Abstract
We use the technique of Ruan (1999) and Li and Ruan (2001) to construct the virtual neighborhoods and show that the Gromov-Witten invariants can be defined as integrals over the top strata of the virtual neighborhoods. We prove that the invariants defined in this way satisfy all the axioms of Gromov-Witten invariants summarized by Kontsevich and Manin (1994).
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References
Berline N, Getzler E, Vergne M. Heat kernels and Dirac Operators. Grundlehren der mathematischen Wissenschaften Berlin-Heidelberg: Springer-Verlag, 2004
Bott R, Tu L W. Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. New York: Springer-Verlag, 1982
Castellano R. Smoothness of Kuranishi atlases on Gromov-Witten moduli spaces. arXiv:1511.04350, 2015
Castellano R. Genus zero Gromov-Witten axioms via Kuranishi atlases. arXiv:1601.04048, 2016
Chen B, Li A-M, Wang B-L. Virtual neighborhood technique for pseudo-holomorphic spheres. arXiv:1306.3276, 2013
Chen B, Li A-M, Wang B-L. Gluing principle for orbifold stratified spaces. In: Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol. 154. Tokyo: Springer, 2016, 15–57
Cox D, Katz S. Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, vol. 68. Providence: Amer Math Soc, 1999
Daskalopoulos G, Mese C. C1 estimates for the Weil-Petersson metric. Trans Amer Math Soc, 2017, 369: 2917–2950
Fabian M, Montesinos V, Zizler V. Smoothness in Banach spaces. Selected problems. RACSAM Rev R Acad Cienc Exactas Fís Nat Ser A Mat, 2006, 100: 101–125
Fukaya K, Oh Y, Ohta H, et al. Technical details on Kuranishi structure and virtual fundamental chain. arXiv:1209.4410, 2012
Fukaya K, Oh Y, Ohta H, et al. Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Singapore: Springer, 2020
Fukaya K, Ono K. Arnold conjecture and Gromov-Witten invariant. Topology, 1999, 38: 933–1048
Gromov M. Pseudo holomorphic curves in symplectic manifolds. Invent Math, 1985, 82: 307–347
Hofer H, Lizan V, Sikorav J-C. On genericity for holomorphic curves in four-dimensional almost-complex manifolds. J Geom Anal, 1997, 7: 149–159
Ivashkovich S, Shevchishin V. Pseudo-holomorphic curves and envelopes of meromorphy of two-spheres in ℂℙ2. arXiv:math/9804014, 1998
Kontsevich M, Manin Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm Math Phys, 1994, 164: 525–562
Li A-M, Ruan Y. Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. Invent Math, 2001, 145: 151–218
Li A-M, Sheng L. A finite rank bundle over J-holomorphic map moduli spaces. arXiv:1711.04228, 2017
Li A-M, Sheng L. The exponential decay of gluing maps for J-holomorphic map moduli space. J Differential Equations, 2019, 266: 2327–2372
Li J, Tian G. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J Amer Math Soc, 1998, 11: 119–174
Liu G, Tian G. Floer homology and Arnold conjecture. J Differential Geom, 1998, 49: 1–74
McDuff D. Notes on Kuranishi atlases. In: Virtual Fundamental Cycles in Symplectic Topology. Mathematical Surveys and Monographs, vol. 237. Providence: Amer Math Soc, 2019, 1–109
McDuff D, Salamon D. J-holomorphic Curves and Symplectic Topology. Providence: Amer Math Soc, 2004
McDuff D, Wehrheim K. Smooth Kuranishi atlases with isotropy. Geom Topol, 2017, 21: 2725–2809
McDuff D, Wehrheim K. The topology of Kuranishi atlases. Proc Lond Math Soc (3), 2017, 115: 221–292
McDuff D, Wehrheim K. The fundamental class of smooth Kuranishi atlases with trivial isotropy. J Topol Anal, 2018, 10: 71–243
Mumford D. Hirzebruch’s proportionality theorem in the noncompact case. Invent Math, 1977, 42: 239–272
Pardon J. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom Topol, 2016, 20: 779–1034
Robbin J, Salamon D. A construction of the Deligne-Mumford orbifold. J Eur Math Soc JEMS, 2006, 8: 611–699
Ruan Y. Topological sigma model and Donaldson-type invariants in Gromov theory. Duke Math J, 1996, 83: 461–500
Ruan Y. Virtual neighborhoods and pseudo-holomorphic curves. Turkish J Math, 1999, 23: 161–231
Ruan Y, Tian G. A mathematical theory of quantum cohomology. J Differential Geom, 1995, 42: 259–367
Ruan Y, Tian G. Higher genus symplectic invariants and sigma models coupled with gravity. Invent Math, 1997, 130: 455–516
Siebert B. Symplectic Gromov-Witten invariants. In: New Trends in Algebraic Geometry. London Mathematical Society Lecture Note Series, vol. 264. Cambridge: Cambridge University Press, 1999, 375–424
Tehrani M, Fukaya K. Gromov-Witten theory via Kuranishi structures. In: Virtual Fundamental Cycles in Symplectic Topology. Mathematical Surveys and Monographs, vol. 237. Providence: Amer Math Soc, 2019, 111–252
Tromba A. Teichmüller Theory in Riemannian Geometry. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser, 1992
Wolpert S. The hyperbolic metric and the geometry of the universal curve. J Differential Geom, 1990, 31: 417–472
Wolpert S. Cusps and the family hyperbolic metric. Duke Math J, 2007, 138: 423–443
Zhang W. Lectures on Chern-Weil Theory and Witten Deformations. Nankai Tracts in Mathematics, vol. 4. Hackensack: World Scientific, 2001
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11890660, 11821001, 11890663, 11871352 and 1196131001). The authors thank Yongbin Ruan, Huijun Fan, Jianxun Hu and Bohui Chen for many helpful discussions.
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In Memory of Professor Zhengguo Bai (1916–2015)
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Li, AM., Sheng, L. Virtual neighborhood technique for moduli spaces of holomorphic curves. Sci. China Math. 64, 1505–1562 (2021). https://doi.org/10.1007/s11425-020-1876-8
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DOI: https://doi.org/10.1007/s11425-020-1876-8