Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 067, 25 pages      arXiv:1301.7632

Minuscule Schubert Varieties and Mirror Symmetry

Makoto Miura
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea

Received August 23, 2016, in final form August 16, 2017; Published online August 23, 2017

We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi-Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety ${\boldsymbol{\Sigma}}$ of the Cayley plane ${\mathbb{OP}}^2$. We calculate topological invariants and BPS numbers of this Calabi-Yau 3-fold and conjecture that it has a non-trivial Fourier-Mukai partner.

Key words: Calabi-Yau; mirror symmetry; minuscule; Schubert variety; toric degeneration.

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  1. Almkvist G., van Enckevort C., van Straten D., Zudilin W., Tables of Calabi-Yau equations, math.AG/0507430.
  2. Almkvist G., Zudilin W., Differential equations, mirror maps and zeta values, in Mirror Symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Amer. Math. Soc., Providence, RI, 2006, 481-515, math.NT/0402386.
  3. Aspinwall P.S., Greene B.R., Morrison D.R., The monomial-divisor mirror map, Int. Math. Res. Not. 1993 (1993), 319-337, alg-geom/9309007.
  4. Baston R.J., Eastwood M.G., The Penrose transform. Its interaction with representation theory, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989.
  5. Batyrev V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493-535, alg-geom/9310003.
  6. Batyrev V.V., Toric degenerations of Fano varieties and constructing mirror manifolds, in The Fano Conference, University Torino, Turin, 2004, 109-122, alg-geom/9712034.
  7. Batyrev V.V., Borisov L.A., Mirror duality and string-theoretic Hodge numbers, Invent. Math. 126 (1996), 183-203, alg-geom/9509009.
  8. Batyrev V.V., Borisov L.A., On Calabi-Yau complete intersections in toric varieties, in Higher-Dimensional Complex Varieties (Trento, 1994), de Gruyter, Berlin, 1996, 39-65, alg-geom/9412017.
  9. Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians, Nuclear Phys. B 514 (1998), 640-666, alg-geom/9710022.
  10. Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., Mirror symmetry and toric degenerations of partial flag manifolds, Acta Math. 184 (2000), 1-39, math.AG/9803108.
  11. Batyrev V.V., Kreuzer M., Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions, Adv. Theor. Math. Phys. 14 (2010), 879-898, arXiv:0802.3376.
  12. Bershadsky M., Cecotti S., Ooguri H., Vafa C., Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993), 279-304, hep-th/9302103.
  13. Bershadsky M., Cecotti S., Ooguri H., Vafa C., Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), 311-427, hep-th/9309140.
  14. Bertram A., Quantum Schubert calculus, Adv. Math. 128 (1997), 289-305, alg-geom/9410024.
  15. Birkhoff G., Lattice theory, American Mathematical Society Colloquium Publications, Vol. 25, 3rd ed., Amer. Math. Soc., Providence, R.I., 1979.
  16. Bondal A., Galkin S.S., Degeneration of minuscule varieties, quiver toric varieties and mirror symmetry, Seminar at IPMU, 2010.
  17. Borisov L.A., Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.
  18. Brown J., Lakshmibai V., Singular loci of Bruhat-Hibi toric varieties, J. Algebra 319 (2008), 4759-4779, arXiv:0707.2392.
  19. Brown J., Lakshmibai V., Singular loci of Grassmann-Hibi toric varieties, Michigan Math. J. 59 (2010), 243-267, math.AG/0612289.
  20. Candelas P., de la Ossa X.C., Green P.S., Parkes L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.
  21. Chaput P.E., Manivel L., Perrin N., Quantum cohomology of minuscule homogeneous spaces, Transform. Groups 13 (2008), 47-89, math.AG/0607492.
  22. Clemens C.H., Double solids, Adv. Math. 47 (1983), 107-230.
  23. Fulton W., Woodward C., On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), 641-661, math.AG/0112183.
  24. Galkin S.S., An explicit construction of Miura's varieties, Talk presented at Tokyo University, Graduate School for Mathematical Sciences, Komaba Campus, February 20, 2014.
  25. Galkin S.S., Apéry constants of homogeneous varieties, arXiv:1604.04652.
  26. Galkin S.S., Kuznetsov A., Movshev M., An explicit construction of Miura's varieties, in prepration.
  27. Gerhardus A., Jockers H., Dual pairs of gauged linear sigma models and derived equivalences of Calabi-Yau threefolds, J. Geom. Phys. 114 (2017), 223-259, arXiv:1505.00099.
  28. Givental A.B., Equivariant Gromov-Witten invariants, Int. Math. Res. Not. 1996 (1996), 613-663, alg-geom/9603021.
  29. Gonciulea N., Lakshmibai V., Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups 1 (1996), 215-248.
  30. Gopakumar R., Vafa C., M-theory and topological strings-II, hep-th/9812127.
  31. Greene B.R., Plesser M.R., Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), 15-37.
  32. Guest M.A., From quantum cohomology to integrable systems, Oxford Graduate Texts in Mathematics, Vol. 15, Oxford University Press, Oxford, 2008.
  33. Hibi T., Distributive lattices, affine semigroup rings and algebras with straightening laws, in Commutative Algebra and Combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., Vol. 11, North-Holland, Amsterdam, 1987, 93-109.
  34. Hibi T., Higashitani A., Smooth Fano polytopes arising from finite partially ordered sets, Discrete Comput. Geom. 45 (2011), 449-461, arXiv:0908.3404.
  35. Hibi T., Watanabe K., Study of three-dimensional algebras with straightening laws which are Gorenstein domains. I, Hiroshima Math. J. 15 (1985), 27-54.
  36. Hori K., Knapp J., A pair of Calabi-Yau manifolds from a two parameter non-Abelian gauged linear sigma model, arXiv:1612.06214.
  37. Hosono S., Konishi Y., Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau 3-folds, Adv. Theor. Math. Phys. 13 (2009), 463-495, arXiv:0704.2928.
  38. Hosono S., Lian B.H., Oguiso K., Yau S.-T., Autoequivalences of derived category of a $K3$ surface and monodromy transformations, J. Algebraic Geom. 13 (2004), 513-545, math.AG/0201047.
  39. Hosono S., Takagi H., Determinantal quintics and mirror symmetry of Reye congruences, Comm. Math. Phys. 329 (2014), 1171-1218, arXiv:1208.1813.
  40. Hosono S., Takagi H., Mirror symmetry and projective geometry of Reye congruences I, J. Algebraic Geom. 23 (2014), 279-312, arXiv:1101.2746.
  41. Inoue D., Ito A., Miura M., Complete intersection Calabi-Yau manifolds with respect to homogeneous vector bundles on Grassmannians, arXiv:1607.07821.
  42. Inoue D., Ito A., Miura M., $I$-functions of Calabi-Yau 3-folds in Grassmannians, arXiv:1607.08137.
  43. Jones B.F., Singular Chern classes of Schubert varieties via small resolution, Int. Math. Res. Not. 2010 (2010), 1371-1416, arXiv:0804.0202.
  44. Kawamata Y., Deformations of canonical singularities, J. Amer. Math. Soc. 12 (1999), 85-92, alg-geom/9712018.
  45. Kim B., Quantum hyperplane section theorem for homogeneous spaces, Acta Math. 183 (1999), 71-99, alg-geom/9712008.
  46. Kresch A., Tamvakis H., Quantum cohomology of orthogonal Grassmannians, Compos. Math. 140 (2004), 482-500, math.AG/0306338.
  47. Lakshmibai V., Mukherjee H., Singular loci of Hibi toric varieties, J. Ramanujan Math. Soc. 26 (2011), 1-29, arXiv:0706.2686.
  48. Lakshmibai V., Musili C., Seshadri C.S., Geometry of $G/P$. III. Standard monomial theory for a quasi-minuscule $P$, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), 93-177.
  49. Lakshmibai V., Weyman J., Multiplicities of points on a Schubert variety in a minuscule $G/P$, Adv. Math. 84 (1990), 179-208.
  50. Lian B.H., Liu K., Yau S.-T., Mirror principle. I, Asian J. Math. 1 (1997), 729-763, alg-geom/9712011.
  51. Morrison D.R., Through the looking glass, in Mirror Symmetry, III (Montreal, PQ, 1995), AMS/IP Stud. Adv. Math., Vol. 10, Amer. Math. Soc., Providence, RI, 1999, 263-277, alg-geom/9705028.
  52. Mukai S., Duality of polarized $K3$ surfaces, in New Trends in Algebraic Geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge University Press, Cambridge, 1999, 311-326.
  53. Nakayama N., Invariance of the plurigenera of algebraic varieties, RIMS preprint 1191, 1998.
  54. Namikawa Y., On deformations of Calabi-Yau $3$-folds with terminal singularities, Topology 33 (1994), 429-446.
  55. Perrin N., Small resolutions of minuscule Schubert varieties, Compos. Math. 143 (2007), 1255-1312, math.AG/0601117.
  56. Perrin N., The Gorenstein locus of minuscule Schubert varieties, Adv. Math. 220 (2009), 505-522, arXiv:0704.0895.
  57. Proctor R.A., Interactions between combinatorics, Lie theory and algebraic geometry via the Bruhat orders, Ph.D. Thesis, Massachusetts Institute of Technology, 1981.
  58. Ravindra G.V., Srinivas V., The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom. 15 (2006), 563-590, math.AG/0511134.
  59. Rødland E.A., The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), 135-149, math.AG/9801092.
  60. Stanley R.P., Two poset polytopes, Discrete Comput. Geom. 1 (1986), 9-23.
  61. Ueda K., Yoshida Y., Equivariant A-twisted GLSM and Gromov-Witten invariants of CY 3-folds in Grassmannians, arXiv:1602.02487.
  62. van Enckevort C., van Straten D., Electronic database of Calabi-Yau equations, available at
  63. van Enckevort C., van Straten D., Monodromy calculations of fourth order equations of Calabi-Yau type, in Mirror symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Amer. Math. Soc., Providence, RI, 2006, 539-559.
  64. Wagner D.G., Singularities of toric varieties associated with finite distributive lattices, J. Algebraic Combin. 5 (1996), 149-165.
  65. Zudilin V.V., Binomial sums associated with rational approximations to $\zeta(4)$, Math. Notes 75 (2004), 594-597, math.CA/0311196.

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