Abstract
In this paper we classify Lagrangian spheres in \(A_n\)-surface singularities up to Hamiltonian isotopy. Combining with a result of Ritter (Geom Funct Anal 20(3):779–816, 2010), this yields a complete classification of exact Lagrangians in \(A_n\)-surface singularities. Our main new tool is the application of a technique which we call ball-swappings and its relative version.
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References
Abouzaid, M.: Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189(2), 251–313 (2012)
Abouzaid, M., Smith, I.: Exact Lagrangians in plumbings (preprint). http://arxiv.org/abs/1107.0129
Borman, M.S., Li, T.-J., Wu, W.: Spherical Lagrangians via ball packings and symplectic cutting. Selecta Math. http://arxiv.org/abs/1211.5952
Evans, J.D.: Symplectic topology of some Stein and rational surfaces. Ph.D. Thesis, Christ’s College, Cambridge (2010)
Evans, J.D.: Lagrangian spheres in del Pezzo surfaces. J. Topol. 3(1), 181–227 (2010)
Evans, J.D.: Symplectic mapping class groups of some Stein and rational surfaces. J. Symplectic Geom. 9(1), 45–82 (2011)
Gompf, R.E.: A new construction of symplectic manifolds. Ann. Math. (2) 142(3), 527–595 (1995)
Hind, R.: Lagrangian unknottedness in Stein surfaces. Asian J. Math. 16(1), 1–36 (2012)
Hind, R., Pinsonnault, M., Wu, W.: Symplectomorphism groups of non-compact manifolds and space of Lagrangians. arXiv:1305.7291
Ishii, A., Uehara, H.: Autoequivalences of derived categories on the minimal resolutions of \(A_n\) singularities on surfaces. J. Differ. Geom. 71, 385–435 (2005)
Ishii, A., Ueda, K., Uehara, H.: Stability conditions on \(A_n\) singularities. J. Differ. Geom. 84(1), 87–126 (2010)
Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203–271 (2002)
Luttinger, K.: Lagrangian tori in \(R^4\). J. Differ. Geom. 42(2), 220–228 (1995)
Lekili, Y., Maydanskiy, M.: The symplectic topology of some rational homology balls. Comm. Math. Helv. arxiv:1202.5625
Lalonde, F., Pinsonnault, M.: The topology of the space of symplectic balls in rational 4-manifolds. Duke Math. J. 122(2), 347–397 (2004)
Li, T.-J., Wu, W.: Lagrangian spheres, symplectic surfaces and the symplectic mapping class group. Geom. Topol. 16(2), 1121–1169 (2012)
McDuff, D.: From symplectic deformation to isotopy. In: Topics in Symplectic 4-Manifolds (Irvine, CA, 1996), First International Press Lecture Series I, pp. 85–99. International Press, Cambridge (1998)
McDuff, D., Opshtein, E.: Nongeneric \(J\)-holomorphic curves in rational manifolds. arXiv:1309.6425
McDuff, D., Polterovich, L.: Symplectic packings and algebraic geometry. Invent. Math. 115(3):405–434 (1994). With appendix by Y. Karshon
McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1998). ISBN 0-19-850451-9
Pinsonnault, M.: Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. J. Mod. Dyn. 2(3), 431–455 (2008)
Ritter, A.: Deformations of symplectic cohomology and exact Lagrangians in ALE spaces. Geom. Funct. Anal. 20(3), 779–816 (2010)
Seidel, P.: Floer homology and the symplectic isotopy problem. University of Oxford, Thesis (1997)
Seidel, P.: Symplectic automorphisms of \(T^*S^2\). arXiv:math/9803084
Seidel, P.: Lagrangian two-spheres can be symplectically knotted. J. Differ. Geom. 52(1), 145–171 (1999)
Seidel, P.: Lectures on four-dimensional Dehn twists. Symplectic 4-Manifolds and Algebraic Surfaces, vol. 1938, pp. 231–267. Lecture Notes in Mathematics. Springer, Berlin (2008)
Seidel, P.: Abstract analogues of flux as symplectic invariants. arXiv:1108.0394
Seidel, P.: Lagrangian homology spheres in \((A_m)\) Milnor fibres via \(C^*\)-equivariant \(A_\infty \) modules. Geom. Topo. 16, 2343–2389 (2012)
Smith, I.: Floer cohomology and pencils of quadrics. Invent. Math. 189(1), 149–250 (2012)
Acknowledgments
The author is deeply indebted to Richard Hind, who inspired the idea in this paper during his stay in Michigan State University. I warmly thank Dusa McDuff for generously sharing preliminary drafts of her recent preprint with Emmanuel Opshtein [18] and for explaining many details, which are very helpful to the current paper. I also thank an anonymous referee for carefully checking many details, as well as suggesting an alternative approach of Lemma 3.4. Many ideas involved in this work were originally due to Jonny Evans in his excellent series of papers [5, 6], and the ball-swapping construction used here stems out from the beautiful ideas exhibited in Seidel’s lecture notes [26]. This work is partially motivated by a talk by Yanki Lekili in MSRI. I would also like to thank the Geometry/Topology group of Michigan State University for providing me a friendly and inspiring working environment, as well as supports for my visitors. The author is supported by NSF Focused Research Grants DMS-0244663.