An extremal problem for generalized Lelong numbers (Q5962173)

Let \(T\) be a closed positive \((1,1)\)-current in an open \(\Omega\subset\mathbb{C}^n\). Then by the Poincaré lemma, locally in each ball \(B\subset\Omega\), we can write \(T:= dd^cu\), where \(u\) is a plurisubharmonic function in \(B\). This function is unique up to addition of a (smooth pluriharmonic) function in \(B\) and is called a local potential of the current \(T\). So the singularities of the current \(T\) in the ball \(B\) are entirely encoded by any local potential \(u\) of \(T\) in \(B\). Since the problem considered here is local we can assueme that \(T\) is globally given by \(T= dd^cu\). Since the singularities of a psh function are known to be at most logarithmic, it is natural to introduce the follwoing numbers. Given a point \(a\in\Omega\) we define the Lelong number of the function \(u\) at the point \(a\) as the limit \[ \nu(u,a):= \liminf_{z\to a}\,{u(z)\over \log|z-a|}. \] Observe that \(\nu(u, a)= 0\) if \(u(a)>-\infty\). The number \(\nu(u, a)\) is the largest positive number \(\mu\) such that \(u(z)\leq\nu\log|z- a|+ O(1)\) near the point \(a\). If \(u:= \log|f|\), where \(f\) is a holomorphic function in some domain \(\Omega\), then \(\nu(\log|f|, a)\) is the multiplicity of the zero \(a\) as a solution of the equation \(f(z)= 0\). We also call \(\nu(u, a)=\nu(T, a)\) the Lelong number of the current \(T\) at the point \(a\). This number measures the logarithmic weight of the singularity \(a\) of the potential \(u\). It can also be defined as the projective mass of the current \(T\) at the point \(a\) according to Lelong's formula: \[ \nu(u,a)= \lim_{r\to 0^+}\, \int_{\varphi_a(z)<\vee r}T\wedge(dd^c \varphi_a)^{n-1}, \] where \(\varphi_a(z):= \log|z-a|\) is the logarithmic weight function at the point \(a\). For some problems it is desirable to introduce a more general weight, replacing the function \(\varphi_a\) in Lelong's formula by a more general weight function \(\varphi\). A weight function \(\varphi\) on \(\Omega\) is a psh function \(\varphi:\Omega\to [-\infty,+\infty[\) which is semi-exhaustive in the sense of Demailly and \(e^\varphi\) is continuous in \(\Omega\). Then Demailly showed that the corresponding formula makes sense and defines a positive number \(\nu_\varphi(u)\) which he called the generalized Lelong number of \(T\) or \(u\) with respect to the weight \(\varphi\). A typical example is the Kiselman weight function depending on the point \(a\in\Omega\) and a real direction \(\alpha(\alpha_1,\dots, \alpha_n)\in \mathbb{R}^n_+\) given a point by \[ \phi_{a,\alpha}(z):= \max_{1\leq k\leq n}\,\alpha^{-1} \log|z- a_k|. \] The corresponding generalized Lelong number is called the Kiselman directional number. Observe that if \(\alpha= (1,\dots,1)\) then we recover the usual Lelong number of \(u\) at the point \(a\). Then the general problem considered here is to find the asymptotic behaviour of \(u\) near the polar \(\varphi^{-1}(-\infty)\) when \(\nu_\varphi(u)\geq c> 0\) for a given superlevel \(c> 0\). This prolem seems to be very difficult even for a maximal weight \(\varphi\), i.e., a weight which satisfies the homogenuous Monge-Ampère equation \((dd^c\varphi)^n= 0\) on \(\Omega\setminus\varphi^{-1}(-\infty)\). It is well-known that the Monge-Ampère measure \((dd^c\varphi)^n\) is a well defined Borel measure with locally finite mass. Then the total mass \[ \tau_\varphi:= \int_{\varphi^{-1}(-\infty)} (dd^c\varphi)^n= \int_\Omega(dd^c\varphi)^n, \] is called the residual Monge-Ampère mass of \(\varphi\). When \(\varphi\) is the Kiselman's weight function, the problem above have a simple solution, namely we have \[ u(z)\leq \tau^{-1}_\varphi \nu_\varphi(u) \varphi(z)+ O(1),\quad\text{as }z\to a. \] But in general we cannot expect to have such an estimate. In order to answer the question in some cases the author considers the following extremal problem: Given a weight function \(\varphi\), find the best upper bound \(d_\varphi\) for functions \(u\) satisfying \(\nu_\varphi(u)\geq c> 0\) such that \[ u(z)\leq c\tau_\varphi(u)+ O(1). \] The extremal function is plurisubharmonic and maximal outside the singular set of \(\varphi\), so it is a fundamental solution for the complex Monge-Ampère operator \((dd^c\cdot)^n\). For a reasonable class of weights, the so-called weights with asymptotic analytic behaviour, the author gives a construction of the extremal function \(d_\varphi\) in terms of the so-called Rees valuations. Moreover for the weights with analytic singularities, it is possible to describe the behaviour of the extremal function. This is related to the Newton polyhedron of the singularities of the analytic function defining the singularities of the weight function. Many other results are given for more special weights.