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A study introduces an axiomatic definition for the Kodaira dimension and classifies Thurston geometries in dimensions ≤ 5. It shows the Kodaira dimension is monotone with respect to maps of nonzero degree between 5-manifolds. The study establishes a connection between the simplicial volume and the holomorphic Kodaira dimension, implying any smooth Kähler 3-fold with nonvanishing simplicial volume has top holomorphic Kodaira dimension.

Theorem 1.1 states that if there is a map of nonzero degree from one closed oriented geometric 5-manifold to another, then the Kodaira dimension of the first manifold is less than or equal to the Kodaira dimension of the second. Theorem 1.3 states that a closed geometric 5-manifold has nonzero simplicial volume if and only if its Kodaira dimension is 5/2.

The study also examines the compatibility of the introduced Kodaira dimension with existing notions, such as the holomorphic Kodaira dimension, and asks whether a smooth complex manifold with nonvanishing simplicial volume has top holomorphic Kodaira dimension.