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Algebraic Solutions of Differential Equations
65
whichpoint by point is a polarization (4.2.0.8). A family of pure Hodge
structure is called
polarizable
if it admits at least one polarization. The
considerations of (4.2.0.1113) apply
mutatis mutandis,
in this context.
(4.2.1.3)
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 sublocal 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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 Fall '11
 NormanKatz

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