A proof of Demailly's strong openness conjecture

In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly. The multiplier ideal sheaf associated to a plurisubharmonic function, which is an invariant of the singularities of the psh function, plays an important role in several complex variables and complex geometry. Various properties about the multiplier ideal sheaves associated to plurisubharmonic functions have been discussed (e.g., see [25], [7], [30], [31]). Demailly’s strong openness conjecture means that the strong openness property about the multiplier ideal sheaf holds. In the present article, we establish such a useful strong openness property on the multiplier ideal sheaf associated to any plurisubharmonic function; i.e., Demailly’s strong openness conjecture is true. 1.1. Organization of the paper. The paper is organized as follows. In the present section, we recall the statement of the strong openness conjecture posed by Demailly and present the main result of the present paper: a solution of the strong openness conjecture. In Section 2, we recall or give some preliminary lemmas used in the proof of the main result. In Section 3, we give the proof of the strong openness conjecture and present some consequences by combining the conjecture with some well-known results.