  We prove a conjecture of Griffiths on the quasi-projectivity of images of
period maps using algebraization results arising from o-minimal geometry.
Specifically, we first develop a theory of analytic spaces and coherent sheaves
that are definable with respect to a given o-minimal structure, and prove a
GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic
spaces. We then combine this with algebraization theorems of Artin to show that
proper definable images of complex algebraic spaces are algebraic. Applying
this to period maps, we conclude that the images of period maps are
quasi-projective and that the restriction of the Griffiths bundle is ample.
