Arakelov-Parshin rigidity of towers of curve fibrations

Arakelov-Parshin rigidity is concerned with varieties mapping rigidly to the moduli stack M_h of canonically polarized manifolds. Affirmative answer for any class of maps implies finiteness of the given class. This article studies Arakelov-Parshin rigidity on an open subspace of M_h, on the locus KF_h of iterated Kodaira fibrations. First, we prove rigidity for all complete curves mapping finitely onto KF_h. Then, for generic affine curves mapping into KF_h, rigidity is shown when deg h =2. The method used in the latter part is showing that the iterated Kodaira-Spencer map is injective.