Resumen de Sur une conjecture de Mukai

Laurent Bonavero, Cinzia Casagrande, Olivier Debarre, Stéphane Druel

• 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