Uniruled Symplectic Divisors

Tian-Jun Li , Yongbin Ruan

Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (2) : 163 -212.

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Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (2) : 163 -212. DOI: 10.1007/s40304-013-0010-x
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Uniruled Symplectic Divisors

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Abstract

In this article, we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry. In particular, we generalize Maulik–Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence. Then, we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.

Keywords

Birational symplectic geometry / Gromov–Witten invariants / Symplectic divisor / Uniruled invariant

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Tian-Jun Li, Yongbin Ruan. Uniruled Symplectic Divisors. Communications in Mathematics and Statistics, 2013, 1(2): 163-212 DOI:10.1007/s40304-013-0010-x

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References

[1]

Chen, B., Li, A.: Symplectic relative virtual localization, in preparation

[2]

Coates T., Givental A. Quantum Riemann—Roch, Lefschetz and Serre. Ann. Math.. 2007, 165 15-53

[3]

Faber C., Pandharipande R. Hodte integrals and Gromov–Witten theory. Invent. Math.. 2000, 139 173-199

[4]

Gathmann, A.: Gromov–Witten invariants of hypersurfaces. Habilitation thesis, University of Kaiserslautern

[5]

Gompf R. A new construction of symplectic manifolds. Ann. Math.. 1995, 142 527-595

[6]

Gompf R. Locally holomorphic maps yield symplectic structures. Commun. Anal. Geom.. 2005, 13 3 511-525

[7]

Gromov M. Pseudo-holomorphic curves in symplectic manifolds. Invent. Math.. 1985, 82 307-347

[8]

Graber T., Vakil R. Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J.. 2005, 30 1 1-37

[9]

Hu J., Li T.-J., Ruan Y. Birational cobordism invariance of symplectic uniruled manifolds. Invent. Math.. 2008, 172 231-275

[10]

Ionel E., Parker T. The symplectic sum formula for Gromov–Witten invariants. Ann. Math. (2). 2004, 159 3 935-1025

[11]

Lerman E. Symplectic cuts. Math. Res. Lett.. 1995, 2 247-258

[12]

Lee Y.P. Quantum Lefschetz hyperplane theorem. Invent. Math.. 2001, 145 121-149

[13]

Lee Y.P., PandharipandeR R. A reconstruction theorem in quantum cohomology and quantum K-theory. Am. J. Math.. 2004, 126 6 1367-1379

[14]

Li B.H., Li T.J. Symplectic genus, minimal genus and diffeomorphisms. Asian J. Math.. 2002, 6 1 123-144

[15]

Li J. Relative Gromov–Witten invariants and a degeneration formula of Gromov–Witten invariants. J. Differ. Geom.. 2002, 60 199-293

[16]

Li T.J., Liu A. Symplectic structures on ruled surfaces and a generalized adjunction inequality. Math. Res. Lett.. 1995, 2 453-471

[17]

Li T.J., Liu A. Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with b +=1. J. Differ. Geom.. 2001, 58 2 331-370

[18]

Lalonde F., McDuff D. The classification of ruled symplectic 4-manifolds. Math. Res. Lett.. 1996, 3 769-778

[19]

Lalonde F., McDuff D. Symplectic structures on fiber bundles. Topology. 2003, 42 309-347

[20]

Li A., Ruan Y. Symplectic surgery and Gromov–Witten invariants of Calabi–Yau 3-folds. Invent. Math.. 2001, 145 151-218

[21]

Li, T.-J., Ruan, Y., Zhang, W.: Symplectic divisoral contractions in dimension 6, in preparation

[22]

Lu G. Finiteness of the Hofer–Zehnder capacity of neighborhoods of symplectic submanifolds. Int. Math. Res. Not.. 2006, 2006 1-33

[23]

Lu G. Gromov–Witten invariants pseudo symplectic capacities. Isr. J. Math.. 2006, 156 1-63

[24]

Lu G. Symplectic capacities of toric manifolds and related results. Nagoya Math. J.. 2006, 181 149-184

[25]

Lian B., Liu K., Yau S.-T. Mirror principle I. Asian J. Math.. 1997, 1997 729-763

[26]

McDuff D. The structure of rational and ruled symplectic 4-manifold. J. Am. Math. Soc.. 1990, 1 3 679-710

[27]

McDuff D. Symplectic manifolds with contact type boundaries. Invent. Math.. 1991, 103 651-671

[28]

McDuff, D.: Hamiltonian S 1 manifolds are uniruled. Preprint (2007)

[29]

McDuff D. Singularities and positivity of intersections of J-holomorphic curves. Holomorphic Curves in Symplectic Geometry. 1994 Basel: Birkhäuser. 191-215

[30]

McDuff D., Salamon D. J-Holomorphic Curves and Symplectic Topology. 2004 Providence: AMS

[31]

Maulik, D., Pandharipande, R.: A topological view of Gromov–Witten theory. Topology, to appear. math.AG/0412503

[32]

Zinger, A.: The reduced genus-one Gromov–Witten invariants of Calabi–Yau hypersurfaces. arXiv:0705.2397

[33]

Zinger A. A comparison theorem for Gromov–Witten invariants in the symplectic category. Adv. Math.. 2011, 228 1 535-574

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