A criterion of algebraicity for Lelong classes and analytic sets
Let \(Y\) be a local and irreducible analytic subset of dimension \(n\) in \(\mathbb C^N\). Let \(\mathcal L(Y)\) denote the closure in \(L^1_{\text{loc}}(Y)\) of the class \(\mathcal L _Y\) of plurisubharmonic functions of ``restricted logarithmic growth'' on \(Y\) defined by \[ \mathcal L_Y:=\text{PSH}(Y)\cap \{v|Y: v\in \mathcal L(\mathbb C^N)\}. \] Here PSH\((Y)\) denotes the class of all functions \(v:Y\to [-\infty,+\infty)\) that are locally bounded above on \(Y\) and plurisubharmonic on \(Y_{\text{reg}}\) the set of regular points of \(Y\); \(\mathcal L(\mathbb C^N)\) is the usual class of plurisubharmonic functions of \(N\) complex variables with logarithmic growth at infinity. Let \(\mathcal A(Y) = \bigcup_{d\geq 1}\mathcal A_d(Y)\) be the graded algebra of holomorphic functions on \(Y\), where \[ \mathcal A_d(Y):= \{f|Y:f \text{ is a polynomial of degree at most } d\},\quad d\geq 1. \] Let \(U\) be a coordinate neighborhood of a point \(a\in Y_{\text{reg}}\) such that there exists a homeomorphism \(\Phi:\overline U \mapsto \overline {\Delta}^n\) onto the closed unit polydisc \(\overline {\Delta}^n\) which is holomorphic on \(U\), and \(\Phi (a) =0\). Put \(\overline U_s: \Phi^{-1}(\overline {\Delta}^n_s)\), where \({\Delta}^n_s\) is the open polydisc with center \(0\) and radius \(0<s<1\). Let \(w\in\text{PSH}(U)\). The number \(\nu(w;a):= \lim_{s\to 0} \frac {\sup_{\overline{U}_s}w }{\log s}\) does not depend on the choice of the coordinate system and is called Lelong number of \(w\) at the point \(a\). Put \(\nu_{\mathcal L(Y)}(a):=\sup\{\nu (w,a); w\in \mathcal L(Y)\}\). Then the number \(\nu(Y):= \inf\{\nu_{\mathcal L(Y)}(a); a\in Y_{\text{reg}}\} \) is called minimal graded Lelong number of \(Y\). For each point \(a\in Y_{\text{reg}}\) define the number \(\mu_{\mathcal A(Y)}(a):= \sup \{m_f(a)/d\); \(f\in \mathcal A_d(Y)\), \(f\neq 0\), \(d\geq 1\}\), where \(m_f(a)\) denotes the order of vanishing of a holomorphic function \(f\neq 0\) at \(a\). The number \(\mu(Y):= \inf \{\mu_{\mathcal A(Y)} (a)\); \(a\in Y_{\text{reg}}\}\) is called the minimal graded multiplicity of \(Y\) at the point \(a\). The main goal of the paper is the following Theorem. Let \(Y\) be a local and irreducible analytic set of dimension \(n\) in \(\mathbb C^N\). Then the following coditions are equivalent: (i) \(Y\) is a piece of an algebraic set in \(\mathbb C^N\); (ii) For every open set \(\Omega \subset Y\) and any subset \(\mathcal M\) of \(\mathcal L(Y)\) the set \(B_{\mathcal M} :=\{x\in\Omega\); \(\sup_{v \in \mathcal M}v(x) <+\infty\}\) is either pluripolar in \(\Omega\), or the family \(\mathcal M\) is locally bounded above in \(\Omega\); (iii) \(\mathcal L(Y)\) contains all real constants, if \(u\in \mathcal L(Y)\) and \(\alpha\) is a real constant then \(u+\alpha \in \mathcal L(Y)\), \(\mathcal L(Y)\) is a closed subset of \(\text{PSH}(Y)\), and \(\mathcal L(Y)\) satisfies (ii) with \(\Omega = Y\); (iv) There exists a compact set \(E\subset Y\) such that for every bounded domain \(\Omega \subset Y\sup \{u(x)\); \(x\in \Omega\), \(u\in {\mathcal L}(Y)u\leq 0\) on \(E\}<+\infty\); (v) There exists a compact set \(K\) in \(Y\) and a relatively compact subdomain \(\Omega\) of \(Y\) such that \(\sup \{u(x)\); \(x\in \Omega u \in \mathcal L(Y) u\leq 0\) on \(K\}<+\infty\); (vi) \(\nu(Y)< +\infty\); (vii) \(\mu(Y)<+\infty\). Moreover, if one of these equivalent conditions is satisfied then \(Y\) is a piece of an irreducible algebraic subvariety \(Z\) of \(\mathbb C^N\) of dimension \(n\), whose degree of algebraicity \(\delta (Z)\) satisfies the estimates \(\mu(Y)\leq\delta(Z)\leq\mu(Y)^n\). Recall that for an arbitrary algebraic subset \(Z\) of \(\mathbb C^N\) of dimension \(n\), the degree of algebraicity \(\delta\) of \(Z\) is the number of points of intersection of \(Z\) with a generic \((N-n)\)-plane of \(\mathbb C^N\). The equivalence (i) \(\Leftrightarrow\) (iv) is known as the local criterion of algebraicity of \textit{A. Sadullaev} [Math. USSR, Izv. 20, 493-502 (1983; Zbl 0582.32023)].
scientific article; zbMATH DE number 1541220
scientific article