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A proof of Demailly’s strong openness conjecture. (English) Zbl 1329.32016

Let \(X\) be a complex manifold and \(\varphi\) be a plurisubharmonic (PSH) function on \(X\). The multiplier ideal sheaf of \(\varphi\) is defined as the subsheaf \({\mathcal I}(\varphi)\) of \({\mathcal O}_X\) consisting of all germs of holomorphic functions \(f\in {\mathcal O}_x\) such that \(|f|^2e^{-\varphi}\) is locally integrable near \(x\in X\).
The authors prove the Strong Openness Conjecture: \({\mathcal I}(\varphi)=\cup_{\varepsilon>0}{\mathcal I}((1+\varepsilon)\varphi)\). It was first conjectured by J.-P. Demailly [in: School on vanishing theorems and effective results in algebraic geometry. Lecture notes of the school held in Trieste, Italy, 2000. Trieste: The Abdus Salam International Centre for Theoretical Physics. 1–148 (2001; Zbl 1102.14300)].
Actually, the authors prove even a stronger result from which the conjecture follows: Let \(\Delta\) denote the unit disc in \(\mathbb C\), let \(\varphi\) be a PSH function on \(\Delta^n\subset {\mathbb C}^n\), and let \(\varphi_0\not\equiv -\infty\) be a PSH function on \(\Delta^n\). Then \({\mathcal I}(\varphi)=\cup_{\varepsilon>0} {\mathcal I}(\varphi+\varepsilon\varphi_0)\).
A weaker version, called the openness conjecture, where it is assumed that \({\mathcal I}(\varphi)={\mathcal O}_X\), was proved by C. Favre and M. Jonsson [J. Am. Math. Soc. 18, No. 3, 655–684 (2005; Zbl 1075.14001)] for the two-dimensional case and by B. Berndtsson in [“The openness conjecture for PSH functions”, Preprint, arXiv:1305.5781] for the general case.

MSC:

32U05 Plurisubharmonic functions and generalizations
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References:

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