o-minimal GAGA and a conjecture of Griffiths

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