Geometric Realizations of Uniformization of Conjugates of Hermitian Locally Symmetric Manifolds

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Let Γ be a bounded symmetric domain, Γ ⊂ Aut(Ω) a torsion-free discrete group of holomorphic automorphisms such that the quotient manifold X = Ω/Γ is of finite volume with respect to the Bergman metric. The manifold X is either algebraic or biholomorphic to a quasi-projective variety, according to Satake, Baily, and Borel [3, 22] for the higher-rank case and to Siu and Yau [24] for the rank-1 case. Fix an embedding of X into a projective space PN and identify X with such a variety. Let σ ∈ Gal(C/Q, and let X σ denote the quasi-projective variety obtained by applying σ to the defining equations of X in P N . By a theorem of Kazhdan [11] in the compact case and a theorem of Borovoy and Kazhdan [5,12] in the general case, X σ ≅ Ω/Γ σ for some torsion-free discrete group of holomorphic automorphisms Γ σ ⊂ Aut(Ω) such that X σ is of finite volume with respect to the Bergmann metric.