The aim of this paper is to prove effectiveness results concerning the strong openness conjecture, the lower semicontinuity of plurisubharmonic functions with multiplier and conjectures of Demailly-Kollár and Jonsson-Mustată.
The Strong openness conjecture posed by J.-P. Demailly [Analytic methods in algebraic geometry. Surveys of Modern Mathematics 1. Somerville, MA: International Press; Beijing: Higher Education Press (2012; Zbl 1271.14001)] is the following: Let \(\varphi\) be a plurisubharmonic function on a complex manifold \(X\), then \[ \mathcal I(\varphi)=\mathcal I_{+}(\varphi):=\bigcup_{\epsilon >0}\mathcal I((1+\epsilon)\varphi), \] where \(\mathcal I(\varphi)\) is the sheaf of germs of holomorphic functions \(f\) such that \(|f|^2e^{-\varphi}\) is locally integrable.
If \(\mathcal I(\varphi)=\mathcal O_X\), then the strong openness conjecture degenerates to the openness conjecture posed by J.-P. Demailly and J. Kollár [Ann. Sci. Éc. Norm. Supér. (4) 34, No. 4, 525–556 (2001; Zbl 0994.32021)].
Let \(D\) be a pseudoconvex domain in \(\mathbb C^n\), \(z_0\in D\). Let \(\varphi\) be a negative plurisubharmonic function on \(D\), and let \(F\) be a holomorphic function on \(D\). Let \[ ||f||_{\varphi}:=\left (\int_D|F|^2e^{-\varphi }d\lambda_n\right)^{\frac 12}, \] where \(\lambda_n\) is the Lebesgue measure in \(\mathbb C^n\), and define the generalized Bergmann kernel as follows
\[ K_{\varphi,F}(z_0):=\left(\inf\Big\{||G||^2_0: G\in \mathcal O(D), (G-F,z_0)\in \mathcal I_{+}(2c_{z_0}^F(\varphi)\varphi)_{z_0} \Big\}\right)^{-1}, \] where
\[ c^F_{z_0}(\varphi):=\sup\big\{c\geq 0: |F|^2e^{-2c\varphi} \text{ is integrable on a neighborhood of } z_0\big\} \]
is the jumping number. Define also the following function \(\theta :(1,\infty)\to \mathbb R\):
\[ \theta(t)=\left (\frac {1}{(t-1)(2t-1)}\right)^{\frac 1t}. \]
The authors establish the following effectiveness of the strong openness conjecture. Let a holomorphic function \(F\) and a negative plurisubharmonic function \(\varphi\) be such that \(||F||_{\varphi}^2\leq C_1\) and \(K_{\varphi,F}^{-1}\geq C_2\), where \(C_1, C_2\) are two positive constants. Then, for any \(p>1\) satisfying \(\theta(p)>\frac {C_1}{C_2}\), one has \((F,z_0)\in \mathcal I(p\varphi)_{z_0}\).
The above result gives a new proof of the strong openness conjecture and gives (for \(F=1\)) a more precise version of the effectiveness result for the openness conjecture of B. Berndtsson [“The openness conjecture for plurisubharmonic functions”, Preprint, arXiv:1305.5781].
A lower semicontinuity property of plurisubharmonic functions with multiplier.
Demailly and Kollár conjectured in [loc. cit.] that, if \(X\) is a complex manifold, \(K\) a compact set and \(L\) an open set such that \(K\subset L\subset X\), then, for every nonzero function \(f\in \mathcal O(X)\), there exists a constant \(\delta=\delta(f,K,L)\) such that, for any \(g\in \mathcal O(X)\),
\[ \sup_L|g-f|<\delta \Rightarrow c_K(\log|g|)\geq c_K(\log |f|), \]
where
\[ c_K(\varphi)=\sup\big\{c\geq 0: e^{-2c\varphi} \text{ is integrable on a neighborhood of } K\big\}. \]
Using the idea from their paper [“Strong openness conjecture and related problems for plurisubharmonic functions”, Preprint, arXiv:1401.7158], the authors prove the following lower semicontinuity property of plurisubharmonic functions with a multiplier. Let \(\{\phi_m\}_{m=1}^{\infty}\) be a sequence of negative plurisubharmonic functions on the polidisc \(\Delta^n\), which is convergent to a negative Lebesgue measurable function \(\phi\) on \(\Delta^n\), in Lebesgue measure. Let \(\{F_m\}_{m=1}^{\infty}\) be a sequence of holomorphic functions on \(\Delta^n\) with uniform bound, which is convergent to a Lebesgue measurable function \(F\) on \(\Delta^n\), in Lebesgue measure. If, for any neighborhood U of \(0\), the pairs \((F_m,\phi_m)\) satisfy \(\inf_{m}K^{-1}_{\phi_m,F_m}(0)>0\), then \(|f|^2e^{-\phi}\) is not integrable near \(0\). In particular, if \(\phi\) is plurisubharmonic and \(F\) is holomorphic, then \((F,0)\notin \mathcal I(\phi)_0\).
Demailly and Kollár posed in [loc. cit.] the following conjecture: Let \(\varphi\) be a plurisubharmonic function on \(\Delta^n\), and let \(K\) be a compact subset of \(\Delta^n\). If \(c_K(\varphi)<\infty\), then \[ \frac{\lambda_n(\{\varphi<\log r\})}{r^{2c_K(\varphi)}} \tag{*} \] has a uniform positive lower bound independent of \(r\in (0,1)\) small enough.
Using the idea from [loc. cit.], the authors prove the following effectiveness of the positive lower bound of (*). Let \(B_0\in (0,1]\), let \(\varphi\) be a negative plurisubharmonic function on a pseudoconvex domain \(D\subset \mathbb C^n\), and let \(F\in \mathcal O(D)\). If \(|F|^2e^{-\varphi}\) is not locally integrable near \(z_0\in D\), then \[ \liminf_{R\to \infty}\frac 1{B_0}\int_D\mathbb I_{\{-(R+B_0)<\varphi<-R\}}|F|^2e^{t_0+B_0}d\lambda_n\geq K^{-1}_{\varphi,F}(z_0). \] In particular if \(F=1\), then \[ \liminf_{R\to \infty}e^{R+B_0}\frac {\lambda_n(\{-(R+B_0)<\varphi<-R\})}{B_0}\geq K^{-1}_{\varphi,1}(z_0)\geq K^{-1}(z_0). \]
M. Jonsson and M. Mustată [J. Inst. Math. Jussieu 13, No. 1, 119–144 (2014; Zbl 1314.32047)] posed the following conjecture: Let \(\psi\) be a plurisubharmonic function on \(\Delta^n\), and let \(I\) be an ideal of \(\mathcal O_{\Delta^n,0}\), generated by \(\{f_j\}_{j=1}^l\). If \(c_0^I(\psi)<\infty\), then \[ \frac {\lambda_n(\{c_0^I(\psi)\psi-\log |I|<\log r\})}{r^2} \tag{**} \] has a uniform positive lower bound independent of \(r\in (0,1)\) small enough, where \(\log |I|=\log (\max _{1\leq j\leq l}|f_j|)\) and \[ c^I_{z_0}(\varphi):=\sup\big\{c\geq 0: |I|^2e^{-2c\varphi} \text{ is integrable on a neighborhood of } 0\big\} \]
is the jumping number.
Using the idea from [loc. cit.], the authors prove the following effectiveness of the positive lower bound of (**). Let \(\delta \in \mathbb N\), let \(\psi\) be a plurisubharmonic function on a pseudoconvex domain \(D\subset \mathbb C^n\) bounded from above, and let \(F\in \mathcal O(D)\). If \(|F|^2e^{-\psi}\) is not locally integrable near \(z_0\in D\), then
\[ \liminf_{R\to \infty} \frac {e^R}{B_0}\lambda_n\Big(\big\{-R-B_0<\psi-\log|F|^2<-R\big\}\Big)\geq \frac {K^{-1}_{\psi+\delta\max (\psi, 2\log |F|), F^{1+\delta}}(z_0)}{(1+\frac 1{\delta})e^{B_0}\sup_De^{(1+\delta)\max(\psi,2\log |F|)}}. \]