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Strong openness conjecture and related problems for plurisubharmonic functions

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmonic functions related to the jumping numbers, which were posed by Demailly-Kollár and Jonsson-Mustată respectively. We give a new proof of a lower semicontinuity conjecture posed by Demailly-Kollár without using the ACC conjecture. Other applications by combining with well-known results are also mentioned.