Concavity of minimal L2 integrals related to multiplier ideal sheaves on weakly pseudoconvex Kähler manifolds

In this paper, we present the concavity of the minimal L 2 integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kähler manifolds which implies the sharp effectiveness results of the strong openness conjecture and a conjecture posed by Demailly and Kollár (2001) on weakly pseudoconvex Kähler manifolds. We obtain the relation between the concavity and the L 2 extension theorem.