Copilot：
What is Demailly's Strong Openness Conjecture
回答：
Demailly's Strong Openness Conjecture is a deep result in complex geometry and analysis, particularly in the theory of multiplier ideal sheaves associated with plurisubharmonic functions.

The Conjecture (Now a Theorem)
Let \varphi be a plurisubharmonic function on a complex manifold X. The conjecture asserts that:
\mathcal{I}_+(\varphi) = \mathcal{I}(\varphi)
where:
- \mathcal{I}(\varphi) is the multiplier ideal sheaf: the sheaf of holomorphic functions f such that |f|^2 e^{-\varphi} is locally integrable.
- \mathcal{I}_+(\varphi) := \bigcup_{\varepsilon > 0} \mathcal{I}((1+\varepsilon)\varphi)
This means that the ideal sheaf stabilizes under small perturbations of the weight function \varphi, reflecting a kind of openness in the space of integrable functions

Historical Context
Jean-Pierre Demailly proposed this conjecture as a strengthening of the earlier Openness Conjecture (by Demailly and Kollár), which was concerned with the integrability of exponentials of plurisubharmonic functions.

✅ Resolution
The conjecture was proven by Qi'an Guan and Xiangyu Zhou in 2015. Their work not only settled the conjecture but also led to further insights into:
- The behavior of singularities of plurisubharmonic functions
- The structure of multiplier ideal sheaves
- Applications to algebraic geometry and complex differential geometry
You can explore the original proof in the Annals of Mathematics article1) or the expanded version on arXiv2).
Would you like to dive into the techniques used in the proof, such as the L^2 extension theorem or curve selection lemma? I think you'd find the analytic machinery quite elegant.

注：
1)Annals of Mathematics article
https://annals.math.princeton.edu/2015/182-2/p05
A proof of Demailly’s strong openness conjecture
Pages 605-616 from Volume 182 (2015), Issue 2 by Qi'an Guan, Xiangyu Zhou
Abstract
In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.

2)arXiv
https://arxiv.org/abs/1401.7158
[Submitted on 28 Jan 2014]
Strong openness conjecture and related problems for plurisubharmonic functions
Qi'an Guan, Xiangyu Zhou