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

We show that a compact K\"ahler 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 $\dim M \not \equiv 3 \pmod 4$. This answers conjectures of Fujita in these dimensions.