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

Christoforos Neofytidis and Weiyi Zhang

Vol. 315 (2021), No. 1, 209–233

DOI: 10.2140/pjm.2021.315.209

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 nonzero 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 nonvanishing simplicial volume has top holomorphic Kodaira dimension.

Keywords

Kodaira dimension, Thurston geometry, simplicial volume, nonzero degree, Gromov's order, 5-manifold, projective manifold, Kähler manifold

Mathematical Subject Classification

Primary: 57M50, 57N16

Milestones

Received: 25 January 2021

Revised: 4 July 2021

Accepted: 13 September 2021

Published: 13 December 2021