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Acta Mathematica

, Volume 184, Issue 1, pp 1–39 | Cite as

Mirror symmetry and toric degenerations of partial flag manifolds

  • Victor V. Batyrev
  • Ionuţ Ciocan-Fontanine
  • Bumsig Kim
  • Duco van Straten
Article

Keywords

Manifold Mirror Symmetry Flag Manifold Toric Degeneration Partial Flag Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Astashkevich, A. &Sadov, V., Quantum cohomology of partial flag manifoldsF n1,...,n k.Comm. Math. Phys., 170 (1995), 503–528.CrossRefMathSciNetGoogle Scholar
  2. [2]
    Batyrev, V. V., On classifications of smooth projective toric varieties.Tôhoku Math. J., 43 (1991), 569–585.MATHMathSciNetGoogle Scholar
  3. [3]
    —, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.J. Algebraic Geom., 3 (1994), 493–535.MATHMathSciNetGoogle Scholar
  4. [4]
    Batyrev, V. V. Toric degenerations of Fano varieties and constructing mirror manifolds.alg-geom/9712034.Google Scholar
  5. [5]
    Batyrev, V. V. &Borisov, L. A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, inMirror Symmetry, Vol. II, pp. 71–86. AMS/IP Stud. Adv. Math., 1. Amer. Math. Soc., Providence, RI, 1997.Google Scholar
  6. [6]
    Batyrev, V. V., Ciocan-Fontanine, I., Kim, B. &Straten, D. van, Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians.Nuclear Phys. B, 514 (1998), 640–666, (alg-geom/9710022).CrossRefMathSciNetGoogle Scholar
  7. [7]
    Batyrev, V. V. &Straten, D. van, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties.Comm. Math. Phys., 168 (1995), 493–533 (alg-geom/9307010).CrossRefMathSciNetGoogle Scholar
  8. [8]
    Behrend, K., Gromov-Witten invariants in algebraic geometry.Invent. Math., 127 (1997), 601–617.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    Borisov, L. A., Towards mirror symmetry of Calabi-Yau complete intersections in Gorenstein toric Fano varieties.alg-geom/9310001.Google Scholar
  10. [10]
    Candelas, P., Ossa, X. de la, Green, P. &Parkes, L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory.Nuclear Phys. B, 359, (1991), 21–74.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Ciocan-Fontanine, I., On quantum cohomology rings of partial flag varieties.Duke Math. J., 98 (1999), 485–524.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    Eguchi, T., Hori, K. &Xiong, C.-S., Gravitational quantum cohomology.Internat. J. Modern Phys. A, 12 (1997), 1743–1782 (hep-th/9605225).CrossRefMathSciNetGoogle Scholar
  13. [13]
    Ehresmann, C., Sur la topologie des certaines espaces homogènes.Ann. of Math., 35 (1934), 396–443.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    Fulton, W.,Introduction to Toric Varieties. Ann. of Math. Stud., 131. Princeton Univ. Press, Princeton, NJ, 1993.Google Scholar
  15. [15]
    Givental, A., Equivariant Gromov-Witten invariants.Iternat. Math. Res. Notices, 1996, 613–663 (alg-geom/9603021).Google Scholar
  16. [16]
    —, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, inTopics in Singularity Theory: V. I. Arnold's 60th Anniversary Collection, pp. 103–115. Amer. Math. Soc. Transl. Ser. 2, 180. Amer. Math. Soc., Providence, RI, 1997 (alg-geom/9612001).Google Scholar
  17. [17]
    —, A mirror theorem for toric complete intersections, inTopological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), pp. 141–175. Progr. Math., 160. Birkhäuser Boston, Boston, MA, 1998 (alg-geom/9701016).Google Scholar
  18. [18]
    Gonciulea, N. &Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties.Transform. Groups, 1 (1996), 215–248.MathSciNetGoogle Scholar
  19. [19]
    —, Schubert varieties, toric varieties, and ladder determinantal varieties.Ann. Inst. Fourier (Grenoble), 47 (1997), 1013–1064.MathSciNetGoogle Scholar
  20. [20]
    Kim, B., Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings.Internat. Math. Res. Notices 1950, 1–16.Google Scholar
  21. [21]
    Kim, B., On equivariant quantum cohomology.Internat. Math. Res. Notices 1996, 841–851.Google Scholar
  22. [22]
    — Quantum cohomology of flag manifoldsG/B and quantum Toda lattices.Ann. of Math., 149 (1999), 129–148.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    — Quantum hyperplane section theorem for homogeneous spaces,Acta Math., 183 (1999), 71–99 (alg-geom/9712008).MATHMathSciNetGoogle Scholar
  24. [24]
    Lakshmibai, V., Degenerations of flag varieties to toric varieties.C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1229–1234.MATHMathSciNetGoogle Scholar
  25. [25]
    Li, J. &Tian, G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties.J. Amer. Math. Soc., 11 (1998), 119–174.CrossRefMathSciNetGoogle Scholar
  26. [26]
    Reid, M., Decomposition of toric morphisms, inArithmetic and Geometry, Vol. II, pp. 395–418. Progr. Math., 36. Birkhäuser Boston, Boston, MA, 1983.Google Scholar
  27. [27]
    Schechtman, V., On hypergeometric functions connected with quantum cohomology of flag spaces.q-alg/9712049.Google Scholar
  28. [28]
    Sturmfels, B.,Gröbner Bases and Convex Polytopes. Univ. Lecture Ser., 8. Amer. Math. Soc., Providence, Ri, 1996.Google Scholar

Copyright information

© Institut Mittag-Leffler 2000

Authors and Affiliations

  • Victor V. Batyrev
    • 1
  • Ionuţ Ciocan-Fontanine
    • 2
  • Bumsig Kim
    • 3
  • Duco van Straten
    • 4
  1. 1.Mathematisches InstitutEberhard-Karls-Universität TübingenTübingenGermany
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonU.S.A.
  3. 3.Department of MathematicsPohang University of Science and Technology (Postech)PohangThe Republic of Korea
  4. 4.FB 17, MathematikJohannes Gutenberg-Universität MainzMainzGermany

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