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 $k(X) \neq 4$ 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.