2017/08/31 by Mark Green, Green, Mark, Phillip Griffiths +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1708.09523
openalex publication_date 2017/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let P be the image of a period map. We discuss progress towards a conjectural Hodge theoretic completion P, an analogue of the Satake-Baily-Borel compactification in the classical case. The set P is defined and given the structure of a compact Hausdorff topological space. We conjecture that it admits the structure of a compact complex analytic variety. We verify this conjecture when dim P ≤ 2. In general, P admits a finite cover S (also a compact Hausdorff space, and constructed from Stein factorizations of period maps). Assuming that S is a compact complex analytic variety, we show that a lift of the augmented Hodge line bundle Λ extends to an ample line bundle, giving P the structure of a projective normal variety. Our arguments rely on refined positivity properties of Chern forms associated to various Hodge bundles; properties that might be of independent interest.