Sum formulas for local GromovWitten invariants of spin curves
Author:
Junho Lee
Journal:
Trans. Amer. Math. Soc. 365 (2013), 459490
MSC (2010):
Primary 53D45; Secondary 14N35
Published electronically:
August 24, 2012
MathSciNet review:
2984064
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Abstract: Holomorphic 2forms on Kähler surfaces lead to ``local GromovWitten 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 MaulikPandharipande formulas that were recently proved by Kiem and Li.
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 D. Abramovich, A. Corti, and A. Vistoli, Twisted bundles and admissible covers, Commun. in Algebra. 31 (2003), 35473618. 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), 97127. MR 1733327 (2001h:14030)
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 M. Cornalba, Moduli of curves and theta charateristics, Lectures on Riemann Surfaces, 560589, World Scientific, Singapore 1989. MR 1082361 (91m:14037)
 [FP]
 C. Faber and R. Pandharipande, Hodge integrals and GromovWitten theory, Invent. Math. 139 (2000), no. 1, 173199. MR 1728879 (2000m:14057)
 [HM]
 J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, 187. SpringerVerlag, New York, 1998. MR 1631825 (99g:14031)
 [IP1]
 E. Ionel and T.H. Parker, Relative GromovWitten invariants, Annals of Math. 157 (2003), 4596. MR 1954264 (2004a:53112)
 [IP2]
 E. Ionel and T.H. Parker, The symplectic sum formula for GromovWitten invariants, Annals of Math. 159 (2004), 9351025. MR 2113018 (2006b:53110)
 [KL1]
 YH. Kiem and J. Li, GromovWitten invariants of varieties with holomorphic forms, preprint, math.AG/07072986.
 [KL2]
 YH. Kiem and J. Li, Low degree GW invariants of spin surfaces, Pure Appl. Math. Q. 7 (2011), no. 4, 14491476. MR 2918169
 [KL3]
 YH. Kiem and J. Li, Low degree GW invariants of surfaces II, Science China Math. 54 (2011), no. 8, 16791706. 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), 385398. MR 1645019 (99k:14040)
 [L]
 J. Lee, Family GromovWitten invariants for Kähler surfaces, Duke Math. J. 123 (2004), no 1, 209233. MR 2060027 (2005d:53141)
 [Lo]
 E. Looijenga, Smooth DeligneMumford compactifications by means of Prym level structures, J. Algebraic Geom. 3 (1994), no. 2, 28329. MR 1257324 (94m:14029)
 [LP1]
 J. Lee and T.H. Parker, A Structure Theorem for the GromovWitten Invariants of Kähler Surfaces, J. Diff. Geom. 77 (2007), 483513. MR 2362322 (2010b:53159)
 [LP2]
 J. Lee and T.H. Parker, An obstruction bundle relating GromovWitten invariants of curves and Kähler surfaces, preprint, arXiv:0909.3610.
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 J. Li and G. Tian, Virtual moduli cycles and GromovWitten invariants of general symplectic manifolds, Topics in symplectic manifolds (Irvine, CA, 1996), 4783, First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 1998. MR 1635695 (2000d:53137)
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 D. Maulik and R. Pandharipande, New calculations in GromovWitten theory, Pure Appl. Math. Q. 4 (2008), no. 2, part 1, 469500. MR 2400883 (2009d:14073)
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 Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259367. 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, 455516. MR 1483992 (99d:58030)
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 L. Tu, Hodge theory and the local Torelli problem, Mem. Amer. Math. Soc. 43 (1983), no. 279. MR 699239 (84k:14008)
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Additional Information
Junho Lee
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email:
junlee@mail.ucf.edu
DOI:
http://dx.doi.org/10.1090/S000299472012056352
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
