Compositio Mathematica

Research Article

Differential equations on complex projective hypersurfaces of low dimension

Simone Diverioa1 p1

a1 Institut Fourier, Université de Grenoble I, BP 74, F-38402, Saint Martin d’Héres, France (email:


Let n=2,3,4,5 and let X be a smooth complex projective hypersurface of $\mathbb {P}^{n+1}$. In this paper we find an effective lower bound for the degree of X, such that every holomorphic entire curve in X must satisfy an algebraic differential equation of order k=n=dim X, and also similar bounds for order k>n. Moreover, for every integer n≥2, we show that there are no such algebraic differential equations of order k<n for a smooth hypersurface in $\mathbb {P}^{n+1}$.

(Received July 23 2007)

(Accepted November 26 2007)

2000 Mathematics subject classification

  • 14F99;
  • 32Q45


  • Kobayashi hyperbolicity;
  • invariant jet differentials;
  • Schur powers;
  • holomorphic Morse inequalities


p1 Current address: Istituto ‘Guido Castelnuovo’, Università di Roma ‘La Sapienza’, P. le Aldo Moro 2, 00185 Roma, Italy