First betti numbers of Kähler manifolds with weakly pseudoconvex boundary

Details First betti numbers of Kähler manifolds with weakly pseudoconvex boundary First betti numbers of Kähler manifolds with weakly pseudoconvex boundary

2011-10-20

arxiv.org/abs/1110.4571v1

Mathematics

Differential Geometry

Scientific paper

We study K\"ahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), it has first betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K$ has $l\ge1$ 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. Some complex 2-manifolds are also shown to have strictly positive Euler number.

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