Torelli theorems

Theorems stating that the Hodge structure (period matrix) in the cohomology spaces of an algebraic or Kähler variety completely characterizes the polarized Jacobi variety of .

The classical Torelli theorem relates to the case of curves (see [1], [2]) and states that a curve is defined up to an isomorphism by its periods. Let be a curve of genus , let be a basis of , let be a basis of the Abelian differentials (cf. Abelian differential) and let the -matrix be the period matrix, where . The intersection of cycles defines a skew-symmetric bilinear form in . Let and be two curves. If bases and can be chosen with respect to which the period matrices and the intersection matrices of the curves are the same, then and are isomorphic. In other words, if the canonically polarized Jacobians of the curves and are isomorphic, then .

Let be a projective variety (or, more generally, a compact Kähler manifold), and let be the Griffiths variety associated with the primitive cohomology spaces (see Period mapping). Then contains the period matrices of primitive -forms on all varieties homeomorphic to . The periods depend on the choice of the isomorphism of into a fixed space . There is a naturally defined group of analytic automorphisms of such that is an analytic space and determines a unique point . In this situation, is called the modular space or the moduli space of Hodge structures.

The global Torelli problem consists in the elucidation of the question whether uniquely determines up to an isomorphism. In the case of an affirmative solution, the problem corresponds to the statement of the so-called (generalized) Torelli theorem. Torelli's theorem holds trivially for Abelian varieties in the case of -forms and in the case of -forms (see [3]). Essentially, the only non-trivial case of a solution of the global Torelli problem (1984) is the case of a -surface. The Torelli theorem has also been generalized to the case of Kähler -surfaces.

The local Torelli problem consists in solving the question when the Hodge structures on the cohomology spaces separate points in the local moduli space (the Kuranishi space) for a variety . Let be a family of polarized algebraic varieties, , and let be the Griffiths variety associated with the periods of primitive -forms on . The period mapping associates with the period matrix of -forms on . This mapping is holomorphic; the corresponding tangent mapping has been calculated (see [3]). The local Torelli problem is equivalent to the question: When is an imbedding? By considering the mapping dual to one obtains a cohomological criterion for the validity of the local Torelli theorem: If the mapping

is an epimorphism, then the periods of the -forms give local moduli for . The local Torelli theorem for curves is equivalent to the fact that quadratic differentials are generated by Abelian differentials. Noether's theorem states that this is true if or if and is not hyper-elliptic. The local Torelli theorem clearly holds in the case if the canonical class is trivial. Such varieties include the Abelian varieties, hypersurfaces of degree in and -surfaces. The validity of the local Torelli theorem has been established for various classes of higher-dimensional varieties. For non-singular hypersurfaces of degree in it has been proved that the period mapping is an imbedding at a generic point except for the case , and, possibly, the cases: divides , and , or and (see [4]).

References

[1]	R. Torelli, Rend. Accad. Lincei V , 22 (1913) pp. 98–103
[2]	A. Weil, "Zum Beweis der Torellischen Satzes" Nachr. Akad. Wiss. Göttingen (1957) pp. 33–53
[3]	P.A. Griffiths, "Periods of integrals on algebraic manifolds I, II" Amer. J. Math. , 90 (1968) pp. 568–626; 805–865
[4]	R. Donagi, "Generic Torelli for projective hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353

References

[a1]	P.A. Griffiths (ed.) , Topics in transcendental algebraic geometry , Princeton Univ. Press (1984)
[a2]	A. van de Ven, "Compact complex surfaces" , Springer (1984)