Completion of the period mappings and ampleness of the Hodge bundles

I. Introduction II. Construction of a completion of the image of a period mapping III. Curvature properties of the extended Hodge line bundle (A) IV. Curvature properties of the extended Hodge line bundle (B) V. Proof that the extended Hodge line bundle is ample VI. Curvature properties of the Hodge vector bundle