Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i(X) and Picard number rho(X). A generalization of a conjecture of Mukai says that rho(X)(i(X) - 1) <= n. We prove that the conjecture holds for a variety X of pseudoindex i(X) >= n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if rho(X) > 1 and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five. (C) Central European Science Journals. All rights reserved.