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Submanifolds with ample normal bundle

Maycol Falla Luza, Frank Loray, Jorge Vitório Pereira

Abstract

We construct germs of complex manifolds of dimension $m$ along projective submanifolds of dimension $n$ with ample normal bundle and without non-constant meromorphic functions whenever $m \geq 2n$. We also show that our methods do not allow the construction of similar examples when $m < 2n$ by establishing an algebraicity criterion for foliations on projective spaces which generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.

Submanifolds with ample normal bundle

Abstract

We construct germs of complex manifolds of dimension along projective submanifolds of dimension with ample normal bundle and without non-constant meromorphic functions whenever . We also show that our methods do not allow the construction of similar examples when by establishing an algebraicity criterion for foliations on projective spaces which generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.
Paper Structure (16 sections, 15 theorems, 10 equations)

This paper contains 16 sections, 15 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Extension
  4. Conventions
  5. Infinitesimal neighborhoods
  6. The field of formal meromorphic functions
  7. Continuation of analytic subvarieties
  8. Construction
  9. Weakly transverse submanifolds
  10. Formal meromorphic functions versus rational first integrals
  11. Existence of weakly transverse submanifolds
  12. Field of rational first integrals of foliations on projective manifolds
  13. Proof of Theorem \ref{['THM:A']}
  14. Algebraicity
  15. Submanifolds transverse to foliations
  16. ...and 1 more sections

Key Result

Theorem 1

For any complex projective manifold $Y$ of dimension $n$ and any pair of natural numbers $(\ell, m)$ such that $m \ge 2n$ and $\ell \le m$, there exists a germ of complex manifold $X$ of dimension $m$ and containing $Y$ such that the normal bundle of $Y$ in $X$ is ample and the transcendence degree

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Theorem 2.6
  • Proposition 2.7
  • proof
  • Proposition 3.1
  • ...and 17 more