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Campana rational connectedness and weak approximation

Qile Chen, Brian Lehmann, Sho Tanimoto

TL;DR

This paper develops a log-geometric framework for Campana orbifolds to study Campana curves, Campana maps, and Campana rational connectedness. It defines stable log maps and their deformation theory, including gluing, smoothing, and splitting of contact orders, to construct Campana sections with prescribed jet data and to analyze freeness properties. Under a strong Campana uniruledness hypothesis on the general fiber, it proves a weak approximation result for Campana sections at places of good reduction, and verifies Campana rational connectedness in the toric setting. The toric and P^1-fibration cases are treated via explicit log-curve parametrizations and lattice-geometry arguments, establishing a robust toric counterpart to the general theory. Overall, the work connects log geometry, moduli stacks of stable log maps, and Campana’s orbifold notions to advance understanding of weak approximation and rational connectivity in Campana settings.

Abstract

Campana introduced a notion of Campana rational connectedness for Campana orbifolds. Given a Campana fibration over a complex curve, we prove that a version of weak approximation for Campana sections holds at places of good reduction when the general fiber satisfies a slightly stronger version of Campana rational connectedness. Campana also conjectured that any Fano orbifold is Campana rationally connected; we verify a stronger statement for toric Campana orbifolds. A key tool in our study is log geometry and moduli stacks of stable log maps.

Campana rational connectedness and weak approximation

TL;DR

This paper develops a log-geometric framework for Campana orbifolds to study Campana curves, Campana maps, and Campana rational connectedness. It defines stable log maps and their deformation theory, including gluing, smoothing, and splitting of contact orders, to construct Campana sections with prescribed jet data and to analyze freeness properties. Under a strong Campana uniruledness hypothesis on the general fiber, it proves a weak approximation result for Campana sections at places of good reduction, and verifies Campana rational connectedness in the toric setting. The toric and P^1-fibration cases are treated via explicit log-curve parametrizations and lattice-geometry arguments, establishing a robust toric counterpart to the general theory. Overall, the work connects log geometry, moduli stacks of stable log maps, and Campana’s orbifold notions to advance understanding of weak approximation and rational connectivity in Campana settings.

Abstract

Campana introduced a notion of Campana rational connectedness for Campana orbifolds. Given a Campana fibration over a complex curve, we prove that a version of weak approximation for Campana sections holds at places of good reduction when the general fiber satisfies a slightly stronger version of Campana rational connectedness. Campana also conjectured that any Fano orbifold is Campana rationally connected; we verify a stronger statement for toric Campana orbifolds. A key tool in our study is log geometry and moduli stacks of stable log maps.
Paper Structure (42 sections, 29 theorems, 123 equations)

This paper contains 42 sections, 29 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Campana curves
  3. Main results
  4. Related works
  5. The plan of the paper
  6. Deformation theory for log maps
  7. Log maps and their stacks
  8. Deformations of log maps
  9. Deformations of log maps relative to $\mathfrak{M}^{\log}_{g,|\varsigma|}$
  10. Deformations of log maps relative to $\mathbf{Log}$
  11. The deformation theory of log immersions
  12. Deformations of log maps with point constraints
  13. Deformation of log maps relative to codimension $1$ boundary
  14. Log gluing along markings
  15. Gluing along non-degenerate point
  16. ...and 27 more sections

Key Result

Theorem 1.6

Assume that $\mathbf k$ is an algebraically closed field of characteristic $0$. Let $\pi: (\mathcal{X},\Delta_\epsilon) \to B$ be a klt Campana fibration over $\mathbf k$ such that a general fiber of $\pi$ is rationally connected and is strongly Campana uniruled (in the sense of Definition defi:stro

Theorems & Definitions (86)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Example 1.5: Campana
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 76 more