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On the complement of nef divisors on projective manifolds

S. Feklistov

TL;DR

The paper characterizes when the complement of a nef effective divisor on a complex projective manifold fails the Hartogs extension property, proving this occurs precisely when the divisor is abundant with Iitaka dimension one. The authors analyze the problem via a case split on κ(D) and ν(D), combining Ueda theory, Lefschetz-type cohomology, and fibration arguments to derive a fiber-structure criterion: X is not Hartogs iff X admits a fibration over an affine curve. The results yield practical tests through linear systems and yield corollaries and examples on ruled surfaces, while suggesting pathways to extend the framework to Kähler manifolds and non-nef settings through divisorial decompositions. Overall, the work links geometric positivity, fibration theory, and complex-analytic extension phenomena to give a precise, transferable criterion for Hartogs in complements of nef divisors.

Abstract

Let $X'$ be a complex projective manifold, $\dim X'>1$, $Z$ a connected analytic subset of codimension one which is the support of a nef effective Cartier divisor $D$ on $X'$, $X:=X'\setminus Z$. Let $κ(D)$ be the Iitaka dimension of $D$. We prove that $X$ is not Hartogs if and only if $D$ is abundant and $κ(D)=1$. In particular, $X$ is not Hartogs if and only if $X$ is a proper fibration over an affine curve.

On the complement of nef divisors on projective manifolds

TL;DR

The paper characterizes when the complement of a nef effective divisor on a complex projective manifold fails the Hartogs extension property, proving this occurs precisely when the divisor is abundant with Iitaka dimension one. The authors analyze the problem via a case split on κ(D) and ν(D), combining Ueda theory, Lefschetz-type cohomology, and fibration arguments to derive a fiber-structure criterion: X is not Hartogs iff X admits a fibration over an affine curve. The results yield practical tests through linear systems and yield corollaries and examples on ruled surfaces, while suggesting pathways to extend the framework to Kähler manifolds and non-nef settings through divisorial decompositions. Overall, the work links geometric positivity, fibration theory, and complex-analytic extension phenomena to give a precise, transferable criterion for Hartogs in complements of nef divisors.

Abstract

Let be a complex projective manifold, , a connected analytic subset of codimension one which is the support of a nef effective Cartier divisor on , . Let be the Iitaka dimension of . We prove that is not Hartogs if and only if is abundant and . In particular, is not Hartogs if and only if is a proper fibration over an affine curve.
Paper Structure (16 sections, 24 theorems, 24 equations)

This paper contains 16 sections, 24 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ueda theory
  4. Lefschetz-type properties
  5. Moving divisors
  6. Proper maps with connected fibers
  7. Proof of Main Theorem
  8. Case $\kappa(D)=\nu(D)=1$
  9. Case $\nu(D)\geq 2$
  10. Case $\kappa(D)=0$, $\nu(D)=1$
  11. Examples
  12. Case of non-splitting of (\ref{['ext']})
  13. Case of splitting of (\ref{['ext']})
  14. Conclusion
  15. Nonsingular projective surfaces.
  16. ...and 1 more sections

Key Result

Corollary 1

Let $X'$ be a complex projective manifold, $\dim X'>1$, $D$ an effective nef divisor with connected support $Z$, $X:=X'\setminus Z$. Then $X$ is not Hartogs if and only if $mD$ is a fiber of a fibration $X'\to C'$ over a projective curve $C'$ for some $m\in\mathbb{Z}_{>0}$.

Theorems & Definitions (43)

  • Definition
  • Corollary
  • Corollary
  • Corollary
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 33 more