2011/10/20 by Weber, Brian
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1110.4571
We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold K has l≥2 boundary components (possibly l=∞), then it has first betti number at least l-1, and the Levi form of any boundary component is zero. If K has l≥1 pseudoconvex boundary components and at least one non-parabolic end, the first betti number of K is at least l. In either case, any boundary component has non-vanishing first betti number. If K has one pseudoconvex boundary component with vanishing first betti number, the first betti number of K is also zero. Especially significant are applications to Kahler ALE manifolds, and to Kahler 4-manifolds. This significantly extends prior results in this direction (eg. Kohn-Rossi), and uses substantially simpler methods.