We study Kähler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold has boundary components (possibly ), then it has the first Betti number at least , and the Levi form of any boundary component is zero. If has pseudoconvex boundary components and at least one nonparabolic end, then the first Betti number of is at least . In either case, any boundary component has a nonvanishing first Betti number. If has one pseudoconvex boundary component with vanishing first Betti number, then the first Betti number of is also zero. Especially significant are applications to Kähler ALE manifolds and to Kähler 4-manifolds. This significantly extends prior results in this direction (e.g., those of Kohn and Rossi) and uses substantially simpler methods.
"Topology of Kähler Manifolds with Weakly Pseudoconvex Boundary." Michigan Math. J. 68 (4) 727 - 742, November 2019. https://doi.org/10.1307/mmj/1563847454