Analog of Satake-Baily-Borel for period maps

TL;DR

This paper introduces a new topological completion method for period maps, inspired by Satake-Baily-Borel compactification, aiming to extend the classical theory to broader contexts with a conjecture on algebraicity.

Contribution

It proposes an analog of the Satake-Baily-Borel compactification for period maps and reduces its algebraicity conjecture to an extension problem.

Findings

Constructed a proper topological completion of period maps.

Formulated a conjecture on the projective algebraic nature of the completion.

Reduced the algebraicity conjecture to a specific extension problem.

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.