Geometric structures, Gromov Norm and Kodaira dimensions

Abstract 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 H 2 × H 2 , H 2 ( C ) or H 4 .