Counterexamples to Torelli theorem in any odd dimension greater than one (Q1075398)

This paper is devoted to the study of smooth hypersurfaces of bidegree (p,2), \(p\geq 1\) in \({\mathbb{P}}^ 1\times {\mathbb{P}}^{2m+1}\). Some topological and analytic invariants of such a hypersurface V are computed, the group Aut(V) is studied and the quadric bundle structure induced by the natural projection \(\pi_ V:\quad V\to {\mathbb{P}}^ 1\) is investigated. The singular fibres of \(\pi_ V\) are \(p(2m+2)\) distinct quadrics of rank \(2m+1\). The variety L(V) parametrizing the linear spaces of dimension m contained in the fibres of \(\pi_ V\) is smooth, and the Stein factorization L(V)\(\to^{f}C\to {\mathbb{P}}^ 1\) of the projection \(L(V)\to {\mathbb{P}}^ 1\) gives a morphism f with connected fibres onto a smooth hyperelliptic curve C which is a double cover of \({\mathbb{P}}^ 1\) branched exactly at those \(P_ i\in {\mathbb{P}}^ 1\) for which \(\pi_ V^{-1}(P_ i)\) is singular. So C parametrizes the rulings of the fibres of \(\pi_ V\) by m-planes. By using a locus of the cycle map it is proved that: J(V)\(\cong J(C)\) as principally polarized abelian varieties; and implicitly that the Abel-Jacobi map Alb(L(V))\(\to^{\sim}J(V)\) is an isomorphism. Since for \(p>1\) the hypersurface V has more moduli than the hyperelliptic curve C, V is not determined by its intermediate Jacobian J(V) and so the Torelli theorem fails, whereas it holds for \(p=1\) (Turin).