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Hyperbolicity of generic hypersurfaces of polynomial degree via Green-Griffiths jet differentials

Benoit Cadorel

Abstract

We give a new version of a recent result of B{é}rczi-Kirwan, proving the Kobayashi and Green-Griffiths-Lang conjectures for generic hypersurfaces in the projective space , with a polynomial lower bound on the degree. Our strategy again relies on Siu's technique of slanted vector fields and the use of holomorphic Morse inequalities to prove the existence of a jet differential equation with a negative twist -- however, instead of using a space of invariant jet differentials, we base our computations on the classical Green-Griffiths jet spaces.

Hyperbolicity of generic hypersurfaces of polynomial degree via Green-Griffiths jet differentials

Abstract

We give a new version of a recent result of B{é}rczi-Kirwan, proving the Kobayashi and Green-Griffiths-Lang conjectures for generic hypersurfaces in the projective space , with a polynomial lower bound on the degree. Our strategy again relies on Siu's technique of slanted vector fields and the use of holomorphic Morse inequalities to prove the existence of a jet differential equation with a negative twist -- however, instead of using a space of invariant jet differentials, we base our computations on the classical Green-Griffiths jet spaces.

Paper Structure

This paper contains 17 sections, 23 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Organization of the article.
  3. Acknowledgments
  4. Weighted projective bundles and Green-Griffiths jet differentials
  5. Weighted projective bundles
  6. Weighted Segre classes
  7. Green-Griffiths jet differentials
  8. Holomorphic Morse inequalities
  9. Nefness of adequate twists
  10. Statement of the main results
  11. Main computations
  12. Annex. Intersection theory on weighted projective bundles
  13. Combinatorial lemmas
  14. Lattices and volumes of fundamental domains
  15. Some integrals
  16. ...and 2 more sections

Key Result

Theorem 1

There exists two sequences of integers $d_{n}, d_{n}'$ such that the following hold:

Theorems & Definitions (41)

  • Conjecture 1.1
  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Remark 2.2
  • Theorem 3: Green-Griffiths GG80, see also Dem12a
  • Proposition 2.3
  • proof
  • Theorem 4
  • Proposition 2.4
  • ...and 31 more