Theta divisors with curve summands and the Schottky problem (Q303587)

A theorem of Riemann says that the theta divisor of the Jacobian of a smooth projective curve of genus \(g\) is given by the image of the \((g-1)\)th symmetric power of the Abel-Jacobi map. The present paper shows the following converse: Let \((A,\theta)\) be an indecomposable principally polarized abelian variety of dimension \(g\geq 2\). If there is a curve \(C\) and a \((g-2)\)-dimensional subvariety \(Y\) such that \(\theta= C+Y\), then \(C\) is smooth and \((A,\theta)\) is the Jacobian of \(C\). The paper also proves a generalization of this, characterizing Jacobians by the existence of subvarieties of smaller dimension with a curve summand whose twisted ideal sheaf is a generic vanishing sheaf in the sense of Pareschi-Popa. The proof of the first theorem uses cohomological and geometric techniques which originated in the work of Ran and Welters as well as the theorem of \textit{L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 10, No. 1, 243--258 (1997; Zbl 0901.14028)] on the singularities of the theta divisor. The proof is then reduced to Matsusaka's characterization of Jacobians by the fact that the cohomology class \({1\over(g-1)}[\theta]^{g-1}\) can be represented by a curve. There are several interesting consequences determining all theta divisors that can be dominated by products of curves.