Symplectic 4-manifolds with Kodaira dimension zero

Li, Tian Jun

doi:10.4310/jdg/1175266207

Cited by 72 publications

(136 citation statements)

References 23 publications

Supporting

Mentioning

136

Contrasting

Order By: Relevance

“…The cases where (t, u, v) = (10, 2, 4) or (16,1,8) can be ruled out as follows. The maximal subgroup D induces a Z 3 -action on the set of isolated fixed points of g, which must be free because D can not act freely and linearly on S 3 .…”

Section: Proof Of Theorem 10mentioning

confidence: 99%

Symmetric symplectic homotopy K 3 surfaces

Chen

Kwasik

2011

Journal of Topology

View full text Add to dashboard Cite

Abstract. A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.

show abstract

Section: Proof Of Theorem 10mentioning

confidence: 99%

Symmetric symplectic homotopy K 3 surfaces

Chen

Kwasik

2011

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…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].…”

Section: Introductionmentioning

confidence: 98%

Geometric structures, Gromov norm and Kodaira dimensions

Zhang

2017

Advances in Mathematics

View full text Add to dashboard Cite

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 ...

show abstract

“…Such manifolds are either CP 2 or an S 2 -bundle over a surface. Minimal symplectic manifolds of Kodaira dimension zero were studied in [8]: it was speculated that they are either K3, Enriques surface or a T 2 -bundle over T 2 ; and it was shown that the χ and σ are bounded if b 1 is bounded by 4.…”

Section: Introductionmentioning

confidence: 99%

Geography of symplectic 4–manifolds with Kodaira dimension one

Baldridge

2005

Algebr. Geom. Topol.

Self Cite

View full text Add to dashboard Cite

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integer called the degeneracy. In this paper we include the degeneracy as an extra parameter in the geography problem and show how to fill out the geography of symplectic 4-manifolds with Kodaira dimension 1 for all admissible triples.

show abstract

Symplectic 4-manifolds with Kodaira dimension zero

Cited by 72 publications

References 23 publications

Symmetric symplectic homotopy K 3 surfaces

Symmetric symplectic homotopy K 3 surfaces

Geometric structures, Gromov norm and Kodaira dimensions

Geography of symplectic 4–manifolds with Kodaira dimension one

Contact Info

Product

Resources

About