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Green functions with singularities along complex spaces. (English) Zbl 1085.32018

Let \(X\) be a connected complex manifold. Recall that for any function \(\alpha:X\to[0,+\infty)\), the pluricomplex Green function with poles \(\alpha\) is defined by the formula \(\widetilde G_\alpha:=\sup\{u: u\in\mathcal F_\alpha\}\), where \(\mathcal F_\alpha\) denotes the family of all negative functions \(u\in\mathcal{PSH}(X)\) such that \(\alpha(a)\leq\nu_u(a)=\) the Lelong number of \(u\) at \(a\in X\). Let \(A\) be a closed complex subspace of \(X\).
The authors define the pluricomplex Green function with singularities along \(A\) by the formula \(G_A:=\sup\{u: u\in\mathcal F_A\}\), where \(\mathcal F_A\) denotes the family of all negative functions \(u\in\mathcal{PSH}(X)\) such that for every point \(a\in X\) there exist local generators \(\psi_1,\dots,\psi_m\) of the sheaf \(\mathcal I_A\) and a constant \(C\) for which \(u\leq\log\| \psi\| +C\) in a neighborhood of \(a\), where \(\psi:=(\psi_1,\dots,\psi_m)\). Then \(G_A\leq\widetilde G_{\tilde\nu_A}\) with \(\widetilde\nu_A:=\nu_{\log\| \psi\| }\) (locally).
The authors present a systematic study of various properties of \(G_A\) (in comparison with corresponding properties of \(\widetilde G_\alpha\)). In particular, they prove the following results: (a) \(G_A\in\mathcal F_A\). (b) \(G_A(x)=\inf\{G_{f^\ast\!A}(0)\:f\in\mathcal O(\overline{\mathbb D},X),\; f(0)=x\}\), \(x\in X\) (Poletsky type disc formula). (c) \(G_{A_1\times A_2}(x_1,x_2)=\max\{G_{A_1}(x_1),G_{A_2}(x_2)\}\), \((x_1,x_2)\in X_1\times X_2\) (product property). Moreover, the authors characterize some cases when \(G_A\equiv\widetilde G_{\tilde\nu_A}\).

MSC:

32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
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