We extend the theory of Pirola and Zucconi (J Algebraic Geom 12(3):535–572, 2003). We introduce the new notion of adjoint quadric for canonical images of irregular varieties. Using this new notion, we obtain the infinitesimal Torelli theorem for varieties whose canonical image is a complete intersection of hypersurfaces of degree > 2 and for Schoen surfaces. Finally, we show that a family with fiberwise liftable holomorphic forms such that the fibers have Albanese morphism of degree 1 is birationally trivial if there exist no adjoint quadrics.
