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

SIGMA 14 (2018), 039, 37 pages      arXiv:1707.08728
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds

Shinobu Hosono and Hiromichi Takagi
Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan

Received September 11, 2017, in final form April 23, 2018; Published online May 02, 2018

We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.

Key words: Calabi-Yau manifolds; mirror symmetry; birational geometry; Hodge theory.

pdf (776 kb)   tex (138 kb)


  1. Alim M., Scheidegger E., Topological strings on elliptic fibrations, Commun. Number Theory Phys. 8 (2014), 729-800, arXiv:1205.1784.
  2. Aspinwall P.S., Greene B.R., Morrison D.R., Multiple mirror manifolds and topology change in string theory, Phys. Lett. B 303 (1993), 249-259, hep-th/9301043.
  3. Batyrev V., Nill B., Combinatorial aspects of mirror symmetry, in Integer Points in Polyhedra - Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, Contemp. Math., Vol. 452, Amer. Math. Soc., Providence, RI, 2008, 35-66, math.CO/0703456.
  4. 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.
  5. 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.
  6. Borisov L., Căldăraru A., The Pfaffian-Grassmannian derived equivalence, J. Algebraic Geom. 18 (2009), 201-222, math.AG/0608404.
  7. Borisov L.A., Li Z., On Clifford double mirrors of toric complete intersections, Adv. Math. 328 (2018), 300-355, arXiv:1601.00809.
  8. Bridgeland T., Stability conditions on triangulated categories, Ann. of Math. 166 (2007), 317-345, math.AG/0212237.
  9. Candelas P., de la Ossa X., Font A., Katz S., Morrison D.R., Mirror symmetry for two-parameter models. I, Nuclear Phys. B 416 (1994), 481-538, hep-th/9308083.
  10. Dolgachev I.V., Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), 2599-2630, alg-geom/9502005.
  11. Fan Y.-W., Hong H., Lau S.-C., Yau S.-T., Mirror of Atiyah flop in symplectic geometry and stability conditions, arXiv:1706.02942.
  12. Favero D., Kelly T.L., Proof of a conjecture of Batyrev and Nill, Amer. J. Math. 139 (2017), 1493-1520, arXiv:1412.1354.
  13. Festi D., Garbagnati A., van Geemen B., van Luijk R., The Cayley-Oguiso automorphism of positive entropy on a K3 surface, J. Mod. Dyn. 7 (2013), 75-97.
  14. Fryers M.J., The movable fan of the Horrocks-Mumford quintic, math.AG/0102055.
  15. Gel'fand I.M., Zelevinskii A.V., Kapranov M.M., Hypergeometric functions and toric varieties, Funct. Anal. Appl. 23 (1989), 94-106.
  16. Griffiths P., Hodge theory and geometry, Bull. London Math. Soc. 36 (2004), 721-757.
  17. Gross M., Siebert B., Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), 169-338, math.AG/0309070.
  18. Gross M., Siebert B., Mirror symmetry via logarithmic degeneration data. II, J. Algebraic Geom. 19 (2010), 679-780, arXiv:0709.2290.
  19. Hosono S., Local mirror symmetry and type IIA monodromy of Calabi-Yau manifolds, Adv. Theor. Math. Phys. 4 (2000), 335-376, hep-th/0007071.
  20. Hosono S., Central charges, symplectic forms, and hypergeometric series in local mirror symmetry, in Mirror Symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Editors S.-T. Yau, N. Yui, J. Lewis, Amer. Math. Soc., Providence, RI, 2006, 405-439, hep-th/0404043.
  21. Hosono S., Moduli spaces of Calabi-Yau complete intersections, Nuclear Phys. B 898 (2015), 661-666.
  22. Hosono S., Klemm A., Theisen S., Yau S.-T., Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nuclear Phys. B 433 (1995), 501-552, hep-th/9406055.
  23. 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.
  24. Hosono S., Lian B.H., Oguiso K., Yau S.-T., Fourier-Mukai number of a K3 surface, in Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes, Vol. 38, Amer. Math. Soc., Providence, RI, 2004, 177-192, math.AG/0202014.
  25. Hosono S., Lian B.H., Yau S.-T., GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Comm. Math. Phys. 182 (1996), 535-577, alg-geom/9511001.
  26. Hosono S., Takagi H., Determinantal quintics and mirror symmetry of Reye congruences, Comm. Math. Phys. 329 (2014), 1171-1218, arXiv:1208.1813.
  27. Hosono S., Takagi H., Mirror symmetry and projective geometry of Reye congruences I, J. Algebraic Geom. 23 (2014), 279-312, arXiv:1101.2746.
  28. Hosono S., Takagi H., Double quintic symmetroids, Reye congruences, and their derived equivalence, J. Differential Geom. 104 (2016), 443-497, arXiv:1302.5883.
  29. Iritani H., Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), 2909-2958, arXiv:1101.4512.
  30. Iritani H., Xiao J., Extremal transition and quantum cohomology: examples of toric degeneration, Kyoto J. Math. 56 (2016), 873-905, arXiv:1504.05013.
  31. Kawamata Y., Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. 127 (1988), 93-163.
  32. Kawamata Y., On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), 665-687, alg-geom/9701006.
  33. Kollár J., Flops, Nagoya Math. J. 113 (1989), 15-36.
  34. Kontsevich M., Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 120-139, alg-geom/9411018.
  35. Kuznetsov A., Homological projective duality for Grassmannians of lines, math.AG/0610957.
  36. Kuznetsov A., Homological projective duality, Publ. Math. Inst. Hautes Études Sci. (2007), 157-220, math.AG/0507292.
  37. Lee Y.-P., Lin H.-W., Wang C.-L., Quantum cohomology under birational maps and transitions, arXiv:1705.04799.
  38. Morrison D.R., Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), 243-271, alg-geom/9304007.
  39. Morrison D.R., Beyond the Kähler cone, in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan University, Ramat Gan, 1996, 361-376, alg-geom/9407007.
  40. Mukai S., Symplectic structure of the moduli space of sheaves on an abelian or $K3$\ surface, Invent. Math. 77 (1984), 101-116.
  41. Oguiso K., Automorphism groups of Calabi-Yau manifolds of Picard number 2, J. Algebraic Geom. 23 (2014), 775-795, arXiv:1206.1649.
  42. Oguiso K., Free automorphisms of positive entropy on smooth Kähler surfaces, in Algebraic Geometry in East Asia - Taipei 2011, Adv. Stud. Pure Math., Vol. 65, Math. Soc. Japan, Tokyo, 2015, 187-199, arXiv:1202.2637.
  43. 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.
  44. Ruddat H., Siebert B., Canonical coordinates in toric degenerations, arXiv:1409.4750.
  45. Strominger A., Yau S.-T., Zaslow E., Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), 243-259, hep-th/9606040.
  46. van Straten D., Calabi-Yau operators, arXiv:1704.00164.

Previous article  Next article   Contents of Volume 14 (2018)