Abstract. We prove a full generalization of the Castelnuovo's free pencil trick. We show its analogies with [RZ1, Theorem 2.1.7]; see also [PZ, Theorem 1.5.1]. Moreover we find a new formulation of the Griffiths's infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic 1-forms.

We prove an equivalence between the infinitesimal Torelli theorem for top forms on a hypersurface contained inside a Grassmannian G and the theory of adjoint volume forms presented in [RZ2]. More precisely, via this theory and a suitable generalization of Macaulay's theorem we show that the differential of the period map vanishes on an infinitesimal deformation if and only if certain explicitly given twisted volume forms go in the generalized Jacobi ideal of X via the cup product homomorphism.

Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

We extend some of the results of [PZ] and we prove a criterion for a family π : X → B of n-dimensional varieties of general type and with Albanese morphism of degree 1 to have birational fibers. We do not assume that the canonical map of the general fiber is a morphism nor that the canonical linear system has no fixed components. We also prove some instances of the generic Torelli theorem under the assumption, natural in this context, that the fibers are minimal and their minimal model is unique. It is trivial to construct counterexamples to generic Torelli without such minimality assumptions.

We solve the infinitesimal Torelli problem for 3-dimensional quasi-smooth Q-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length 3 Gushel-Mukai chain of prime smooth Fanos of coindex 3 and degree 10.