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Gurjar has related this question to the Shafarevitch conjecture [17] which asks: is X holomorphically convex ? F. Campana and J. Kollar have studied, for examples, non negativity of a variant Kodaira dimension [3] and some nonvanishing properties for plurigenera [12]. The author [23] has shown recently that the Shafarevitch varieties sh(X) in the semisimple representations case are always of general type and all nonconstant morphisms form all curves of genus < 1 into sh(X) must be contained in a proper subvariety of sh(X). In a sense, Theorem 1 can be thought as a sharp version of this property.

A finiteness property of representations of $\pi_1$ of algebraic surfaces