Schottky problem

Remarks

By the Torelli theorem, a curve is determined by its Jacobian with its polarization, so $\mathcal{J}_g$ has the dimension $3g - 3$ of the moduli of curves when $g \ge 2$. The space $\mathcal{A}_g$ has dimension $g(g + 1)/2$. For $g \le 3$ the closure of $\mathcal{J}_g$ is all of $\mathcal{A}_g$, and for $g \ge 4$ it is a proper closed subset. For $g = 4$, Schottky found a modular form, built from theta functions, that vanishes on $\mathcal{J}_4$. Igusa showed that its zero locus is exactly the closure of $\mathcal{J}_4$. In 1986 Shiota proved a conjecture of Novikov on this problem. It says that an indecomposable principally polarized abelian variety is a Jacobian if and only if its theta function solves the KP equation. Other characterizations, for example by trisecants of the Kummer variety, are also known.