A note on abelian varieties embedded in quadrics (Q2504921)

\textit{A. Van de Ven} showed [Ann. Mat. Pura Appl., IV. Ser. 103, 127--129 (1975; Zbl 0304.14025)] using numerical methods that an abelian \(d\)-fold can be embedded in \({\mathbb P}^{2d}\) only if \(d=1\) (smooth cubic curves) or \(d=2\) (Horrocks-Mumford surfaces). This paper proves a similar result for abelian \(d\)-folds in \(2d\)-dimensional quadrics. The only possibility turns out to be \(d=1\) (curves of bidegree \((2,2)\)). Numerical methods are not quite sufficient, though. Using the self-intersection formula and Riemann-Roch reduces to \(d\leq 3\), but the cases \(d=2\) and \(d=3\) are eliminated by using results of Lazarsfeld and of Iyer on line bundles of certain types on abelian varieties.