A fibration of an algebraic surface S over a curve B, with fibres of genus at least 2, has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group G. A variety isogenous to a (higher) product is the quotient of a product of curves of genus at least 2 by the free action of a finite group. Theorem B gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the Euler number. Theorem C classifies the groups thus occurring and shows that, after fixing the group and the Euler number, one obtains an irreducible moduli space. The result of Theorem B is extended to higher dimension in Theorem G, thus generalizing (cf. also Theorem H) results of Jost-Yau and Mok concerning varieties whose universal cover is a polydisk. Theorem A shows that fibrations where the fibre genus and the genus of the base B are at least 2 are invariants of the oriented differentiable structure. The main Theorems D and E characterize surfaces carrying constant moduli fibrations as surfaces having a Zariski open set satisfying certain topological conditions (e.g., having the right Euler number, the right fundamental group and the right fundamental group at infinity).


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