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Irregularities of special C-pairs

Stefan Kebekus, Erwan Rousseau, Frédéric Touzet

TL;DR

The paper addresses irregularity-type invariants for special $\mathcal{C}$-pairs in Campana’s framework, proving a universal bound $q^+(X,D)\le \dim X$ under mild singularities and broadening Campana’s classic results. It develops a robust toolkit: an extension theorem for adapted reflexive differentials, a foliated-albanese approach via covers, and a Bogomolov-type theory built from strict wedge subspaces to control invariants across covers. Central to the analysis is the construction of $G$-invariant Bogomolov sheaves and the use of Kodaira-Iitaka dimensions of adapted differential sheaves, all culminating in the main bound for augmented irregularity. The results illuminate the interplay between differential-geometric invariants and birational geometry in the minimal model program, with implications for “special” vs. non-special behavior and potential density phenomena in arithmetic and analytic contexts.

Abstract

This paper studies irregularity-type invariants of special C-pairs, or "geometric orbifolds" in the sense of Campana. Under mild assumptions on the singularities, we show that the augmented irregularity of a C-pair (X,D) is bounded by its dimension. This generalizes earlier results of Campana, and strengthens known results even in the classic case where X is a projective manifold and D = 0. The proof builds on new extension results for adapted forms, analysis of foliations on Albanese varieties, and constructions of Bogomolov sheaves using strict wedge subspaces of adapted forms.

Irregularities of special C-pairs

TL;DR

The paper addresses irregularity-type invariants for special -pairs in Campana’s framework, proving a universal bound under mild singularities and broadening Campana’s classic results. It develops a robust toolkit: an extension theorem for adapted reflexive differentials, a foliated-albanese approach via covers, and a Bogomolov-type theory built from strict wedge subspaces to control invariants across covers. Central to the analysis is the construction of -invariant Bogomolov sheaves and the use of Kodaira-Iitaka dimensions of adapted differential sheaves, all culminating in the main bound for augmented irregularity. The results illuminate the interplay between differential-geometric invariants and birational geometry in the minimal model program, with implications for “special” vs. non-special behavior and potential density phenomena in arithmetic and analytic contexts.

Abstract

This paper studies irregularity-type invariants of special C-pairs, or "geometric orbifolds" in the sense of Campana. Under mild assumptions on the singularities, we show that the augmented irregularity of a C-pair (X,D) is bounded by its dimension. This generalizes earlier results of Campana, and strengthens known results even in the classic case where X is a projective manifold and D = 0. The proof builds on new extension results for adapted forms, analysis of foliations on Albanese varieties, and constructions of Bogomolov sheaves using strict wedge subspaces of adapted forms.
Paper Structure (37 sections, 27 theorems, 97 equations)

This paper contains 37 sections, 27 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Outline of the proof
  4. Acknowledgements
  5. Notation and known results
  6. Global assumptions and standard notation
  7. C-pairs and adapted morphisms
  8. Linear systems in reflexive sheaves
  9. SNC morphisms
  10. Forms on fibre spaces
  11. Foliations defined by meromorphic maps
  12. Rational maps induced by generically generated sheaves
  13. Functoriality
  14. Extension of adapted reflexive differentials, Proof of Theorem \ref{['thm:1-6']}
  15. Definition
  16. ...and 22 more sections

Key Result

Theorem 1.3

Let $(X, D)$ be a $\mathcal{C}$-pair that satisfies one of the following conditions. If $(X,D)$ is special, then $q⁺(X, D) ≤ \dim X$.

Theorems & Definitions (72)

  • Conjecture 1.1: Irregularities of special $\mathcal{C}$-pairs, orbiAlb1
  • Definition 1.2: Irregularity, augmented irregularity, orbiAlb1
  • Theorem 1.3: Boundedness of augmented irregularity
  • Remark 1.4: Novelty of the result
  • Remark 1.5: Earlier results on Albanese irregularities
  • Theorem 1.6: Extension of adapted forms on dlt pairs
  • Corollary 1.7: Adapted one-forms on covers of spacial pairs
  • Definition 2.1: Big and small sets
  • Definition 2.2: $q$-morphisms
  • Remark 2.6
  • ...and 62 more