The openness conjecture for plurisubharmonic functions

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We give a proof of the openness conjecture of Demailly and Ko llár.

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The weighted log canonical threshold (2014, background): “In 2013, Berndtsson [2] completely proved it in arbitrary dimension.”
A sharp effectiveness result of Demailly's strong openness conjecture (2017, background): “ I(ϕ) = I+(ϕ) := ∪p>1I(pϕ). The two-dimensional case of SOC was proved by Jonsson-Musta¸t˘a [19]. An important case of SOC so-called the openness conjecture (OC for short) was proved by Berndtsson [2] and the two-dimensional case of OC was proved by Favre-Jonsson [10, 9]. Recently, SOC was proved”
The openness conjecture and complex Brunn-Minkowski inequalities (2014, background): “the Green potential of ∆u, but the higher dimensional case is much more subtle. The openness conjecture was ﬁrst proved in dime nsion 2 by Favre and Jonsson (see [11]), and then for all dimensions in [2]. After that, simpler proofs and generalizations to the so called strong openness conjecture (see”