A simply connected surface of general type withpg= 0 andK2= 4

Park, Heesang; Park, Jong-Il; Shin, Dong Yun

doi:10.2140/gt.2009.13.1483

Cited by 27 publications

(32 citation statements)

References 9 publications

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“…Let us take M = CP 2 #8CP 2 and N is the Barlow surface [2] which is a complex surface of general type homeomorphic to M . One can also take M = CP 2 #5CP 2 and N is the complex surface of general type homeomorphic to M constructed in [43]. Then M × Σ g is diffeomorphic to N × Σ g .…”

Section: 2mentioning

confidence: 99%

Geometric structures, Gromov norm and Kodaira dimensions

Zhang

2017

Advances in Mathematics

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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 ...

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Section: 2mentioning

confidence: 99%

Geometric structures, Gromov norm and Kodaira dimensions

Zhang

2017

Advances in Mathematics

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“…The choice of four quadrics form a Gr (4,16). This together with the projective transformations of ℙ 2…”

Section: Deformation Of Surface With = = and = ℤmentioning

confidence: 99%

Godeaux, Campedelli, and surfaces of general type with χ=4 and 2≤K2≤8

Iqbal

2017

Math. Nachr.

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We construct simply connected surfaces of general type with invariants χ(O)=4 and 2≤K2≤8. We use double-struckQ‐Gorenstein deformations in conjunction with explicit constructions that express the canonical rings by generators and relations. The canonical rings of the surfaces are described as projections. The whole construction is simplified by the use of key varieties based on Steiner 3‐folds. As a consequence of the construction we find two families, each family in a different connected component of the moduli stack scriptM¯2,1, and each linking a Campedelli surface with a Godeaux surface.

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“…In order to construct smooth surface of general type with p g = q = 0 for a given self-intersection number of the canonical class, Y. Lee, H. Park, J. Park, and D. Shin generate rational elliptic surfaces with nef canonical classes and quotient singularities of class T and then take their Q-smoothings in [26], [33], [34]. Many such surfaces are presented in [27].…”

Section: Mainmentioning

confidence: 99%