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Transactions of the American Mathematical Society

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On certain vanishing identities for Gromov-Witten invariants


Author: Xiaobo Liu
Journal: Trans. Amer. Math. Soc. 363 (2011), 2939-2953
MSC (2010): Primary 53D45; Secondary 14N35
DOI: https://doi.org/10.1090/S0002-9947-2011-05091-9
Published electronically: January 27, 2011
MathSciNet review: 2775793
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Abstract: In this paper we study certain vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove their conjectures when the summation range is big compared to the genus. In such cases, we actually obtained vanishing identities which are stronger than their conjectures. We also prove these conjectures in low genus cases.


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  • 1. K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45-88. MR 1437495 (98e:14022)
  • 2. T. Eguchi and C. Xiong, Quantum cohomology at higher genus: Topological recursion relations and Virasoro conditions, Adv. Theor. Math. Phys. 2 (1998), no. 1, 219-229. MR 1635867 (99i:14033)
  • 3. C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173-199. MR 1728879 (2000m:14057)
  • 4. E. Getzler, Topological recursion relations in genus $ 2$, Integrable systems and algebraic geometry (Kobe/kyoto, 1997) 73-106. MR 1672112 (2000b:14028)
  • 5. T. Kimura and X. Liu, A genus-$ 3$ topological recursion relation, Comm. Math. Phys. 262 (2006), no. 3, 645-661. MR 2202306 (2006i:14060)
  • 6. J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47-83. MR 1635695 (2000d:53137)
  • 7. K. Liu and H. Xu, A proof of the Faber intersection number conjecture, J. Differential Geom. 83 (2009), 313-335. MR 2577471
  • 8. X. Liu, Quantum product on the big phase space and Virasoro conjecture, Advances in Mathematics, 169 (2002), 313-375. MR 1926225 (2003j:14075)
  • 9. X. Liu, Relations among universal equations for Gromov-Witten invariants, ``Frobenius Manifolds, Quantum Cohomology and Singularities'', C. Hertling and M. Marcolli (Eds.), Aspects of Mathematics, A publication of the Max-Planck-Institute for Mathematics, Bonn, (2004), pp. 169 - 180. MR 2115770 (2005m:14106)
  • 10. X. Liu, Quantum product, topological recursion relations, and the Virasoro conjecture, ``Surveys on Geometry and Integrable Systems'', Advanced Studies in Pure Mathematics, vol. 51, (2008), pp. 235-257. MR 2509795 (2010b:53160)
  • 11. X. Liu and R. Pandharipande, New topological recursion relations, preprint, arXiv:0805.4829. To appear in J. Alg. Geom.
  • 12. X. Liu and G. Tian, Virasoro constraints for quantum cohomology, J. Diff. Geom. 50 (1998), 537 - 591. MR 1690740 (2000f:14090)
  • 13. E. Witten, Two dimensional gravity and intersection theory on Moduli space, Surveys in Diff. Geom., 1 (1991), 243-310. MR 1144529 (93e:32028)

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Additional Information

Xiaobo Liu
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: xliu3@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05091-9
Received by editor(s): January 8, 2009
Published electronically: January 27, 2011
Additional Notes: This research was partially supported by NSF grant DMS-0505835
Article copyright: © Copyright 2011 American Mathematical Society

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