On hyperbolicity and tautness modulo an analytic subset of Hartogs domains (Q2845437)

Let \(X\) be a complex space and let \(H:= X\times\mathbb{C}^m\to[-\infty,\infty)\) be an upper semicontinuous function such that \(H(z,w)\geq 0\) and \(H(z,\lambda w)=|\lambda|H(z, w)\) for \(\lambda\in\mathbb{C}\), \(z\in X\) and \(w\in\mathbb{C}^m\).NEWLINENEWLINE Let \(\Omega_H(X):= \{(z, w)\in X\times\mathbb{C}^m\mid H(z, w)< 1\}\), let \(S\) be an analytic subset of \(X\) and let \(\widetilde S:= S\times\mathbb{C}^m\). In this paper, the authors show that:NEWLINENEWLINE a) (Theorem 2.1) \(\Omega_H(X)\) is hyperbolic modulo \(\widetilde S\) (in the sense of Kobayashi) iff \(X\) is hyperbolic modulo \(S\) and \(H\) satisfies the followind condition NEWLINENEWLINENEWLINENEWLINE (1) If \(\{z_k\}_{k\geq 1}\subset X\setminus S\) with \(\lim_{k\to\infty}z_k\in X\setminus S\) and \(\{w_k\}_{k\geq 1}\) with \(\lim_{k\to\infty}w_k\not=0\), then \(\liminf_{k\to \infty} H(z_k,w_k)=0\).NEWLINENEWLINENEWLINENEWLINE b) (Therem 2.3 (i)) If \(X\) is taut modulo \(\widetilde S\), then \(X\) is taut modulo \(S\) and \(\log H\) is continuous and plurisubharmonic on \(X\setminus S\times\mathbb{C}^m\). NEWLINENEWLINENEWLINENEWLINE On the other hand, they show that, if \(X\) is taut modulo \(S\), \(H\) is continuous on \(X\setminus \mathbb{C}^m\) and \(\log H\) is plurisubharmonic on \(X\times\mathbb{C}^m\), then conversely, \(\Omega_H(X)\) is taut modulo \(\widetilde S\) (Theorem 2.3. (iii)).NEWLINENEWLINE In the special case when \(H(z, w)= |w|e^{\phi(z)}\), where \(\phi:X\to [-\infty,\infty)\) is an upper semicontinuous function on \(X\), one recovers previously known results for the Hartogs domains NEWLINE\[CARRIAGE_RETURNNEWLINE\Omega_\phi(X):= \big\{(x, z)\in X\times\mathbb{C}\,\big|\,|z|< e^{-\phi(x)}\big\}.CARRIAGE_RETURNNEWLINE\]NEWLINE In this special case, the technical condition (1) mentioned above is equivalent to the local lower boundedness of \(\phi\) (Corollary 2.2) on \(X\setminus S\).