Nonhomeomorphic conjugates of connected Shimura varieties


We show that conjugation by an automorphism of C may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups. Let V be an algebraic variety over C, and let τ be an automorphism of C (as an abstract field). On applying τ to the coefficients of the polynomials defining V, we obtain a conjugate algebraic variety τV over C. The cohomology groups H i (V an, Q) and H i ((τV) an, Q) have the same dimension, and hence are isomorphic, because, when tensored with Qℓ, they become isomorphic to the étale cohomology groups which are unchanged by conjugation. Similarly, the profinite completions of the fundamental groups of V an and (τV) an are isomorphic because they are isomorphic to the étale fundamental groups. However, Serre [Se] gave an example in which the fundamental groups themselves are not isomorphic (see [EV] for a discussion of this and other examples). It seems to have been known (or, at least, expected) for some time that the theory of Shimura varieties provides many more examples, but, as far we know, the details have not been written dow

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