Analog of Satake-Baily-Borel for period maps

Abstract

We propose an analog of the Satake--Baily--Borel compactification and Borel's extension theorem for arbitrary period maps. The proposed analog is constructed as a proper topological completion of the period map. It is conjectured that the construction is projective algebraic, and the conjecture is reduced to a certain extension problem.

This is a major revision. The previous version has been separated into two parts; this is Part 1. The previous version erroneously claimed that certain line bundles $\det (F^p)$ descend to the SBB completion. We thank Ben Bakker for providing a counter-example. We are able to show that the product ${\prod }_p\det (F^p{)}^{\otimes m_p}$ descends, for suitable choice of $m_p>0$