Geometric structures, Gromov norm and Kodaira dimensions

Abstract

A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.uk GEOMETRIC STRUCTURES, GROMOV NORM AND KODAIRA DIMENSIONS WEIYI ZHANGAbstract. 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. IntroductionComplex Kodaira dimension κ h (M, J) provides a very successful classification scheme for complex manifolds. This notion is generalized by several authors (c.f. [31,39,40,26,27]) to symplectic manifolds, especially of dimension two and four. In these two dimensions, this symplectic Kodaira dimension is independent of the choice of symplectic structures [31]. In other words, it is a smooth invariant of the manifold which is thus denoted by κ s (M ). In dimension four, the smaller the symplectic Kodaira dimension, the more we know. Symplectic 4-manifolds with κ s = −∞ are diffeomorphic to rational or ruled surfaces [36]. When κ s = 0, all known examples are K3 surface, Enrique surface and T 2 bundles over T 2 . Moreover, it is shown in [31] that a symplectic manifold with κ s = 0 has the same homological invariants as one of the manifolds listed above. When κ s = 1 or 2, no classification is possible since symplectic manifolds in both categories could admit arbitrary finitely presented group as their fundamental group [19].In [9], the authors prove that complex and symplectic Kodaira dimensions are compatible with each other. More precisely, when a 4-manifold M admits at the same time both complex and symplectic structures (but the structures are not necessarily compatible with each other), then κ s (M ) = κ h (M, J). In [35], a general framework of "additivity of Kodaira dimension" is provided to further understand the compatibility of various Kodaira dimensions in possibly different dimensions. In particular, it is shown that the Kodaira dimensions are additive for fiber bundles, Lefschetz fibrations and coverings.Higher dimensional generalizations of Kodaira dimension, e.g. symplectic Kodaira dimension in dimensions six or higher, are less understood except for a proposed definition in [33]. Like complex Kodaira dimension, it will no longer be a smooth invariant. Hence, the study of this notion in higher dimensions will be associated to the study of deformation classes of symplectic structures and symplectic birational geometry.As suggested by the additivity framework, dimension three should also attach certain counterpart of Kodaira dimension. In this paper, we give a definition of Kodaira dimension κ t (M ) in dimension three through Thurston's eight ...

Ordering Thurston’s geometries by maps of nonzero degree

“…Zhang [25] defined the Kodaira dimension of 3-manifolds as follows: Divide the eight 3dimensional Thurston geometries into four categories assigning a value to each category: −∞ : S 3 , S 2 × R 0 : R 3 , Nil 3 , Sol 3 1 : H 2 × R, SL 2 3 2 : H 3 . Let M be a closed oriented 3-manifold.…”

“…We call this a T -decomposition of M. The Kodaira dimension of M is then defined as follows: Definition 6.1. ( [25]). The Kodaira dimension κ t of a closed oriented 3-manifold M is (1) κ t (M) = −∞, if for any T -decomposition each piece belongs to the category −∞;…”

“…For instance, 1-domination defines a partial order on the set of closed Hopfian aspherical manifolds of a given dimension (see [18] for 3manifolds). Other special cases have been studied by several authors; see for example [3,1,2,25].…”

“…The Kodaira dimension is a significant invariant in the classification scheme of manifolds; we refer to [12] for a recent survey on the various notions of Kodaira dimension of low-dimensional manifolds. Zhang [25] defined the Kodaira dimension κ t for 3-manifolds and showed ([25, Theorem 1.1]) that it is monotone with respect to the domination relation; see Theorem 6.1. Moreover, Zhang [25] defined the Kodaira dimension κ g for geometric 4-manifolds and suggested that it should also be monotone with respect to the domination relation (we will give both definitions of κ t and κ g in Section 6).…”

“…Zhang [25] defined the Kodaira dimension κ t for 3-manifolds and showed ([25, Theorem 1.1]) that it is monotone with respect to the domination relation; see Theorem 6.1. Moreover, Zhang [25] defined the Kodaira dimension κ g for geometric 4-manifolds and suggested that it should also be monotone with respect to the domination relation (we will give both definitions of κ t and κ g in Section 6). In this paper we confirm Zhang's suggestion:…”