Geometric structures, Gromov norm and Kodaira dimensions

TL;DR

This paper introduces a new way to classify 3-dimensional manifolds using Kodaira dimension based on Thurston's geometries, and explores its relation to geometric structures, Gromov norm, and higher-dimensional cases.

Contribution

It defines the Kodaira dimension for 3-manifolds via Thurston's geometries and studies its compatibility with existing invariants and mapping orders, extending ideas to higher dimensions.

Findings

Kodaira dimension classification aligns with existing invariants.

Closed geometric 4-manifolds have non-zero Gromov norm only in specific geometries.

Established relations between geometric structures and mapping orders in higher 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 $\mathbb H^2\times \mathbb H^2$, $\mathbb H^2(\mathbb C)$ or $\mathbb H^4$.