The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X, D) with D\neq 0 and with the log Fano pseudoindex \geq 2, we prove that the restriction homomorphism Pic(X)\to Pic(D_1) of Picard groups is injective for any irreducible component D_1\subset D.The strategy of our proof is to run a certain minimal model program and is similar to the argument of Casagrande’s one. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).