An algebraic approach to the openness conjecture of Demailly and Kollár

By Mattias Jonsson, M. Mustaţă. Journal of the Institute of Mathematics of Jussieu, 2012

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Abstract We reduce the openness conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement.

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A sharp effectiveness result of Demailly's strong openness conjecture (2017, background): “Recall that co (φ) := sup{c ≥ 0 : |F | (2)e is L(1) on a neighborhood of o} is the jumping number (see [20]).”
The weighted log canonical threshold (2014, background): “In 2005, this conjecture was proved in dimension 2 by Favre and Jonsson ([14], [20], [21]).”
Effectiveness of Demailly’s strong openness conjecture and related problems (2014, background): “We generalize the Bergman kernel K on D to Kφ,F as follows: Kφ,F (z0) := 1 inf{‖F1‖(2)0|(F1 − F, z0) ∈ I+(2cF z0(φ)φ)z0 & F1 ∈ O(D)} where cF z0(φ) := sup{c ≥ 0 : |F |2e−2cφ is L1 on a neighborhood of z0} is the jumping number (see [26]).”