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

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 $\dim M\equiv ̸3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. This answers conjectures of Fujita in these dimensions.