Continuity properties of certain weighted log canonical thresholds
Let \(\Omega\subset\mathbb C^n\) be a domain with \(0\in\Omega\). For a function \(\varphi\in\mathcal{PSH}(\Omega)\) and a non-negative Radon measure \(\mu\) on a neighborhood of \(0\in\mathbb C^n\), let \[ c_\mu(\varphi):=\sup\big\{c\geq0: e^{-2c\varphi}\in L^1(\mu)\text{ on a neighborhood of }0\big\}. \] Let \(c(\varphi):=c_{dV_{2n}}(\varphi)\), where \(dV_{2n}\) denotes the Lebesgue measure in \(\mathbb C^n\). The main results of the paper are the following two theorems: -- Let \((\varphi_j)_{j=0}^\infty\subset\mathcal{PSH}(\Omega)\) be such that \(\varphi_j\to\varphi_0\) in \(L^1_{\mathrm{loc}}(\Omega)\). Then \[ \liminf_{j\to +\infty}c^{}_{\| z\|^{2t}dV_{2n}}(\varphi_j)\geq c^{}_{\| z\|^{2t}dV_{2n}}(\varphi_0),\quad t\in (-n,1]. \] -- Let \(\varphi\in\mathcal{PSH}(\Omega)\), \(1\leq k\leq n\). Then \[ c^{}_{\| z\|^{2(k-n)}dV_{2n}}(\varphi)=\sup\big\{c(\varphi|_H):H\text{ is a }k\text{-dimensional linear subspace of }\mathbb C^n\big\} \] \[ =\sup\big\{c(\varphi|_H):H\text{ is a germ at }0\text{ of a smooth }k\text{-dimensional submanifold}\big\}. \]
scientific article; zbMATH DE number 6673223
scientific article