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

## TL;DR

This paper connects solutions to several conjectures in complex analysis, demonstrating their implications for the strong openness property of multiplier ideal sheaves and establishing new positivity results for vector bundles.

## Contribution

It introduces a matrix version of Demailly's strong openness conjecture and proves twisted versions, linking them to optimal $L^{2}$ extension and positivity of vector bundles.

## Key findings

- Matrix version of strong openness conjecture proved
- Twisted versions of the strong openness conjecture established
- Positivity of vector bundles from $L^{2}$ extension confirmed

## Abstract

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