Geometric structures, the Gromov order, Kodaira dimensions and simplicial volume

Abstract

We introduce an axiomatic definition for the Kodaira dimension and classify Thurston geometries in dimensions $\le 5$ in terms of this Kodaira dimension. We show that the Kodaira dimension is monotone with respect to the partial order defined by maps of non-zero degree between 5-manifolds. We study the compatibility of our definition with traditional notions of Kodaira dimension, especially the highest possible Kodaira dimension. To this end, we establish a connection between the simplicial volume and the holomorphic Kodaira dimension, which in particular implies that any smooth Kähler 3-fold with non-vanishing simplicial volume has top holomorphic Kodaira dimension.

23 pages; v2: minor changes, to appear in Pacific Journal of Mathematics

References in corpus (7)

• The simplicial volume of closed manifolds covered by H^2 x H^2
• On three-manifolds dominated by circle bundles
• Kodaira dimensions of almost complex manifolds II
• Minimal models for Kähler threefolds
• 5-dimensional geometries III: the fibered geometries
• 5-dimensional geometries I: the general classification
• 5-dimensional geometries II: the non-fibered geometries