Strong openness of multiplier ideal sheaves and optimal $L^{2}$ extension

摘要

In this note, we reveal that our solution of Demailly's strong opennessconjecture implies a matrix version of the conjecture; our solutions of twoconjectures of Demailly-Koll\'{a}r and Jonsson-Mustat\u{a} implies the truth oftwisted versions of the strong openness conjecture; our optimal $L^{2}$extension implies Berndtsson's positivity of vector bundles associated toholomorphic fibrations over a unit disc.