Simply connected symplectic 4-manifolds with b2+=1 and c12=2

Park, Jong-Il

doi:10.1007/s00222-004-0404-1

Cited by 74 publications

(80 citation statements)

References 16 publications

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“…(e.g. [1,3,7,16,21,22,25,24,26,27,31,41,42,53].) Resting on spectacular results of Taubes on Seiberg-Witten invariants of symplectic 4-manifolds, these constructions have demonstrated that minimal symplectic 4-manifolds not only constitute small exotic 4-manifolds (which are homeomorphic but not diffeomorphic to standard ones), but also resource them in almost all the known instances.…”

Section: Introductionmentioning

confidence: 98%

Small Lefschetz fibrations and exotic 4-manifolds

Baykur

Korkmaz

2016

Math. Ann.

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Abstract. We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP 2 #p CP 2 for p = 7, 8, 9, and to 3CP 2 #q CP 2 for q = 12, . . . , 19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b + ≤ 3, with one possible exception when b + = 3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of CP 2 #4CP 2 and 3CP 2 #6CP 2 from small Lefschetz fibrations over surfaces of higher genera.

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Section: Introductionmentioning

confidence: 98%

Small Lefschetz fibrations and exotic 4-manifolds

Baykur

Korkmaz

2016

Math. Ann.

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“…Recently, new methods were developed [Pa05,ABP07] to construct symplectic 4-manifolds with small topology and exotic smooth structures. Moreover, the method proposed by J.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the method proposed by J. Park [Pa05] was later refined [LePa07,PPS07] to produce interesting examples of minimal complex surfaces of general type. In this paper we show how we can use these constructions in regard to the existence or non-existence of Einstein metrics.…”

Section: Introductionmentioning

confidence: 99%

“…Based on ideas from [Pa05], Lee and Park [LePa07] and more recently Park, Park, and Shin [PPS07] constructed new examples of simply connected, minimal surfaces of general type with p g = 0, K 2 = 1, 2 or 3. We show that their examples satisfy the ampleness condition: Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…In his paper, Park [Pa05], used his examples and LeBrun's Obstruction Theorem to exhibit a manifold homeomorphic, non-diffeomorphic to CP 2 #8CP 2 which does not admit an Einstein metric. We extend his results to CP 2 #kCP 2 , k = 6, 7, and we also exhibit an infinite family of homeomorphic manifolds for which no Einstein metric exists.…”

Section: Introductionmentioning

confidence: 99%

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Smooth structures and Einstein metrics on

Răsdeaconu¹,

Suvaina²

2009

Math. Proc. Camb. Phil. Soc.

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ABSTRACT. We show that each of the topological 4-manifolds CP 2 #kCP 2 , for k = 6, 7 admits a smooth structure which has an Einstein metric of scalar curvature s > 0, a smooth structure which has an Einstein metric with s < 0 and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to CP 2 #5CP 2 which don't admit an Einstein metric. We also exhibit new examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.

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Introduction

Silva¹

Lecture Notes in Mathematics

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The word symplectic in mathematics was coined in the late 1930's by Weyl [142, p.165] who substituted the Latin root in complex by the corresponding Greek root in order to label the symplectic group (first studied be Abel). An English dictionary is likely to list symplectic as the name for a bone in a fish's head.2 The name lagrangian manifold was introduced by Maslov [93] in the 1960's, followed by lagrangian plane, etc., introduced by Arnold [2].14 Whitney Extension Theorem: Let M be a manifold and X a submanifold of M . Suppose that at each p ∈ X we are given a linear isomorphism Lp : TpM ≃ −→ TpM such that Lp| TpX = Id TpX and Lp depends smoothly on p. Then there exists an embedding h : N → M of some neighborhood N of X in M such that h| X = id X and dhp = Lp for all p ∈ X. A proof relies on a tubular neighborhood model. Lagrangian Submanifolds First Lagrangian SubmanifoldsLet (M, ω) be a symplectic manifold.Definition 2.1 A submanifold X of (M, ω) is lagrangian (respectively, isotropic and coisotropic) if, at each p ∈ X, the space T p X is a lagrangian (respectively, isotropic and coisotropic) subspace of (T p M, ω p ).65 Even if the action is not free, the orbit through p is a compact submanifold of M . In that

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Simply connected symplectic 4-manifolds with b2+=1 and c12=2

Cited by 74 publications

References 16 publications

Small Lefschetz fibrations and exotic 4-manifolds

Small Lefschetz fibrations and exotic 4-manifolds

Smooth structures and Einstein metrics on

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