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Picard groups of completed period images and the Deng-Robles problem

Badre Mounda, Dongzhe Zheng

Abstract

A basic problem in the geometry of degenerating period maps is to determine whether their completed images admit an intrinsic algebraic description. For polarized variations of Hodge structure over smooth quasi-projective surfaces, Deng and Robles formulated such a problem in terms of the Kato-Nakayama-Usui completion of the period image and a conjectural Proj description involving the augmented Hodge line bundle and the boundary divisor on a smooth compactification of the base. We show that the essential obstruction to this description is divisor-theoretic: it may be expressed as a Picard-generation statement on the completed mixed period image. We prove this statement when the pure period image is one-dimensional, and consequently obtain the Deng-Robles Proj description in this case.

Picard groups of completed period images and the Deng-Robles problem

Abstract

A basic problem in the geometry of degenerating period maps is to determine whether their completed images admit an intrinsic algebraic description. For polarized variations of Hodge structure over smooth quasi-projective surfaces, Deng and Robles formulated such a problem in terms of the Kato-Nakayama-Usui completion of the period image and a conjectural Proj description involving the augmented Hodge line bundle and the boundary divisor on a smooth compactification of the base. We show that the essential obstruction to this description is divisor-theoretic: it may be expressed as a Picard-generation statement on the completed mixed period image. We prove this statement when the pure period image is one-dimensional, and consequently obtain the Deng-Robles Proj description in this case.
Paper Structure (21 sections, 19 theorems, 75 equations)

This paper contains 21 sections, 19 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Kato--Usui and Kato--Nakayama--Usui Completion
  5. Quasi-projectivity and Theta Positivity of Mixed Period Map Images
  6. SBB-type Completion and Positivity of Augmented Hodge Line Bundle
  7. GGR Local Structure: LMHS Fibers and Theta--Boundary Formula
  8. Deng--Robles Completion and Contraction
  9. Picard, Néron--Severi, and Horizontal/Vertical Decomposition
  10. Structure of Question 1.8 and Reduction to Picard Generation
  11. Augmented Hodge Line Bundle on the Image Space
  12. Relation between Λ_Y and the Theta Line Bundle
  13. Picard Generation Condition (HPic)
  14. From (HPic) to the Span Condition (*Span)
  15. Picard Generation When Pure Period Image Has Dimension One
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $D$ be the period domain parametrizing polarized pure Hodge structures of a given type, and $\Gamma\subset G(\mathbb Q)$ a discrete subgroup. Given a rational fan $\Sigma$ satisfying strong compatibility with $\Gamma$, there exists a complex space with log structure and a natural embedding $D\hookrightarrow D_\Sigma$, and for each polarized VHS with quasi-unipotent monodromy at the boundary,

Theorems & Definitions (29)

  • Theorem 3.1: Kato--Usui, pure case KU09
  • Theorem 3.2: Kato--Nakayama--Usui, mixed case KNU10KNU13
  • Theorem 3.3: Usui Usu06
  • Theorem 3.4: Deng--Robles DengRobles23
  • Theorem 3.5: BBKT, BBT BBKT24BBT23BBT23GAGA
  • Theorem 3.6: GGLR, GGR GGLR20GGR20SBBGGR25
  • Theorem 3.7: Fibers are complex tori, GGR25
  • Theorem 3.8: Theta--boundary formula, GGR25
  • Theorem 3.9: Extension rigidity, GGR25
  • Theorem 3.10: Deng--Robles DengRobles23
  • ...and 19 more