Some remarks on the uniqueness of the complex projective spaces (Q2801732)

In [J. Math. Pure Appl. 36, 201--216 (1957; Zbl 0090.38601)] \textit{F. Hirzebruch} and \textit{K. Kodaira} investigated the uniqueness of the Kähler structure of \(\mathbb{CP}^n\), and more precisely the question: if a compact Kähler manifold \(X\) is diffeomorphic to \(\mathbb{CP}^n\), must it be biholomorphic to it?NEWLINENEWLINEThey answered this affirmatively, except possibly when \(n\) is even and \(-c_1(X)\) is represented by a Kähler form. This case was finally ruled out by \textit{S.-T. Yau} [Proc. Nat. Acad. Sci. U.S.A. 74, 1798--1799 (1977; Zbl 0355.32028)], thus completely answering the question above (and in fact he showed that diffeomorphic may be replaced with homeomorphic). He also proved that when \(\dim X=2\) the Kähler assumption on \(X\) can be dropped, and that in this case it is enough to assume that \(X\) is homotopy equivalent to \(\mathbb{CP}^2\), thus answering a long-standing conjecture of Severi. See also the notes by the reviewer [``Uniqueness of \(\mathbb{CP}^n\)'', Preprint, \url{arXiv:1508.05641}] for a detailed exposition of these results.NEWLINENEWLINEIn this paper the author relaxes the assumption of homeomorphism in the theorem of Hirzebruch-Kodaira and Yau, to just assuming that the compact Kähler manifold \(X\) has the same cohomology ring and Pontrjagin classes as \(\mathbb{CP}^n\) (and if \(n\) is even he also needs to assume that \(\pi_1(X)\) is finite). When \(n=4\) he shows that the assumption on the Pontrjagin classes can be dropped.NEWLINENEWLINEThe main idea of the proof is the following: the only place where the stronger hypothesis of homeomorphism was used was to determine the second Stiefel-Whitney class of \(X\), and hence the first Chern class mod \(2\). He shows that the existing proof already in fact determines the first Chern class using an extra elementary argument.