Remarks on the holomorphic convexity of the universal covering space of a projective manifold
Let \((X,w)\) be a compact Hermitian complex manifold with associated \((1,1)\)-form \(w\). Let \((L,h)\) be a holomorphic line bundle over \(X\) with Hermitian metric \(h\). Further let \(\pi : \widetilde X \to X\) be an infinite unramified Galois covering from a complex manifold \(\widetilde X\). By \((\widetilde L, \widetilde h)\) denote the pullback. Denote by \(H^0_{(2)} (\widetilde X, \widetilde L^{\otimes k})\) the space of holomorphic \(L^2\)-sections of \(\widetilde{L}^{\otimes m}\) over \(\widetilde X\). Fix an \(\sim\) origin \(x_0 \in \widetilde{X}\). It can be shown that there exists \(x_r \in \widetilde X\) such that \(|\tau(x_r)|= \max_{x \in \widetilde{X}} |\tau(x)|\) for each non-zero \(\tau \in H^0_{(2)} (\widetilde{X}, \widetilde{L}^{\otimes k})\). Theorem 1. Assume that \[ \lim_{k \to +\infty} \sup\{d(x_\tau, (\tau)_0);\quad \tau \in H^0_{(2)} (\widetilde{X}, \widetilde{L}^{\otimes k})\} = +\infty.\tag \(*\) \] Then \(\widetilde{L}\) admits a flat Hermitian structure. In particular, if \(\widetilde{X}\) is the universal cover then \(\widetilde{L} \cong G_{\widetilde{X}}\). Here \((\tau)_0\) is the zero locus of \(\tau\).
scientific article; zbMATH DE number 861925