Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Sum formulas for local Gromov-Witten invariants of spin curves

Author: Junho Lee
Journal: Trans. Amer. Math. Soc. 365 (2013), 459-490
MSC (2010): Primary 53D45; Secondary 14N35
Published electronically: August 24, 2012
MathSciNet review: 2984064
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Holomorphic 2-forms on Kähler surfaces lead to ``local Gromov-Witten invariants'' of spin curves. This paper shows how to derive sum formulas for such local GW invariants from the sum formula for GW invariants of certain ruled surfaces. These sum formulas also verify the Maulik-Pandharipande formulas that were recently proved by Kiem and Li.

References [Enhancements On Off] (What's this?)

  • [ACV] D. Abramovich, A. Corti, and A. Vistoli, Twisted bundles and admissible covers, Commun. in Algebra. 31 (2003), 3547-3618. MR 2007376 (2005b:14049)
  • [AC] E. Arbarello and M. Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. No. 88 (1998), 97-127. MR 1733327 (2001h:14030)
  • [C] M. Cornalba, Moduli of curves and theta charateristics, Lectures on Riemann Surfaces, 560-589, World Scientific, Singapore 1989. MR 1082361 (91m:14037)
  • [FP] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173-199. MR 1728879 (2000m:14057)
  • [HM] J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998. MR 1631825 (99g:14031)
  • [IP1] E. Ionel and T.H. Parker, Relative Gromov-Witten invariants, Annals of Math. 157 (2003), 45-96. MR 1954264 (2004a:53112)
  • [IP2] E. Ionel and T.H. Parker, The symplectic sum formula for Gromov-Witten invariants, Annals of Math. 159 (2004), 935-1025. MR 2113018 (2006b:53110)
  • [KL1] Y-H. Kiem and J. Li, Gromov-Witten invariants of varieties with holomorphic $ 2$-forms, preprint, math.AG/07072986.
  • [KL2] Y-H. Kiem and J. Li, Low degree GW invariants of spin surfaces, Pure Appl. Math. Q. 7 (2011), no. 4, 1449-1476. MR 2918169
  • [KL3] Y-H. Kiem and J. Li, Low degree GW invariants of surfaces II, Science China Math. 54 (2011), no. 8, 1679-1706. MR 2824966
  • [KM] M. Kontsevich and Y.I. Manin, Relations between the correlators of the topological sigma model coupled to gravity, Commun. Math. Phys. 196 (1998), 385-398. MR 1645019 (99k:14040)
  • [L] J. Lee, Family Gromov-Witten invariants for Kähler surfaces, Duke Math. J. 123 (2004), no 1, 209-233. MR 2060027 (2005d:53141)
  • [Lo] E. Looijenga, Smooth Deligne-Mumford compactifications by means of Prym level structures, J. Algebraic Geom. 3 (1994), no. 2, 283-29. MR 1257324 (94m:14029)
  • [LP1] J. Lee and T.H. Parker, A Structure Theorem for the Gromov-Witten Invariants of Kähler Surfaces, J. Diff. Geom. 77 (2007), 483-513. MR 2362322 (2010b:53159)
  • [LP2] J. Lee and T.H. Parker, An obstruction bundle relating Gromov-Witten invariants of curves and Kähler surfaces, preprint, arXiv:0909.3610.
  • [LT] 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, First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 1998. MR 1635695 (2000d:53137)
  • [MP] D. Maulik and R. Pandharipande, New calculations in Gromov-Witten theory, Pure Appl. Math. Q. 4 (2008), no. 2, part 1, 469-500. MR 2400883 (2009d:14073)
  • [RT1] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259-367. MR 1366548 (96m:58033)
  • [RT2] Y. Ruan and G. Tian, Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math. 130 (1997), no. 3, 455-516. MR 1483992 (99d:58030)
  • [T] L. Tu, Hodge theory and the local Torelli problem, Mem. Amer. Math. Soc. 43 (1983), no. 279. MR 699239 (84k:14008)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53D45, 14N35

Retrieve articles in all journals with MSC (2010): 53D45, 14N35

Additional Information

Junho Lee
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Received by editor(s): May 26, 2009
Received by editor(s) in revised form: January 7, 2010, September 28, 2010, and May 9, 2011
Published electronically: August 24, 2012
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society