A ug 1 99 7 On the period map of Calabi – Yau manifolds

If X is a smooth complex projective n-fold, Hodge-Lefschetz theory provides a filtration on the (primitive) cohomology H(X,C)0 by complex subspaces, satisfying certain compatiblity conditions with a bilinear form Q on cohomology. The Torelli problem, proposed in this form by Griffiths, is to decide whether given X, Y in a certain class C of varieties, the existence of an isomorphism H(X,C)0 ∼= H(Y,C)0 respecting the bilinear forms and filtrations implies that X and Y are isomorphic. The answer for curves is classically known to be affirmative, whereas for higher dimensions there are various positive results for different classes C: ‘most’ generic hypersurfaces in projective or weighted projective spaces ([Do], [DT], [Sa]), high degree hyperplane sections of varieties with very ample canonical bundle ([Gre]), and others. (See [Gr4] for a review.) The problem appears to be difficult for varieties with trivial canonical bundle, more specifically for Calabi-Yau varieties – an n-dimensional complex projective variety X is called Calabi-Yau if it has trivial canonical bundle and satisfies H (X,OX) = 0 for 0 < i < n. The proof [PS] for n = 2 (K3 surfaces) is one of the greatest achievements of the theory. For n = 3, the only result known to the author is the recent work of Voisin [Vo], for generic quintic hypersurfaces in P. In this paper Calabi-Yau n-folds for n ≥ 3 will be considered, and it will be proved that the filtration determines the varieties up to finitely many possibilities among Calabi-Yaus possessing ample sheaves with bounded self-intersection. In fact, the first piece of the filtration will only be used; for n = 3 this is equivalent by [BG]. Fix once and for all the dimension n ≥ 3 of the varieties, a lattice VZ (of even dimension for odd n, otherwise arbitrary) together with a non-degenerate bilinear form Q, symmetric for n even, skew for n odd. Let V = VZ⊗C be the corresponding