Generic vanishing subschemes of codimension three

Source abstract

We classify geometrically nondegenerate GV subschemes of codimension three in indecomposable principally polarized complex abelian varieties of dimension $g\ge6$. Every such subscheme is a translate of $\pm W_{g-3}(C)$ in the Jacobian of a smooth curve $C$. The proof studies the theta-dual surface: tangent cones and theta Hessians give a curve summand in the singular case, while Hodge gradings and characteristic cycles give one in the smooth case. Theta duality yields the corresponding classification of GV surfaces. We also compute the cohomology of all topologically trivial twists of the ideal sheaf twisted by the principal polarization, and characterize the hyperelliptic case by a cohomology jump.