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Baily--Borel compactifications of period images and the b-semiampleness conjecture

Benjamin Bakker, Stefano Filipazzi, Mirko Mauri, Jacob Tsimerman

TL;DR

The article develops a comprehensive Hodge-theoretic framework to realize Baily–Borel-type compactifications for period images and proves a broad semiampleness phenomenon for Griffiths and Hodge line bundles, via definable GAGA and CY variations. By constructing moderate-growth sections of Griffiths/Hodge bundles and leveraging CY boundaries, it establishes a functorial BB compactification Y^{BB} for period images and a CY-oriented BBH compactification for moduli with Hodge bundles extending amply. The work ties together Borel-type extension theorems, moduli problems for Calabi–Yau varieties, and lc-trivial fibrations, delivering a robust bridge between Hodge theory and birational geometry (via the b-semiampleness conjecture). It further clarifies how these compactifications interplay with adjunction, stratifications, and the structure of lc centers, offering new tools for moduli theory and the MMP in settings enriched by Hodge-theoretic data.

Abstract

We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which the Griffiths bundle extends amply. In particular the Griffiths bundle is semiample. We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture and the existence of a functorial Hodge-theoretic compactification of moduli spaces of polarized Calabi--Yau varieties. The semiampleness results (and the construction of the Baily--Borel compactifications) crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Kollár on the geometry of minimal lc centers to verify the extra conditions.

Baily--Borel compactifications of period images and the b-semiampleness conjecture

TL;DR

The article develops a comprehensive Hodge-theoretic framework to realize Baily–Borel-type compactifications for period images and proves a broad semiampleness phenomenon for Griffiths and Hodge line bundles, via definable GAGA and CY variations. By constructing moderate-growth sections of Griffiths/Hodge bundles and leveraging CY boundaries, it establishes a functorial BB compactification Y^{BB} for period images and a CY-oriented BBH compactification for moduli with Hodge bundles extending amply. The work ties together Borel-type extension theorems, moduli problems for Calabi–Yau varieties, and lc-trivial fibrations, delivering a robust bridge between Hodge theory and birational geometry (via the b-semiampleness conjecture). It further clarifies how these compactifications interplay with adjunction, stratifications, and the structure of lc centers, offering new tools for moduli theory and the MMP in settings enriched by Hodge-theoretic data.

Abstract

We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which the Griffiths bundle extends amply. In particular the Griffiths bundle is semiample. We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture and the existence of a functorial Hodge-theoretic compactification of moduli spaces of polarized Calabi--Yau varieties. The semiampleness results (and the construction of the Baily--Borel compactifications) crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Kollár on the geometry of minimal lc centers to verify the extra conditions.

Paper Structure

This paper contains 64 sections, 67 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Baily--Borel compactifications of images of period map
  3. Griffiths bundle
  4. Past work
  5. Applications
  6. b-semiampleness conjecture
  7. Past work
  8. Applications
  9. Proof outline
  10. Paper outline
  11. Hodge-theoretic preliminaries
  12. Definable analytic spaces
  13. Trivializing covers of local systems
  14. DR-neighborhoods
  15. Griffiths and Hodge bundles
  16. ...and 49 more sections

Key Result

Theorem 1.1

Let $Y$ be a period image. Then there exists a functorial projective compactification $Y^{\mathrm{BB}}$ such that for any log smooth algebraic space $(\Theta,D_\Theta)$, any morphism $\Theta\backslash D_\Theta\to Y$ for which the resulting morphism $(\Theta\backslash D_\Theta)^\mathrm{an}\to \Gamma\

Theorems & Definitions (161)

  • Theorem 1.1
  • Theorem 1.2: \ref{['thm:bailyborel']}
  • Corollary 1.3
  • Corollary 1.4: of \ref{['thm:bailyborel']}
  • Theorem 1.5: b-semiampleness
  • Theorem 1.6: \ref{['thm:semiample Hodge']}
  • Theorem 1.7: \ref{['thm:BBHodge']}
  • Corollary 1.8: cf. \ref{['cor:CYmod']}
  • Theorem 1.9: \ref{['thm:bbextensino']}
  • Theorem 2.1: Definable GAGA, BBT23
  • ...and 151 more