Sections of Hodge bundles I: Global theory and applications to period maps

We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of the image of the period map, which provides an explicit Euclidean realization. Combining these perspectives, we prove that the image of the lifted period map on the universal cover is contained in a complex Euclidean subspace of the period domain, thereby giving a partial solution to a conjecture of Griffiths on the global behavior of period maps. As an application, we construct a global complex affine structure on the Teichmüller space of Calabi–Yau type manifolds.