Compactification of homology cells, Fujita's conjectures and the complex projective space
Ping Li, Thomas Peternell
TL;DR
This work resolves Fujita’s conjectures for most dimensions by showing that a smooth divisor D in a projective n-fold M, with M\setminus D a homology cell, forces M to be the projective space in all cases where n is not 3 modulo 4. The authors combine cohomological rigidity from Lefschetz-type results, Chern-number relations via Hirzebruch–Riemann–Roch, and the Kobayashi–Ochiai criterion to obtain a sharp dichotomy on the first Chern class, yielding the standard projective case; in the remaining congruence class, they derive strong topological constraints and outline a program to complete the classification. A second main result leverages Poincaré–Lefschetz duality to extend the conclusions to a broader setting where the open complement is a homology cell, and a fibration application yields a dimension bound. The paper also sketches a numerical approach for the still-open case n ≡ 3 (mod 4) using the χy-genus to formulate Chern-number equations that could rule out nonstandard examples.
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 $\dim M \not \equiv 3 \pmod 4$. This answers conjectures of Fujita in these dimensions.