The Openness Conjecture and Complex Brunn-Minkowski Inequalities

Abstract

ABSTRACT. We discuss recent versions of the Brunn-Minkowski inequality in the complex setting, and use it to prove the openness conjecture of Demailly and Kollár. Dedicated to Bradley Manning and Edward Snowden in recognition of their work for openness

Convergence results for two Kähler–Ricci flows

“…The above statement is more quantitative comparing with the orginal statement in [1] and certainly a direct consequence of the argument there. In application, we look at the function − ∂u ∂t + u ∈ P SH F ω (X) and − ∂u ∂t + u C. The integral bound is clear from global cohomology information: The assumption on Ricci form can be removed by observing that the result in [9] quoted before allows us to replace ∂u ∂t by a constant (the average for example) up to a control term, which can be absorbed by the volume form.…”

Cauchy–Riemann meet Monge–Ampère

“…We will start with the following recent result of Berndtsson [10] (proved by Favre and Jonsson [54] in dimension 2) confirming the openness conjecture of Demailly-Kollár [46].…”

“…The original proof of Theorem 3.1 from [10] was more complicated but Berndtsson [11] extracted the following simple one from the method of Guan-Zhou [58] who showed a more general strong openness conjecture, where instead of e − pϕ one is interested in local integrability of | f | 2 e − pϕ for some fixed holomorphic f . The proof of this was simplified by Hiep [64].…”

“…The simple proof of the Siu theorem was found by Demailly [45] who devised a special approximation of an arbitrary plurisubharmonic function by smooth ones with possibly analytic singularities. The openness conjecture was first proved by Berndtsson [10] who subsequently simplified the proof in [11] using an approach of Guan and Zhou [58].…”

Strong openness of multiplier ideal sheaves and optimal L 2 extension

“…When I(ϕ) is trivial, Demailly's strong openness conjecture degenerates to the openness conjecture posed in [15] which was proved by Berndtsson [5] (dimension two case was proved by Favre and Jonsson [18]). …”