The Riemann-Schottky problem via singularities of theta divisors

The Riemann-Schottky problem is the problem of determining which principally polarized abelian varieties (PPAV) arise as Jacobian of curves. Riemann showed that the theta divisor on the Jacobian of a hyperelliptic curve has singularity of codimension three. Beauville conjectured that any irreducible PPAV whose theta divisors have singularity of codimension three actually come from hyperelliptic curves. In this talk, I will discuss a refinement of this problem and a partial solution. To achieve this, we develop a theory of higher multiplier ideals for Q-divisors using the theory of complex Hodge modules of Sabbah-Schnell, building on M.Saito’s theory of rational Hodge modules. This is joint work with Christian Schnell.