Constructions of generalized complex structures in dimension four. (English) Zbl 1260.53136
Summary: Four-manifold theory is employed to study generalized complex structures. We drastically enlarge the number of available known examples of generalized complex four-manifolds by building on recent constructions of different smooth structures. Generalized complex structures that have multiple type change loci are constructed.
MSC:
| 53D18 | Generalized geometries (à la Hitchin) |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
References:
| [1] | Akbulut S.: On 2-dimensional homology classes of 4-manifolds. Math. Proc. Camb. Phil. Soc. 82, 99–106 (1977) · Zbl 0355.57013 · doi:10.1017/S0305004100053718 |
| [2] | Akbulut S., Yasui K.: Corks, Plugs and exotic structures. J. Gökova Geom. Topol. 2, 40–82 (2008) · Zbl 1214.57027 |
| [3] | Akhmedov A., Park B.D.: Exotic smooth structures on small 4-manifolds with odd signatures. Invent. Math. 181(3), 577–603 (2010) · Zbl 1206.57029 · doi:10.1007/s00222-010-0254-y |
| [4] | Auroux D., Donaldson S.K., Katzarkov L.: Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves. Math. Ann. 326, 185–203 (2003) · Zbl 1026.57020 · doi:10.1007/s00208-003-0418-9 |
| [5] | Baldridge S., Kirk P.: On symplectic 4-manifolds with prescribed fundamental group. Comment. Math. Helv. 82(4), 845–875 (2007) · Zbl 1155.57024 · doi:10.4171/CMH/112 |
| [6] | Baldridge S., Kirk P.: A symplectic manifold homeomorphic but not diffeomorphic to $${\(\backslash\)mathbb{CP}\^2 \(\backslash\)# 3 \(\backslash\)overline{\(\backslash\)mathbb{CP}}\^2}$$ . Geom. & Topol. 12, 919–940 (2008) · Zbl 1152.57026 · doi:10.2140/gt.2008.12.919 |
| [7] | Baldridge S., Kirk P.: Constructions of Small Symplectic 4-Manifolds using Luttinger Surgery. J. Diff. Geom. 82(2), 317–362 (2009) · Zbl 1173.53042 |
| [8] | Barth, W.P., Hulek, K., Peters, C.A.M., van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 4, Berlin: Springer, Second Enlarged Edition, 2004 · Zbl 1036.14016 |
| [9] | Cavalcanti, G.R.: Introduction to generalized complex geometry. Publicações Matemáticas do IMPA. 26o Colóquio Brasileiro de Matemática. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2007 |
| [10] | Cavalcanti G.R., Gualtieri M.: A surgery for generalized complex structures on 4-manifolds. J. Diff. Geom. 76(1), 35–43 (2007) · Zbl 1124.57011 |
| [11] | Cavalcanti G.R., Gualtieri M.: Blow-up of generalized complex 4-manifolds. J. Topology 2, 840–864 (2009) · Zbl 1197.53090 · doi:10.1112/jtopol/jtp031 |
| [12] | Donaldson S.: An application of gauge theory to four dimensional topology. J. Diff. Geom. 18, 279–315 (1983) · Zbl 0507.57010 |
| [13] | Donaldson S.: Seiberg-Witten equations and 4-manifold topology. Bull. Amer. Math. Soc. 33, 45–70 (1996) · Zbl 0872.57023 · doi:10.1090/S0273-0979-96-00625-8 |
| [14] | Fintushel R., Stern R.: Knots, links, and 4-manifolds. Invent. Math. 134, 363–400 (1998) · Zbl 0914.57015 · doi:10.1007/s002220050268 |
| [15] | Fintushel R., Park B.D., Stern R.: Reverse Engineering small 4-manifolds. Alg. & Geom. Topol. 7, 2103–2116 (2007) · Zbl 1142.57018 · doi:10.2140/agt.2007.7.2103 |
| [16] | Freedman M.: The topology of four-dimensional manifolds. J. Diff. Geom. 17(3), 357–453 (1982) · Zbl 0528.57011 |
| [17] | Gompf R.E.: Nuclei of elliptic surfaces. Topology 30, 479–511 (1991) · Zbl 0732.57010 · doi:10.1016/0040-9383(91)90027-2 |
| [18] | Gompf R.E.: A new construction of symplectic manifolds. Ann. of Math. (2) 142(3), 527–595 (1995) · Zbl 0849.53027 · doi:10.2307/2118554 |
| [19] | Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, 20. Providence, RI: Amer. Math. Soc., 1999 · Zbl 0933.57020 |
| [20] | Gualtieri M.: Generalized complex geometry. Ann. of Math. 174, 75–123 (2011) · Zbl 1235.32020 · doi:10.4007/annals.2011.174.1.3 |
| [21] | Gualtieri, M.: Generalized complex geometry. PhD thesis, Oxford University, 2004, http://arxiv.org/abs/math/0401221v1 [math. DG], 2004 · Zbl 1079.53106 |
| [22] | Hambleton I., Kreck M.: Cancellation, Elliptic Surfaces and the Topology of Certain Four-Manifolds. J. Reine Angew. Math. 444, 79–100 (1993) · Zbl 0779.57015 |
| [23] | Hambleton I., Teichner P.: A Non-Extended Hermitian form over $${\(\backslash\)mathbb{Z} [\(\backslash\)mathbb{Z}]}$$ . Manuscripta Mathematica 94, 435–442 (1997) · Zbl 0890.57034 · doi:10.1007/BF02677483 |
| [24] | Hitchin N.: Generalized Calabi-Yau manifolds. Quart. J. Math. Oxford 54, 281–308 (2003) · Zbl 1076.32019 · doi:10.1093/qmath/hag025 |
| [25] | Hitchin, N.: Lectures on generalized geometry. In: Surveys in Differential Geometry Vol. XVI, Geometry of special holonomy and related topics, 2011, pp. 79–124, http://arxiv.org/abs/1008.0973v1 [math.DG], 2010 |
| [26] | Hughes, M.C.: Branched covering constructions and the symplectic geography problem. PhD Thesis, University of Waterloo, 2008 |
| [27] | Kodaira K.: On the structure of compact complex analytic surfaces. I. Amer. J. Math. 86, 751–798 (1964) · Zbl 0137.17501 · doi:10.2307/2373157 |
| [28] | Kotschick D.: All fundamental groups are almost-complex. Proc. Amer. Math. Soc. 134, 3081–3083 (2006) · Zbl 1094.57004 · doi:10.1090/S0002-9939-06-08352-3 |
| [29] | Kronheimer P.B., Mrowka T.S.: The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1, 797–808 (1994) · Zbl 0851.57023 · doi:10.4310/MRL.1994.v1.n6.a14 |
| [30] | Luttinger K.M.: Lagrangian Tori in $${\(\backslash\)mathbb{R}\^4}$$ . J. Diff. Geom. 42, 220–228 (1995) |
| [31] | McCarthy J., Wolfson J.: Symplectic normal connect sum. Topology 33, 729–764 (1994) · Zbl 0812.53033 · doi:10.1016/0040-9383(94)90006-X |
| [32] | Morgan J.W., Mrowka T.S., Szabó Z.: Product formulas along T 3 for Seiberg-Witten invariants. Math. Res. Lett. 4, 915–929 (1997) · Zbl 0892.57021 · doi:10.4310/MRL.1997.v4.n6.a11 |
| [33] | Szabó Z.: Simply-connected irreducible 4-manifolds with no symplectic structures. Invent. Math. 132, 457–466 (1998) · Zbl 0906.57014 · doi:10.1007/s002220050230 |
| [34] | Taubes C.H.: The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. 1(6), 809–822 (1994) · Zbl 0853.57019 · doi:10.4310/MRL.1994.v1.n6.a15 |
| [35] | Witten E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994) · Zbl 0867.57029 · doi:10.4310/MRL.1994.v1.n6.a13 |
| [36] | Yazinski, J.: A new bound on the size of symplectic 4-manifolds with prescribed fundamental group to appear. J. of Symplectic Geom (2010) |
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