Gromov-Witten invariants for abelian and nonabelian quotients
Authors:
Aaron Bertram, Ionuţ Ciocan-Fontanine and Bumsig Kim
Journal:
J. Algebraic Geom. 17 (2008), 275-294
DOI:
https://doi.org/10.1090/S1056-3911-07-00456-0
Published electronically:
October 1, 2007
MathSciNet review:
2369087
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Abstract |
References |
Additional Information
Abstract: Conjectural formulas are given expressing the genus zero Gromov-Witten invariants of the quotient of a complex projective manifold $X$ by a reductive group $G$ (the nonabelian quotient) in terms of the invariants of the quotient by a maximal torus $T$ (the abelian quotient). The “$J$-function” version of the formulas is proved when $X$ is a (generalized) flag manifold.
References
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References
- K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495 (98e:14022)
- A. Bertram, Another way to enumerate rational curves with torus actions, Invent. Math. 142 (2000), no. 3, 487–512. MR 1804158 (2001m:14077)
- A. Bertram, I. Ciocan-Fontanine and B. Kim, Two proofs of a conjecture of Hori and Vafa, Duke Math. J. 126 (2005), 101–136. MR 2110629 (2006e:14077)
- M. Brion, The push-forward and Todd class of flag bundles, in “Parameter spaces (Warsaw, 1994)”, 45–50, Banach Center Publ., 36, Polish Acad. Sci., Warsaw, 1996. MR 1481478 (98h:14059)
- T. Coates and A. Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53. MR 2276766 (Review)
- G. Ellingsrud and S. A. Strømme, On the Chow ring of a geometric quotient, Ann. of Math. 130 (1989), 159-187. MR 1005610 (90h:14019)
- A mirror theorem for toric complete intersections, in Topological field theory, primitive forms and related topics (Kyoto, 1996), 141–175, Progr. Math. vol. 160, Birkhäuser, Boston, MA, 1998. MR 1653024 (2000a:14063)
- A. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), no. 4, 551–568, 645. MR 1901075 (2003j:53138)
- K. Hori and C. Vafa, Mirror symmetry, preprint (2000), hep-th/0002222.
- B. Kim, Quantum hyperplane section theorem for homogeneous spaces, Acta Math. 4 (1999), 71–99. MR 1719555 (2001i:14076)
- B. Kim, Quantum hyperplane section principle for concavex decomposable vector bundles, J. Korean Math. Soc. 37(2000), no. 3, 455–461. MR 1760373 (2002b:14070)
- J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants, Jour. Amer. Math. Soc. 11(1998), 119–174. MR 1467172 (99d:14011)
- Y. P. Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145(2001), 121–149. MR 1839288 (2002i:14049)
- C. H. Liu, K. Liu and S. T. Yau, $S^1$-fixed-points for hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram, preprint(2004), math.AG/0401367.
- S. Martin, Symplectic quotients by a nonabelian group and by its maximal torus, preprint (2000), math.SG/0001002.
Additional Information
Aaron Bertram
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
MR Author ID:
246391
Email:
bertram@math.utah.edu
Ionuţ Ciocan-Fontanine
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
MR Author ID:
365502
Email:
ciocan@math.umn.edu
Bumsig Kim
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, Korea
MR Author ID:
359696
Email:
bumsig@kias.re.kr
Received by editor(s):
February 1, 2006
Received by editor(s) in revised form:
April 6, 2006
Published electronically:
October 1, 2007
Additional Notes:
The first two authors were partially supported by NSF grants DMS-0200895 and DMS-0303614, respectively. The third author was supported by KOSEF R01-2004-000-10870-0.
