Some remarks on the uniqueness of the complex projective spaces

TL;DR

This paper refines classical results to characterize complex projective spaces among Kähler manifolds using cohomology and Pontrjagin classes, and extends these characterizations to certain homotopy types, especially for $ =4$.

Contribution

It provides new criteria for identifying complex projective spaces based on cohomology and Pontrjagin classes, including a case for $ =4$ without Pontrjagin assumptions.

Findings

Characterization of $ =2k+1$ complex projective spaces via cohomology and Pontrjagin classes.

Extension of the characterization to simply-connected cases for even $ $.

Identification of $ =4$ case without Pontrjagin class assumptions.

Abstract

We first notice in this article that if a compact K\"{a}hler manifold has the same integral cohomology ring and Pontrjagin classes as the complex projective space $\mathbb{C}P^n$, then it is biholomorphic to $\mathbb{C}P^n$ provided $n$ is odd. The same holds for even $n$ if we further assume that $M$ is simply-connected. This technically refines a classical result of Hirzebruch-Kodaira and Yau. This observation, together with a result of Dessai and Wilking, enables us to characterize all $\mathbb{C}P^n$ in terms of homotopy type under mild symmetry. When $n=4$, we can drop the requirement on Pontrjagin classes by showing that a simply-connected compact K\"{a}hler manifold having the same integral cohomology ring as $\mathbb{C}P^4$ is biholomorphic to $\mathbb{C}P^4$, which improves on results of Fujita and Libgober-Wood.