Abstract
This chapter includes a gentle introduction to the Gromov-Witten invariants, and then explores the relationship between these and McDuff’s results on rational/ruled surfaces, including a complete proof that the latter are the only symplectic 4-manifolds that are uniruled.
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- 1.
We are being intentionally vague here about the meaning of the words “perturbations of the nonlinear Cauchy-Riemann equation”. This can mean various things depending on the context and the precise definition of the intersection number (7.3) that one adopts, e.g. in the approaches of [MS12, CM07], the usual equation \(\bar {\partial }_J u = 0\) is generalized to allow generic dependence of J on points in the domain of u, while in [RT97, Ger], one instead introduces a generic nonzero term (inhomogeneous perturbation) on the right hand side of the equation. A more abstract functional-analytic approach is taken in [HWZa].
- 2.
For slightly different reasons, the genus 0 invariants \(\operatorname {GW}_{0,m}^{(M,\omega )}(\alpha _1,\ldots ,\alpha _m;\beta )\) for m ≥ 3 are also integers whenever α 1, …, α m and β are all integral classes and (M, ω) is semipositive, see [MS12]. The semipositivity condition is always satisfied when \(\dim M \le 6\).
- 3.
With a little more effort and intersection theory, one can show that \({\mathcal M}_S^{\operatorname {bad}}(J)\) is actually finite for generic J, but we will not need this.
- 4.
“Indefinite” means that Q(e.e) is positive for some \(e \in {\mathbb Z}^2\) and negative for others. Equivalently, one could stipulate that for the induced real-valued nondegenerate quadratic form on \({\mathbb R}^2\), the maximum dimensions of subspaces on which this form is positive- or negative-definite are each 1.
References
G.E. Bredon, Topology and Geometry (Springer, New York, 1993)
K. Cieliebak, K. Mohnke, Symplectic hypersurfaces and transversality in Gromov-Witten theory. J. Symplectic Geom. 5(3), 281–356 (2007)
J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, 3rd edn. (Springer, New York, 1999). With additional contributions by E. Bannai, R.E. Borcherds, J. Leech, S.P. Norton, A.M. Odlyzko, R.A. Parker, L. Queen, B.B. Venkov
O. Debarre, Higher-Dimensional Algebraic Geometry. Universitext (Springer, New York, 2001)
C. Gerig, C. Wendl, Generic transversality for unbranched covers of closed pseudoholomorphic curves. Commun. Pure Appl. Math. 70(3), 409–443 (2017)
A. Gerstenberger, Geometric transversality in higher genus Gromov-Witten theory (2013). Preprint arXiv:1309.1426
A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)
M.W. Hirsch, Differential Topology (Springer, New York, 1994)
H. Hofer, K.Wysocki, E. Zehnder, Applications of polyfold theory I: the polyfolds of Gromov-Witten theory (2011). Preprint arXiv:1107.2097
M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994)
P.D. Lax, Linear Algebra and Its Applications. Pure and Applied Mathematics (Hoboken), 2nd edn. (Wiley-Interscience, Hoboken, NJ, 2007)
J.M. Lee, Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218 (Springer, New York, 2003)
D. McDuff, The structure of rational and ruled symplectic 4-manifolds. J. Am. Math. Soc. 3(3), 679–712 (1990)
D. McDuff, Immersed spheres in symplectic 4-manifolds. Ann. Inst. Fourier (Grenoble) 42(1–2), 369–392 (1992) [English, with French summary]
D. McDuff, D. Salamon, J-Holomorphic Curves and Quantum Cohomology. University Lecture Series, vol. 6 (American Mathematical Society, Providence, RI, 1994)
D. McDuff, D. Salamon, A survey of symplectic 4-manifolds with b + = 1. Turk. J. Math. 20(1), 47–60 (1996)
D. McDuff, D. Salamon, J-Holomorphic Curves and Symplectic Topology. American Mathematical Society Colloquium Publications, vol. 52, 2nd edn. (American Mathematical Society, Providence, RI, 2012)
D. McDuff, D. Salamon, Introduction to symplectic topology, 3rd edn. (Oxford University Press, Oxford, 2017)
J.W. Milnor, Topology from the Differentiable Viewpoint. Princeton Landmarks in Mathematics (Princeton University Press, Princeton, NJ, 1997). Based on notes by David W. Weaver; Revised reprint of the 1965 original
J. Milnor, D. Husemoller, Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73 (Springer, New York, 1973)
J.W. Milnor, J.D. Stasheff, Characteristic Classes. Annals of Mathematics Studies, vol. 76 (Princeton University Press, Princeton, NJ, 1974)
J. Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom. Topol. 20(2), 779–1034 (2016)
Y. Ruan, G. Tian, Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130, 455–516 (1997)
C.H. Taubes, The Seiberg-Witten and Gromov invariants. Math. Res. Lett 2(2), 221–238 (1995)
C.H. Taubes, Counting pseudo-holomorphic submanifolds in dimension 4. J. Differ. Geom. 44(4), 818–893 (1996)
C.H. Taubes, SW ⇒ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Am. Math. Soc. 9(3), 845–918 (1996)
R. Thom, Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28, 17–86 (1954) [French]
C. Wendl, Transversality and super-rigidity for multiply covered holomorphic curves (2016). Preprint arXiv:1609.09867v3
A. Zinger, Pseudocycles and integral homology. Trans. Am. Math. Soc. 360(5), 2741–2765 (2008)
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Wendl, C. (2018). Uniruled Symplectic 4-Manifolds. In: Holomorphic Curves in Low Dimensions. Lecture Notes in Mathematics, vol 2216. Springer, Cham. https://doi.org/10.1007/978-3-319-91371-1_7
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