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We first note in this article that if a compact Kähler manifold has the same integral cohomology ring and Pontrjagin classes as the complex projective space    C   P  n    , then it is biholomorphic to    C   P  nprovided   n   is odd. The same holds for even   n   if we further assume that the fundamental group of   M   is finite. 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    C   P  nin terms of homotopy type under some symmetry condition. When    n  =  4   , we can drop the requirement on Pontrjagin classes by showing that a compact Kähler manifold with finite fundamental group and having the same integral cohomology ring as    C   P  4is biholomorphic to    C   P  4    , which improves on results of Fujita and Libgober–Wood.

Some remarks on the uniqueness of the complex projective spaces