Topological characterization and Hodge structures of some rationally elliptic projective fourfolds

Abstract

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.