Some remarks on the uniqueness of the complex projective spaces

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.