Arithmeticity of the monodromy of some Kodaira fibrations

TL;DR

This paper proves that the monodromy groups of certain algebraic surfaces called Atiyah-Kodaira manifolds are arithmetic, using topological methods focused on the geometric monodromy in the mapping class group.

Contribution

It provides a positive answer to Griffiths-Schmid's question for Atiyah-Kodaira manifolds, establishing their monodromy groups as arithmetic.

Findings

Monodromy groups of Atiyah-Kodaira manifolds are arithmetic.

Topological methods analyze geometric monodromy in the mapping class group.

Resolution of Griffiths-Schmid question for this class of algebraic surfaces.

Abstract

A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the "geometric" monodromy, valued in the mapping class group of the fiber.