  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.
