JournalscmhVol. 78 , No. 3DOI 10.1007/s00014-003-0765-x

Sur une conjecture de Mukai

  • Laurent Bonavero

    Université Grenoble I, Saint-Martin-d'Hères, France
  • Stéphane Druel

    Université Grenoble I, Saint-Martin-d'Hères, France
  • Olivier Debarre

    École Normale Supérieure, Paris, France
  • Cinzia Casagrande

    Università di Pisa, Italy
Sur une conjecture de Mukai cover

Abstract

Generalizing a question of Mukai, we conjecture that a Fano manifold X with Picard number ρX\rho_X and pseudo-index ιX\iota_X satisfies ρX\rho_X (ιX\iota_X - 1) <= dim(X). We prove this inequality in several situations: X is a Fano manifold of dimension <= 4, X is a toric Fano manifold of dimension <= 7 or X is a toric Fano manifold of arbitrary dimension with ιX\iota_X >= dim(X) / 3 + 1. Finally, we offer a new approach to the general case.