Arakelov-Parshin rigidity of towers of curve fibrations

TL;DR

This paper investigates the rigidity properties of certain algebraic varieties mapping into the moduli space of canonically polarized manifolds, establishing rigidity results for specific classes of curves and fibrations.

Contribution

It proves Arakelov-Parshin rigidity for all complete curves and for generic affine curves with degree 2 mappings into the locus of iterated Kodaira fibrations.

Findings

Rigidity for all complete curves mapping finitely onto KF_h.

Rigidity for generic affine curves with deg h = 2.

Injectivity of the iterated Kodaira-Spencer map in key cases.

Abstract

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.