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 ${\mathbb{H}}^2\times {\mathbb{H}}^2$, ${\mathbb{H}}^2(\mathbb{C})$ or ${\mathbb{H}}^4$.

arXiv comment33 pages. v3, more references added

References in corpus (5)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• A simply connected surface of general type with p_g=0 and K^2=2
• The simplicial volume of closed manifolds covered by H^2 x H^2
• Minimal entropy and geometric decompositions in dimension four

Cited by in corpus (1)

• The Kodaira dimension of contact 3-manifolds and geography of symplectic fillings