论文标题
某些理性椭圆形射射线的拓扑表征和霍奇结构四倍
Topological characterization and Hodge structures of some rationally elliptic projective fourfolds
论文作者
论文摘要
在本文中，我们考虑了将荷花体形态嵌入复杂的射击八个空间$ \ mathbb {p}^8 $中的理性椭圆形的射射线四倍。事实证明，通过使用Euler特征和Chern数字公式的holomorphic bundle ulomorphic embb，p $ imorphic embbbbbbbbbbbbbbbbb {p}^8 $通过使用Euler特征和CHERN数字公式为Holomorphic ended $ impheD $ i：x \ i i;在证明结果的过程中，我们偶然发现$ \ mathbb {q} $ - 同源投影4空间$ x $带有kodaira dimension $ k（x）\ neq 4 $是同构至$ \ mathbb {p}^4 $。这一发现为威尔逊提出的问题提供了积极的答案，在尺寸$ n = 4 $的情况下。使用类似的方法，我们表明，Hodge的猜想适用于理性椭圆形的四倍$ x \ subset \ mathbb {p}^8 $，而理性椭圆四倍$ x \ subset \ subset \ subset \ mathbb {p}^8 $具有非阳性的hodge水平。
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.
