Analog of Satake-Baily-Borel for period maps
Mark Green, Phillip Griffiths, Colleen Robles
TL;DR
This work generalizes the Satake–Baily–Borel compactification to arbitrary period maps by constructing a topological completion ${\overline{\wp}}$ of the period map and a corresponding extension ${\Phi}_{\mathrm{SBB}}:{\overline{B}}\to{\overline{\wp}}$. It develops a framework of limiting mixed Hodge structures at infinity, with monodromy cones $\sigma_I$ and strata $Z_I^*$, to control degenerations and define extended period maps $\Phi_I$. A Stein-factorization analogue yields ${\overline{\wp}}'$ and connected fibres, and the boundary has a projective-analytic structure in certain cases; a conjectural algebraic structure (GGLR) is connected to holomorphic extension problems. Key technical achievements include a topological completion with a proper, continuous ${\Phi}_{\mathrm{SBB}}$, a reduced limit period map compatible with the LMHS, descent results for determinant line bundles via a monodromy character $\chi_\infty$, and norm-one/root-of-unity properties for $\chi_\infty$ that enable descent to boundary quotients. The paper also links these constructions to a semi-ampleness conjecture and discusses the implications for algebraicity in low dimensions, offering a path toward a broader SBB-type compactification beyond Hermitian symmetric domains.
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.