Genus-decreasing relation of Gromov-Witten invariants for surfaces under blow-up

Abstract

Using the degeneration formula, we study the change of Gromov-Witten invariants under blow-up for symplectic 4-manifolds and obtain a genus-decreasing relation of Gromov-Witten invariant of symplectic four manifold under blow-up.

Keywords

Gromov-Witten invariant / blow-up / degeneration formula

References

[1] Clemens C H. Degeneration of Kähler manifolds. Duke Math J, 1977, 44(2): 215–290

[2] Donaldson S K, Smith I. Lefschetz pencils and the canonical class for symplectic fourmanifolds. Topology, 2003, 42(4): 743–785

[3] Fintushel R, Stern R. Immersed spheres in 4-manifolds and the immersed Thom conjecture. Turkish J Math, 1995, 19: 145–157

[4] Fukaya K, Ono K. Arnold conjecture and Gromov-Witten invariant. Topology, 1999, 38(5): 933–1048

[5] Graber T, Pandharipande R. Localization of virtual classes. Invent Math, 1999, 135: 487–518

[6] He W Q, Hu J X, Ke H Z, Qi X X. Blow-up formulae of high genus Gromov-Witten invariants for threefolds. Math Z, 2018, 290(3-4): 857–872

[7] Hu J X. Gromov-Witten invariants of blow-ups along points and curves. Math Z, 2000, 233: 709–739

[8] Hu J X, Li T-J, Ruan Y B. Birational cobordism invariance of uniruled symplectic manifolds. Invent Math, 2008, 172: 231–275

[9] Ionel E, Parker T. The symplectic sum formula for Gromov-Witten invariants. Ann of Math, 2004, 159(3): 935–1025

[10] Kontsevich M, Mannin Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm Math Phys, 1994, 164(3): 525–562

[11] Li A-M, Ruan Y B. Symplectic surgery and Gromov-Witten invariants of Calabi 3-folds. Invent Math, 2001, 145(1): 151–218

[12] Li J. A degeneration formula of GW-invariants. J Differential Geom, 2002, 60: 199–293

[13] Li J, Tian G. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J Amer Math Soc, 1998, 11(1): 119–174

[14] Li J, Tian G. Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds. In: Stern R J, ed. Topics in Symplectic 4-Manifolds. First International Press Lecture Ser, Vol 1. Boston: International Press, 1998, 47–83

[15] Li T-J, Liu A. The equivalence between SW and GR in the case where b⁺=1. Int Math Res Not IMRN, 1999, 7: 335–345

[16] McDuff D. The symplectomorphism group of a blow up. Geom Dedicata, 2008, 132: 1–29

[17] McDuff D, Salamon D. J-holomorphic Curves and Symplectic Topology. Amer Math Soc Colloq Publ, Vol 52. Providence: Amer Math Soc, 2012

[18] McDuff D, Tehrani M, Fukaya K, Joyce D. Virtual Fundamental Cycles in Symplectic Topology. Math Surveys Monogr, Vol 237. Providence: Amer Math Soc, 2019

[19] Pandharipande R. Hodge integrals and degenerate contributions. Comm Math Phys, 1999, 208: 489–506

[20] Pardon J. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom Topol, 2016, 20(2): 779–1034

[21] Qi X X. A blowup formula of high-genus Gromov-Witten invariants of symplectic 4-manifolds. Adv Math (China), 2014, 43: 603–607 (in Chinese)

[22] Ruan Y B, Tian G. A mathematical theory of quantum cohomology. J Differential Geom, 1995, 42(2): 259–367

[23] Ruan Y B, Tian G. Higher genus symplectic invariants and sigma model coupled with gravity. Invent Math, 1997, 130: 455–516

[24] Taubes C. Seiberg-Witten and Gromov Invariants for Symplectic 4-Manifolds. First International Press Lecture Ser, Vol 2. Boston: International Press, 2000

RIGHTS & PERMISSIONS

2021 Higher Education Press