Symplectic surgeries and normal surface singularities

Gay, David T¹ ; Stipsicz, András I²

¹Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa
²Hungarian Academy of Sciences, Renyi Institute of Mathematics, Reáltanoda utca 13–15, Budapest, 1053, Hungary, Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027, USA

Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2203-2223

Résumé

We show that every negative definite configuration of symplectic surfaces in a symplectic 4–manifold has a strongly symplectically convex neighborhood. We use this to show that if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic 4–manifolds.

Détail

DOI : 10.2140/agt.2009.9.2203

Keywords: symplectic rational blow-down, surface singularity, symplectic neighborhood

Bibliographie

[1] F Bogomolov, Fillability of contact pseudoconvex manifolds, Göttingen Univ. preprint Heft 13 (1993) 1

[2] C Caubel, A Némethi, P Popescu-Pampu, Milnor open books and Milnor fillable contact 3–manifolds, Topology 45 (2006) 673

[3] C Caubel, P Popescu-Pampu, On the contact boundaries of normal surface singularities, C. R. Math. Acad. Sci. Paris 339 (2004) 43

[4] Y Eliashberg, On symplectic manifolds with some contact properties, J. Differential Geom. 33 (1991) 233

[5] T Etgü, B Ozbagci, Explicit horizontal open books on some plumbings, Internat. J. Math. 17 (2006) 1013

[6] J B Etnyre, Symplectic convexity in low-dimensional topology, from: "Symplectic, contact and low-dimensional topology (Athens, GA, 1996)" (editors G Matić, C McCrory), Topology Appl. 88 (1998) 3

[7] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[8] D T Gay, A I Stipsicz, Symplectic rational blow-down along Seifert fibered 3–manifolds, Int. Math. Res. Not. (2007)

[9] H Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962) 331

[10] P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007)

[11] H B Laufer, Normal two-dimensional singularities, 71, Princeton Univ. Press (1971)

[12] Y Lee, J Park, A simply connected surface of general type with pg = 0 and K2 = 2, Invent. Math. 170 (2007) 483

[13] E Looijenga, J Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986) 261

[14] J D Mccarthy, J G Wolfson, Symplectic resolution of isolated algebraic singularities, from: "Geometry, topology, and dynamics (Montreal, PQ, 1995)" (editor F Lalonde), CRM Proc. Lecture Notes 15, Amer. Math. Soc. (1998) 101

[15] A Némethi, Five lectures on normal surface singularities, from: "Low dimensional topology (Eger, 1996/Budapest, 1998)" (editors K Böröczky Jr., W Neumann, A Stipsicz), Bolyai Soc. Math. Stud. 8, János Bolyai Math. Soc. (1999) 269

[16] A Némethi, P Popescu-Pampu, On the Milnor fibers of sandwiched singularities

[17] M Symington, Generalized symplectic rational blowdowns, Algebr. Geom. Topol. 1 (2001) 503

[18] M Symington, Four dimensions from two in symplectic topology, from: "Topology and geometry of manifolds (Athens, GA, 2001)" (editors G Matić, C McCrory), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 153

[19] C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809

[20] J Wahl, Topology, geometry, and equations of normal surface singularities, from: "Singularities and computer algebra" (editors C Lossen, G Pfister), London Math. Soc. Lecture Note Ser. 324, Cambridge Univ. Press (2006) 351