This paper deals with an original approach to the openness conjecture in the line of previous work by the same authors [Ann. Inst. Fourier 62, No. 6, 2145–2209 (2012; Zbl 1272.14016)].
The openness conjecture, formulated by J-P. Demailly and J. Kollár in [Ann. Sci. Éc. Norm. Supér. (4) 34, No. 4, 525–556 (2001; Zbl 0994.32021)], states that for any plurisubharmonic function \(\varphi\) such that \(e^{-\varphi}\) is locally integrable, there exists a number \(c>1\) such that \(e^{-c \varphi}\) is still locally integrable.
The authors propose a (set of) conjecture(s) of purely algebraic nature that they show to imply the openness conjecture, in an even strengthened form. More precisely, the conjectures assert that the log canonical threshold of appropriate graded sequences of ideals can be computed by one quasi-monomial valuation.
The rough strategy of the proof is as follows: first operate an algebraic reduction using Demailly’s approximation theorem that allows to compute \(c_x(\varphi)\), the complex singularity exponent of \(\varphi\), as a log canonical threshold of the sequence of multiplier ideals \(\mathfrak b_{\bullet}:=(\mathcal I(k\varphi))_{k \geq 1}\). By the conjecture, there exists a quasi-monomial valuation \(\nu\) computing the lct of this sequence. The core of the effort is then devoted to attach to \(\nu\) Kiselman numbers (or directional Lelong numbers) computing \(\nu(\mathfrak b_{\bullet})\). Then, it remains to carefully evaluate \(e^{-c_x(\varphi) \varphi}\) on the birational model provided by \(\nu\) using the properties of Kiselman numbers.
After the paper had been published, the openness conjecture was proved in [B. Berndtsson, “The openness conjecture for plurisubharmonic functions”, Preprint, arXiv:1305.5781] and [Q. Guan and X. Zhou, “Strong openness conjecture for plurisubharmonic functions”, Preprint, arXiv:1311.3781].