The Torelli theorem (cf.[Tor13], [Andr58], [Weil-57]) states
that two algebraic curves are isomorphic if and only if their Jacobian varieties are
isomorphic as polarized abelian varieties. ¶
André Weil ([Weil-CPII]) set up a program for doing arithmetics on K3 surfaces,
based on a Torelli type theorem, which was later proven through the effort of several
authors (for this long story and related references we refer to Chapter VIII of the
book [BPV]). This result is crucial for answering questions about the existence of K3
surfaces or families thereof possessing certain curve configurations. ¶
The general Torelli question, as set up by Griffiths ([Grif68], [Grif70], cf. also
[Grif-Schmid], [Grif84]), is to associate to each projective variety X of general type
its Hodge structure of weight n = dim(X), and ask whether the corresponding "period
map" ψ n is injective on the local moduli space (or Kuranishi space of X). ¶
It was known since long time that, as soon as the dimension is at least two,
there are families of varieties without Hodge structures, which are not rigid. Surfaces
of general type with q = pg = 0 were constructed in the 30's by Campedelli and
Godeaux ([Cam32], [God35]), and for instance, in the case of the Godeaux surfaces,
the Kuranishi family has dimension 8. ¶
A natural question which arises is: under which hypothesis on X is a local Torelli
theorem valid for the Hodge structure of weight n = dim X? ¶
In other words, when is the local period map ψ n a local embedding?