几何结构、Gromov范数和Kodaira维数
Geometric structures, Gromov norm and Kodaira dimensions
摘要：
我们定义的Kodaira尺寸$3$维流形通过瑟斯顿的八个几何形状，以及在此Kodaira尺寸的分类。我们表明，这是兼容的与其他现有的Kodaira尺寸和非零次地图定义的偏序。对于更高的维度，我们探索的几何结构和映射顺序与各种Kodaira维数和其他不变量的关系。特别地，我们证明了闭几何$4$ -流形具有非零Gromov范数当且仅当它具有几何$\mathbb H^2\times \mathbb H^2$，$\mathbb H^2(\mathbb C)$或$\mathbb H^4$ .
英文摘要：
We define the Kodaira dimension for $3$-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. For higher dimensions, we explore the relations of geometric structures and mapping orders with various Kodaira dimensions and other invariants. Especially, we show that a closed geometric $4$-manifold has nonvanishing Gromov norm if and only if it has geometry $\mathbb H^2\times \mathbb H^2$, $\mathbb H^2(\mathbb C)$ or $\mathbb H^4$.
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