In this note, we reveal that our solution of Demaillys strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll{a}r and Jonsson-Mustatu{a} implies the truth of twist
In this note, we present the concavity of the minimal $L^2$ integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal $L^{2}$ extension problem to hold.
In this article, we establish a sharp effectiveness result of Demaillys strong openness conjecture. We also establish a sharp effectiveness result related to a conjecture posed by Demailly and Kollar.