Vol. 315, No. 1, 2021

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

Christoforos Neofytidis and Weiyi Zhang

Vol. 315 (2021), No. 1, 209–233

We introduce an axiomatic definition for the Kodaira dimension and classify Thurston geometries in dimensions 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.

Kodaira dimension, Thurston geometry, simplicial volume, nonzero degree, Gromov's order, 5-manifold, projective manifold, Kähler manifold
Mathematical Subject Classification
Primary: 57M50, 57N16
Received: 25 January 2021
Revised: 4 July 2021
Accepted: 13 September 2021
Published: 13 December 2021
Christoforos Neofytidis
Department of Mathematics
Ohio State University
Columbus, OH
United States
Weiyi Zhang
Mathematics Institute
University of Warwick
United Kingdom