Plurisubharmonic functions with singularities and affine invariants for finite sets in \(\mathbb C^n\) (Q1598160)

The authors associate two numbers \(\gamma(S)\geq \widetilde \gamma(S)\) to every finite subset \(S\) of \(\mathbb C^n\) and relate them to algebraic geometric properties of the set \(S\). These numbers are defined as follows: For every \(u\in \text{PSH}(\mathbb C^n)\) let \(\gamma_u=\limsup_{\|z\|\to \infty}(u(z)/\log\|z\|)\), \(\widetilde\gamma(S)=\inf\{\gamma_u: u\in \widetilde E(S)\}\), and \(\gamma(S)=\inf\{\gamma_u: u\in E(S)\}\), where \(\widetilde E(S)\) consists of all \(u\in \text{PSH}(\mathbb C^n)\cap L^\infty_{loc}(\mathbb C^n\setminus S)\) such that \(\gamma_u\in (0,+\infty)\) and \(u(z)-\log\|z-p\|=O(1)\) as \(z\to p\) for all \(p\in S\), and \(E(S)\) is the subclass of \(\widetilde E(S)\) consisting of \(u\) such that \((dd^cu)^n=\sum_{p\in S}\delta_p\), where \(\delta_p\) is the Dirac measure at the point \(p\). Let me only mention two of the many interesting results proved in the paper. If \(|S|\) denotes the number of points in \(S\), then \(|S|^{1/n}\leq\widetilde\gamma(S)\leq\gamma(S)\leq |S|\), and \(\widetilde\gamma(S)=|S|\) if and only if \(S\) is contained in a complex line. If \(V\subset\mathbb C^n\) is an algebraic variety of pure dimension \(m\), then \(|S\cap V|\leq \sum_{p\in S\cap V}\nu(V,p)\leq \widetilde \gamma(S)^m\text{deg} V\), where \(\nu(V,p)\) is the Lelong number of \(V\) at the point \(p\). Many interesting examples are treated in the case \(n=2\). In all the examples \(\widetilde\gamma(S)=\gamma(S)\) and it remains an open question if this equality always holds.