Compactification of homology cells, Fujita's conjectures and the complex projective space

Abstract

We show that a compact Kähler manifold $M$ containing a smooth connected divisor $D$ such that $M\setminus D$ is a homology cell, e.g., contractible, must be projective space with $D$ a hyperplane, provided $\dim M\not\equiv 3(\mathrm{mod}4)$. This answers conjectures of Fujita in these dimensions.

final version, to appear in Mathematische Annalen