarized abelian varieties. We review van Geemen's (1984) and Donagi's (1987) work in that direction. The loci they get contain the Jacobian locus as an irreducible component. In the “geometrical approach”, one tries to give geometric properties that are satisfied only by Jacobians. We review the following: singularities of the theta divisor (Andreotti and Mayer 1967); reducibility of intersections of a theta divisor with a translate (Welters 1984) and trisecants to the Kummer variety (Welters 1983, Beauville and Debarre 1986, Debarre 1992); the K-P equation and Novikov's conjecture (Shiota 1986, Arbarello and De Concini 1984); double translation hypersurfaces (Little 1989); the van Geemen-van der Geer conjectures on the base locus of the set of second order theta functions that vanish with multiplicity ≥ 4 at the origin (Beauville and Debarre 1989, Izadi 1993); subvarieties with minimal class (Ran 1981, Debarre 1992); the Buser-Sarnak approach (1993).

Introduction

For g ≥ 2, the moduli space Ag of principally polarized abelian varieties of dimension g has dimension

and the moduli space Mg of smooth connected projective algebraic curves of genus g has dimension 3g — 3.