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

TL;DR

The paper proves that certain compact Kähler manifolds with a divisor whose complement is contractible are isomorphic to complex projective space, confirming Fujita's conjectures in specific dimensions.

Contribution

It establishes a classification result for Kähler manifolds with a contractible complement of a divisor, confirming Fujita's conjectures in dimensions not congruent to 3 mod 4.

Findings

Such manifolds are projective spaces with a hyperplane divisor

Confirms Fujita's conjectures in specified dimensions

Provides conditions under which a homology cell complement implies projective space

Abstract

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.