Effectiveness of Demailly’s strong openness conjecture and related problems

Abstract

Abstract. In this article, stimulated by the effectiveness in Berndtsson's solution of the openness conjecture and continuing our solution of Demailly's strong openness conjecture, we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result. We explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier, which is implicitly contained in [20]. We also obtain optimal effectiveness of the conjectures of Demailly-Kollár and Jonsson-Mustatȃ respectively.

The Openness Conjecture and Complex Brunn-Minkowski Inequalities

“…The original openness conjecture is thus the case of strong openness when h = 1. The strong openness conjecture was proved by Guan-Zhou in [13] and Hiep, [15]. Here we will discuss how this problem might be related to the methods described above, in the simpler case when we look at integrability over a compact manifold instead of some neighbourhood of the origin.…”

“…It should be stressed that the recent proofs of Guan-Zhou and Hiep are actually simpler than the method to prove the openness conjecture presented here, but we hope that the original proof still has some interest and that the exposition here also may serve as an introduction to how our 'complex Brunn-Minkowski inequalities' can be applied. The method here also gives for free a bound on the exponents p, that can also be obtained in the setting of Guan-Zhou, [14], but with more work.…”

“…The openness conjecture was first proved in dimension 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 below), have been given by Guan-Zhou, [13] and Hiep, [15], see also [20]. The proof of the openness conjecture from [2] was based on positivity properties of certain vector bundles from [4], that in some ways can be seen as a complex variables generalization of the Brunn-Minkowski theorem.…”

Strong openness of multiplier ideal sheaves and optimal L 2 extension

“…In [22], we proved Demailly's strong openness conjecture posed in [11,12] (related effectiveness result see [25]). This means that the multiplier ideal sheaf has strong openness property.…”

“…In [23,25], we prove two conjectures posed by Demailly-Kollár (see [15]) and Jonsson-Mustatȃ (see [28]) respectively by using the following result. have positive lower bounds independent of r ∈ (0, 1).…”