Algebraic Solutions of Differential Equations 65 which-point by point is a polarization (126.96.36.199). A family of pure Hodge structure is called polarizable if it admits at least one polarization. The considerations of (188.8.131.52-13) apply mutatis mutandis, in this context. (184.108.40.206) Proposition. Let 5 P be a connected, locally arcwise connected, locally arcwise simply connected topological space (so that 6~' "has" a universal covering). Let (H z, F) be a polarizable family of pure Hodge structures. Suppose that the filtration F is locally constant, in the sense that it comes from a filtration by sub-local systems of the complexified local system H c. Then there exists a finite ~tale covering n: d,~'-* 6 e such that the inverse image n* (H z, F) oj" (Hz, F) on ~' is a constant family of pure Hodge structures. Proof. Fix a point s o ~ 5<. The (topological) fundamental group 7q (~ so) acts on the stalk (Hz)so, and by hypothesis it preserves the filtration F~io of (He)so. A polarization (,) on the family (H z, F) induces a polarization
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Topological space, Hodge, Connected space, Hodge structures, pure Hodge