  In this paper, we consider the rationally elliptic K\"ahler fourfolds which
are holomorphically embedding 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 $i:X \to\mathbb{P}^8$. Wilson asked whether a
$\mathbb Q$-homological projective $n$-space $X$ with Kodaira dimension
$k(X)\neq n$ is isomorphic to $\mathbb{P}^n$. We give a positive answer for
this question in dimension $4$. Using the same idea, 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.
