Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections
Nonsingular complete intersections X in \({\mathbb{P}}^ n({\mathbb{C}})\) are considered of the form \(F_ 0=...=F_ m=0\) where \(F_ i\) is homogeneous of degree \(d\geq 2\) and \(m\leq n-2\). The Hodge structure on the middle primitive cohomology \(H^{n-m-1}(X)\) defines a period map p: \(M\to D/\Gamma\) where M is the moduli space of X, D the period domain and \(\Gamma\) a discrete subgroup of Aut(D). The question dealt with is the weak global Torelli problem: is p a map of degree one onto its image? It was known that p is an immersion [cf. \textit{C. A. M. Peters}, Math. Ann. 217, 1-16 (1975; Zbl 0293.14004) and \textit{S. Usui}, Japan J. Math., New Ser. 2, 411-418 (1976; Zbl 0347.14007)]. The main theorem says that p has degree one in case \(d\leq 2((m+1)d- (n+1))\leq \min (md,(d-1)m+(d-2)).\)- The proof has the following ingredients. First there is an algebraic correspondence between X as above and the hypersurface \({\mathfrak X}\subset {\mathbb{P}}^ n\times {\mathbb{P}}^ m\) with equation \(u_ 0F_ 0+...+u_ mF_ m=0\), establishing an isomorphism of Hodge structures \(H_{prim}^{n+m- 1}({\mathfrak X},{\mathbb{Q}})\to H_{prim}^{n-m-1}(X,{\mathbb{Q}})(-m)\). Then the infinitesimal variation of Hodge structure of X is expressed in terms of a bigraded ring R, and it is shown that weak global Torelli would follow if for a suitable \(X'\) a natural map \(H^ 0(\omega_{X'}^{\otimes 2})\to Hom(H^ 1(\Theta_{X'})^{\otimes (n-m-1)},H^{n-m-1,n-m- 1}(X'))\) is injective. The injectivity of this map is finally checked for complete intersections of Fermat type.
infinitesimal variation of Hodge structure
scientific article; zbMATH DE number 4210310