Period Mappings and Ampleness of the Hodge line bundle

We discuss progress towards a conjectural Hodge theoretic completion of a period map. The completion is defined, and we conjecture that it admits the structure of a compact complex analytic variety. The conjecture is proved when the image of the period map has dimension 1,2. Assuming the conjecture holds, we then prove that the augmented Hodge line bundle extends to an ample line bundle on the completion. In particular, the completion is a projective algebraic variety that compactifies the image, analogous to the Satake-Baily-Borel compactification.Comment: 62 pages. v2 significant revision of the initial submission (v1); v3 further improvements and new references adde

Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.Comment: 26 pages, 9 figure