On the generalized toroidal completion of period mappings

TL;DR

The paper develops a generalized toroidal approach to completing period mappings on quasi-projective bases by introducing the combinatorial monodromy complex (CMCX). This framework enables constructing a completion $ar B$ of the base and a corresponding target $arp$ so that the extended period map records boundary data through nilpotent-orbit limiting mixed Hodge structures, even when the period domain is not Hermitian symmetric. It connects Hodge-theoretic degeneration with logarithmic/toridal geometry to produce a holomorphic, often projective, completion and applies this to give an alternative algebraicity proof for weakly special subvarieties. The work also clarifies the relationship with existing approaches (e.g., KU08, KNU10, DR23, BFMT25) and demonstrates the feasibility of projective completions in concrete cases (notably in dimension two). The combination of CMCX construction, analytic extension, and finiteness results for hyperintersections provides a robust toolkit for boundary behavior of period mappings in broad Hodge-theoretic settings.

Abstract

Given a period map defined over a quasi-projective variety, we construct a completion with rich geometric and Hodge-theoretic meaning. This result may be regarded as an analog of Mumford's toroidal compactification for locally symmetric Hodge varieties as well as a realizable alternative of Kato--Nakayama--Usui's construction.

Figures (2)

• Figure 1: Semicomplex from subdividing complex
• Figure 2: The canonical complexification

Theorems & Definitions (99)

• Theorem 1.1
• Theorem 2.1: BB66AMRT10
• Theorem 2.2: Nam76AB12
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• Theorem 2.8
• Remark 2.9