Commentarii Mathematici Helvetici
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Volume 78, Issue 3, 2003, pp. 601–626
DOI: 10.1007/s00014-003-0765-x
Sur une conjecture de Mukai
Laurent Bonavero (1), Cinzia Casagrande (2), Olivier Debarre (3) and Stéphane Druel (4)
(1) Institut Fourier, UFR de Mathématiques, Université Grenoble I, UMR 5582, B.P. 74, F-38402, SAINT MARTIN D'HERES CEDEX, FRANCE(2) Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5,, 56127, PISA, ITALY
(3) Département de Mathématiques et Applications , École Normale Supérieure, 45 rue d'Ulm, 75230, PARIS CEDEX 05, FRANCE
(4) Institut Fourier, UFR de Mathématiques, Université Grenoble I, UMR 5582, B.P. 74, F-38402, SAINT MARTIN D'HERES CEDEX, FRANCE
Generalizing a question of Mukai, we conjecture that a Fano manifold X with Picard number $\rho_X$ and pseudo-index $\iota_X$ satisfies $\rho_X$ ($\iota_X$ - 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 $\iota_X$ >= dim(X) / 3 + 1. Finally, we offer a new approach to the general case.
Keywords: Variétés de Fano, théorie de Mori, géométrie torique