Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems
If \(M\) is a connected topological space, an end is either an element of \(\lim \pi_0(M\setminus K)\) where the limit is taken over all compact subsets \(K\) of \(M\), or it is a connected component of \(M\setminus K\) for some compact subset \(K\) which is not relatively compact in \(M\). The ends of a smooth manifold say something about its structure `at infinity'. A complete Riemannian manifold is said to have bounded geometry if it has bounded curvature and derivatives thereof and has positive injectivity radius. A complex manifold is said to be weakly 1-complete if it admits a continuous plurisubharmonic exhaustion function. For the main result of this article, \(M\) is supposed to be a connected Kähler manifold which has either bounded geometry or is weakly 1-complete. The conclusions are that if \(H^1(M,\mathbb{R})=0\), then \(M\) has at most one end whilst if \(M\) has three or more ends, then it admits a proper holomorphic mapping onto a Riemann surface. As shown by a classical example of Cousin, this second conclusion is sharp -- there is a Kähler surface which has both bounded geometry and is 1-complete but admits no nonconstant holomorphic functions. There are several other interesting theorems proved along the way and a version of the Lefschetz theorem due to Nori (by completely different methods) follows. The proof applies the methods of abstract potential theory and techniques introduced by Gromov, Li, Nakai, Sario, Sullivan, and others.
scientific article; zbMATH DE number 847578
scientific article