Arithmeticity of the monodromy of some Kodaira fibrations

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In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties. The topology of such a bundle is completely encoded in its monodromy representation (a homomorphism to a mapping class group), and it is a fundamental problem to understand precisely how the topology of the bundle is reflected in algebraic properties of the monodromy. The main result of this paper is that the Atiyah–Kodaira bundles have arithmetic monodromy groups. This addresses a question of Griffiths-Schmid. As an application, using work of N. Salter and L. Chen, we show that Atiyah-Kodaira bundles fiber in exactly two ways.

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