From the author's introduction: In the present paper, we study the canonical embedding of \(\mathbb{C}P^n\) in the \(U(n+1)\)-space \(\mathbb{R}P^{n^2+2n}\). The fact that this embedding is natural allows one to hope that complex projective geometry can be invariantly characterized in terms of real projective geometry. We prove that the embedding \(\sigma :\mathbb{C}P^n_{\mathbb{R}}\to\mathbb{R}P^{n^2+2n}\) can be naturally described in terms of holomorphic bivectors in the realification of the complex vector space. The image of the embedding \(\sigma\) is called a \(\mathbb{C}P^n\)-surface. We prove that a \(\mathbb{C}P^n\)-surface is a flat section of a real Grassmanian. In particular, this implies that the image of the embedding \(\sigma\) is a real algebraic subvariety. We also prove that the embedding constructed internally determines the canonical Kähler structure on \(\mathbb{C}P^n_{\mathbb{R}}\). In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on \(\mathbb{C}P^n_{\mathbb{R}}\) is completely defined by its second fundamental form; therefore this embedding is said to be canonical. The embedding \(\sigma\) allows one to model the geometry of complex projective spaces on real-analytic surfaces. For example, we prove that completely geodesic invariant and anti-invariant submanifolds on \(\mathbb{C}P^n_{\mathbb{R}}\) are flat sections of a \(\mathbb{C}P^n\)-surface. In addition, we discuss an alternative approach to the construction of the embedding \(\sigma\). This approach is based on the adjoint representation of the Lie group \(U( n+1)\) and on the fact that in the Lie algebra \(u( n+1)\equiv\mathbb{R}P^{n^2+2n+1}\) there exists an orbit of the adjoint representation diffeomorphic to \(\mathbb{C}P^n_{\mathbb{R}}\). On this orbit, a canonical Kähler structure exists [see \textit{A. L. Besse}, Einstein manifolds. Berlin etc.: Springer-Verlag (1987; Zbl 0613.53001)]. We prove that the image of this orbit under the projectivisation mapping is a \(\mathbb{C}P^n\)-surface. This approach allows us to state a geometric characterization of the canonical decomposition of the Lie algebra \(u( n+1)\).