Ensembles analytiques complexes définis comme ensembles de densité et contrôles de croissance
Let G be the space \({\mathbb{C}}^ n\) or an n-dimensional Stein manifold imbedded in \({\mathbb{C}}^{2n+1}\), and \(M^ P_+(G)\) the cone of closed positive (n-p,n-p)-currents on G, \(0\leq p\leq n-1.\) Denote by \(\nu\) (x,T) the Lelong number of a current \(T\in M^ p_+(G)\) at a point \(x\in G\). By a result of \textit{Y. T. Siu} [Invent. Math. 27, 53-156 (1974; Zbl 0289.32003); see also Bull. Am. Math. Soc. 79 (1973), 1200-1205 (1974; Zbl 0282.32006)], the set \(E(c,T):=\{x\in G:\nu (x,T)\geq c\},\) \(c>0\), is analytic of dimension \(\leq p\). The main problem considered in the paper is the following: Given \(T\in M^ p_+({\mathbb{C}}^ n)\) and \(c>0\), can the growth of the k-dimensional component \(E^ k(c,T)\), \(0\leq k\leq p\), of E(c,T) be estimated by the one of T? Here the growth of a current \(T\in M^ p_+({\mathbb{C}}^ n)\) is measured by the function \(\nu_ T(r):=(\pi r^ 2)^{-p}T(\chi_ r\omega^ p)\) (the growth indicatrix) where \(\chi_ r\) is the characteristic function of the ball \(| z| <r\) and \(\omega\) is the standard Kähler form of \({\mathbb{C}}^ n\); the indicatrix of \(E^ k(c,T)\) is the one of the current \([E^ k(c,T)]\) of integration. An irreducible analytic subset Z of G (cycle) is said to be extremal with respect to a current \(T\in M^ p_+(G)\) if \(N_ T(Z):=\inf \{\nu (x,T):x\in Z\}>0\) and if there does not exist a cycle \(Z'\supset Z,\) \(Z'\neq Z\) such that \(N_ T(Z')=N_ T(Z).\) The author studies the series \(S^ k=\sum_{q}N_ T(Z^ k_ q)[Z^ k_ q]\) where \(T\in M^ p_+(G)\) and \(\{Z^ k_ q\}\) is the family of all k- dimensional extremal cycles with respect to T (they are shown to be countably many), \(0\leq k\leq p.\) In contrast to the case \(k<p\), the series \(S^ p\) is always convergent and the indicatrix \(\omega_ p(c,r)\) of \(E^ p(c,T)\) can be estimated by above: \(\omega_ p(c,r)\leq c^{- 1}\nu_ T(r).\) The sets E(c,T) are shown to be locally stable on the left, i.e. for every domain \(D\subset \subset G\) and \(c>0\) there exists \(\gamma >0\) such that \(E(c,T)\cap D=E(c',T)\cap D\) for \(c-\gamma \leq c'\leq c.\) If the global stability condition at c is satisfied, a control on the growth of functions defining E(c,T) is obtained. It is also proved that for given \(T\in M^ p_+({\mathbb{C}}^ n)\) and \(c>0\) there exists a hypersurface \(y_ c:=F^{-1}(0)\), F holomorphic function on \({\mathbb{C}}^ n\), with asymptotic control on the growth containing E(c,T). If the indicatrix \(\nu_ T(r)\) is bounded, then the sets E(c,T) are algebraic.
scientific article; zbMATH DE number 3882722
scientific article