Holomorphic neighbourhood retractions of ample hypersurfaces (Q1359510)

Let \(X\) be a hypersurface in a compact complex manifold \(M\) with \(\dim M\geq 2\). We consider the neighbourhood structure of \(X\) and of coverings of \(X\); specifically, the existence of holomorphic neighbourhood retractions. We show that if \(M\) has no nontrivial holomorphic vector fields and \(X\) is sufficiently ample, then no neighbourhood of \(X\) retracts holomorphically onto \(X\). Let \(U\) be a tubular neighbourhood of \(X\) and \(\pi:V \to U\) be a covering space. We show that if \(X\) is ample and linearly equivalent to some other hypersurface in \(M\), and bounded holomorphic functions separate points locally on the universal covering of \(X\), then any holomorphic retraction \(V\to \pi^{-1} (X)\) is a lifting of a retraction \(U\to X\). This implies that if \(X\) is a sufficiently ample curve in a surface \(M\) of general type with universal covering \(\pi:\widetilde M\to M\), then \(\pi^{-1} (U)\) does not retract holomorphically onto \(\pi^{-1} (X)\) for any neighbourhood \(U\) of \(X\).