Introduction A differentiable manifold, if it admits a complex structure, can usually be given more than one complex structures. Speaking in general terms the Torelli problem asks whether it is possible to distinguish the complex structures at the cohomology level here by cohomology we mean the Dolbeault cohomology HP' . For example, if f : M M~ is a diffeomorphism of compact Kahler manifolds and f preserves the Hodge decomposition in the sense that f H^'^CM^) = H^'^(M ), are M and M~ isomorphic as complex manifolds? A classical result in this direction is the Torelli theorem for algebraic curves. It asserts that if the Jacobians of two curves are isomorphic as polarized Abelian varieties, then the two curves are isomorphic as complex manifolds. This may be phrased in terms of moduli: the assignment of the Jacobian to a smooth curve gives a map, called the period map, from the moduli space of all smooth curves of genus g to the space of all polarized Abelian varieties, and in this context the Torelli theorem is equivalent to the injectivity of the period map. For higher-dimensional varieties, one defines the period map in terms of the Hodge decomposition by associating to a variety the matrices of the periods of (p,q)-forms. Because the diffeomorphism type of a smooth variety of dimension greater than one is not determined by a single number such as the genus or by a finite set of known invariants, the Torelli problem is more properly stated in relative terms: given a family of diffeomorphic algebraic manifolds parametrized by an algebraic variety B, is the period map from B into the period matrix domain injective? A weaker version is the local Torelli problem, which asks whether the period map is locally one-to-one, or relaxing the problem even further, whether it is a branched covering. Note that the period map is a branched covering if and only if it has no positive dimensional fibers in B. It is this aspect of the Torelli problem that we will be concerned with. The main results of this paper fall into two parts. In the first part we reprove the following local Torelli theorem three times: if a family of curves, possibly with singular members, is such that the smooth members all have the same Hodge structure, then the smooth fibers are isomorphic as v
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