Monodromy of Kodaira Fibrations of Genus $3$

Abstract

A Kodaira fibration is a non-isotrivial fibration f : S → B from a smooth algebraic surface S to a smooth algebraic curve B such that all fibers are smooth algebraic curves of genus g. Such fibrations arise as complete curves inside the moduli space Mg of genus g algebraic curves.We investigate here the possible connected monodromy groups of a Kodaira fibration in the case g = 3 and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of M3 parametrizing curves whose Jacobians have extra endomorphisms.

Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

“…Since the construction of the first examples of such fibrations by Atiyah and Kodaira [5,44], these complex surfaces have been widely studied, either from the point of view of complex geometry or from the point of view of the theory of surface bundles. See for instance [8,15,17,18,20,31,32,36,46,60] for a few works on this topic. In this text we study several geometric and group theoretical problems related to these complex surfaces.…”