Algebraicity, Hodge Gradings, and the Geometry of Period Images

We prove the algebraicity of period images in arithmetic quotients without using o-minimality. The proof uses infinitesimal neighborhoods and formal algebraization to recover the images and the maps from their algebraic sources. Hodge positivity and an algebraic ampleness criterion yield the Griffiths polarization and quasi-projectivity. For lifted images, we construct Stein neighborhoods and holomorphic matrix realizations of the Hodge filtration. On closed horizontal subspaces, we bound canonical Bergman density below by the Hodge volume form. We compare the Bergman and Griffiths--Schmid metrics near one-parameter boundaries with nontrivial unipotent monodromy and nondegenerate graded period maps, and in all directions near general one- and two-node quintic threefolds. A period surface arising from abelian threefolds shows that the upper metric comparison fails in general.