Topological characterization and Hodge structures of some rationally elliptic projective fourfolds

In this paper, we consider the rationally elliptic projective fourfolds that are holomorphically embedded into the complex projective eight-space ${\mathbb{P}}^8$. It is proved that a simply-connected $\mathbb{Q}$-homological projective four-space $X\subset {\mathbb{P}}^8$ is biholomorphic to ${\mathbb{P}}^4$ by using Euler characteristic and Chern numbers formulae of the normal bundle for a holomorphic embedding $i:X\to {\mathbb{P}}^8$. During the process of proving the result, we incidentally discovered that a $\mathbb{Q}$-homological projective 4-space $X$ with Kodaira dimension $k(X)\mathrm{\ne }4$ is isomorphic to ${\mathbb{P}}^4$. This finding provides a positive answer to a question posed by Wilson in the case where the dimension $n=4$. Using a similar approach, we show that the Hodge conjecture holds for the rationally elliptic fourfold $X\subset {\mathbb{P}}^8$, and the rationally elliptic fourfold $X\subset {\mathbb{P}}^8$ has non-positive Hodge level.