Abstract
We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.
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References
Abbondandolo A., Schwarz M.: On the Floer homology of cotangent bundles. Comm. Pure Appl. Math. 59(2), 254–316 (2006)
M. Abouzaid, A cotangent fibre generates the Fukaya category, preprint (2010); arXiv:1003.4449
Abouzaid M., Seidel P.: An open string analogue of Viterbo functoriality. Geom. Topol. 14(2), 627–718 (2010)
Anosov D.V.: Some homotopies in a space of closed curves. Math. USSR-Izv 17, 423–453 (1981)
F. Bourgeois, T. Ekholm, Y. Eliashberg, Effect of legendrian surgery, preprint (2009); arXiv:SG/0911.0026
Bourgeois F., Eliashberg Y., Hofer H., Wysocki K., Zehnder E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)
K. Cieliebak, Y. Eliashberg, Symplectic geometry of Stein manifolds, in preparation.
Cieliebak K., Floer A., Hofer H., Wysocki K.: Applications of symplectic homology II:stability of the action spectrum. Math. Z 223, 27–45 (1996)
Dostoglou S., Salamon D.: Self-dual instantons and holomorphic curves. Ann. of Math. (2) 139, 581–640 (1994)
Y. Eliashberg, M. Gromov, Convex symplectic manifolds, in “Several Complex Variables and Complex Geometry Part 2”, Proc. Sympos. Pure Math. 52, Amer. Math. Soc., Providence, RI (1991), 135–162.
Y. Félix, S. Halperin, J.-C. Thomas, The homotopy Lie algebra for finite complexes, Inst. Hautes Études Sci. Publ. Math. 56 (1983), 179–202. 1982.
Y. Félix, S. Halperin, J.-C. Thomas, Rational Homotopy Theory, Springer Graduate Texts in Mathematics 205 (2001).
Félix Y., Thomas J.-C.: The radius of convergence of Poincaré series of loop spaces. Invent. Math. 68(2), 257–274 (1982)
Floer A., Hofer H., Salamon D.: Transversality in elliptic Morse theory for the symplectic action. Duke Math. J. 80, 251–292 (1995)
Gromov M.: Homotopical effects of dilatation. J. Differential Geom. 13(3), 303–310 (1978)
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203, 205–326.
Lambrechts P.: The Betti numbers of the free loop space of a connected sum. J. London Math. Soc. (2) 64(1), 205–228 (2001)
D. McDuff, D. Salamon, Introduction to Symplectic Topology. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, second edition, 1998.
McLean M.: Lefschetz fibrations and symplectic homology. Geom. Topol. 13(4), 1877–1944 (2009)
A. Oancea, A survey of Floer homology for manifolds with contact type boundary or symplectic homology, Ensaios Mat. 7, Soc. Brasil. Mat., Rio de Janeiro (2004), 51–91.
A. Ritter, Topological quantum field theory structure on symplectic cohomology, preprint (1998); arXiv:SG/1003.1781
Salamon D.A., Weber J.: Floer homology and the heat flow. Geom. Funct. Anal. 16(5), 1050–1138 (2006)
Seidel P.: A biased view of symplectic cohomology. Current Developments in Mathematics 2006, 211–253 (2008)
Totaro B.: Complexifications of nonnegatively curved manifolds. J. Eur. Math. Soc. (JEMS) 5(1), 69–94 (2003)
Vigué-Poirrier M.: Homotopie rationnelle et croissance du nombre de géodésiques fermées. Ann. Sci. École Norm. Sup. (4) 17(3), 413–431 (1984)
Viterbo C.: Functors and computations in Floer homology with applications. part II, preprint (1996)
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McLean, M. The Growth Rate of Symplectic Homology and Affine Varieties. Geom. Funct. Anal. 22, 369–442 (2012). https://doi.org/10.1007/s00039-012-0158-7
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DOI: https://doi.org/10.1007/s00039-012-0158-7