Characterization of products of theta divisors (Q2921071)

Let \((A,\Theta )\) be a principally polarized abelian variety so that \(\Theta \) is ample and \(h^0(\mathcal O_A (\Theta ))=1\). If \(\Theta \) is irreducible, then by [\textit{L. Ein} and \textit{R. Lazarsfeld}, J. Am. Math. Soc. 10, No. 1, 243--258 (1997; Zbl 0901.14028)], it is normal with rational singularities and so it is easy to see that if \(X\to \Theta \) is a desingularization, then \(\chi (X,\omega _X)=\chi (\Theta,\omega _\Theta )= h^0(\mathcal O_A (\Theta ))=1\). In this paper the authors show that this property characterizes theta divisors, namely they show: \textit{Let \(Y\) be a subvariety of an abelian variety \(A\) and \(X\to Y\) a desingularization, then \(Y\) is isomorphic to a product of of theta divisors if and only if \(Y\) is normal and \(\chi (X,\omega _X)=1\).} They also study smooth projective varieties \(X\) of maximal Albanese dimension (i.e. such that the Albanese map is generically finite but not necessarily birational) and \(\chi (X,\omega _X)=1\). By [\textit{C. D. Hacon} and \textit{R. Pardini}, Math. Res. Lett. 12, No. 1, 129--140 (2005; Zbl 1070.14043)], it is known that in this case \(\dim X \leq q(X):=h^0(\Omega ^1 _X)\leq 2\dim X\) and if \(q(X)= 2\dim X\), then \(X\) is birational to a product of curves of genus \(2\). The authors give a complete characterization of the case \(q(X)= 2\dim X-1\).