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A survey of Torelli and monodromy results for holomorphic-symplectic varieties

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Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

We survey recent results about the Torelli question for holomorphicsymplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup WExc, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kähler cone as a fundamental domain for the WExc action on the positive cone. A proof of a weak version of Morrison’s movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV.

Mathematics Subject Classification (2010) 53C26, 14D20, 14J28, 32G20.

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References

  1. Alekseevskij, D. V., Vinberg, E. B., Solodovnikov, A. S.: Geometry of spaces of constant curvature. Geometry, II, 1–138, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993.

    Google Scholar 

  2. Baily, W. L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84 (1966), 442–528.

    Article  MathSciNet  Google Scholar 

  3. Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706.

    MATH  MathSciNet  Google Scholar 

  4. Beauville, A.: Variétés Kähleriennes dont la premiere classe de Chern est nulle. J. Diff. Geom. 18, p. 755–782 (1983).

    MATH  MathSciNet  Google Scholar 

  5. Beauville, A.: Holomorphic symplectic geometry: a problem list. Preprint, arXiv:1002.4321.

    Google Scholar 

  6. Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact complex surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4. Springer-Verlag, Berlin, 2004.

    Google Scholar 

  7. Borel, A.: Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geometry 6 (1972), 543–560.

    MATH  MathSciNet  Google Scholar 

  8. Borcea, C.: Diffeomorphisms of a K3 surface. Math. Ann. 275 (1986), no. 1, 1–4.

    Article  MATH  MathSciNet  Google Scholar 

  9. Boucksom, S.: Le cone Kählerien d’une variété hyperkählerienne. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 10, 935–938.

    MATH  MathSciNet  Google Scholar 

  10. Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76.

    MATH  MathSciNet  Google Scholar 

  11. Burns, D., Rapoport, M.: On the Torelli problem for kählerian K3 surfaces. Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 235–273.

    MATH  MathSciNet  Google Scholar 

  12. Dolgachev, I.: Reflection groups in algebraic geometry. Bull. Amer. Math. Soc. (N.S.) 45 (2008). no. 1, 1–60.

    Article  MATH  MathSciNet  Google Scholar 

  13. Druel, S.: Quelques remarques sur la decomposition de Zariski divisorielle sur les varietes dont la premire classe de Chern est nulle. Math. Z., DOI 10.1007/s00209-009-0626-4 (2009).

    Google Scholar 

  14. Debarre, O., Voisin, C.: Hyper-Kähler fourfolds and Grassmann geometry. Preprint, arXiv:0904.3974.

    Google Scholar 

  15. Fujiki, A.: A theorem on bimeromorphic maps of Kähler manifolds and its applications. Publ. Res. Inst. Math. Sci. 17 (1981), no. 2, 735–754.

    Article  MATH  MathSciNet  Google Scholar 

  16. Gritsenko, V.; Hulek, K.; Sankaran, G.K.: Moduli spaces of irreducible symplectic manifolds. Compositio Math. 146 (2010) 404–434.

    Article  MATH  MathSciNet  Google Scholar 

  17. Gritsenko, V.; Hulek, K.; Sankaran, G.K.: Moduli of K3 Surfaces and Irreducible Symplectic Manifolds. Preprint arXiv:1012.4155.

    Google Scholar 

  18. Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286 (1990), 193–207.

    Article  MATH  MathSciNet  Google Scholar 

  19. Hassett, B.: Special cubic fourfolds. Compositio Math. 120 (2000), no. 1, 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  20. Huybrechts, D, Lehn, M.: The geometry of moduli spaces of sheaves. Second edition. Cambridge University Press, Cambridge, 2010.

    Google Scholar 

  21. Huybrechts, D., Nieper-Wisskirchen, N.: Remarks on derived equivalences of Ricci-flat manifolds. Math. Z., DOI: 10.1007/s00209-009-0655 (2010).

    Google Scholar 

  22. Hassett, B., Tschinkel, Y.: Rational curves on holomorphic symplectic fourfolds. Geom. Funct. Anal. 11, no. 6, 1201–1228, (2001).

    Article  MATH  MathSciNet  Google Scholar 

  23. Hassett, B., Tschinkel, Y.: Moving and ample cones of holomorphic symplectic fourfolds. Geom. Funct. Anal. 19, no. 4, 1065–1080, (2009).

    Article  MATH  MathSciNet  Google Scholar 

  24. Hassett, B., Tschinkel, Y.: Monodromy and rational curves on holomorphic symplectic fourfolds. Preprint 2009.

    Google Scholar 

  25. Hassett, B., Tschinkel, Y.: Intersection numbers of extremal rays on holomorphic symplectic varieties. Preprint, arXiv:0909.4745 (2009).

    Google Scholar 

  26. Huybrechts, D.: Compact Hyperkähler Manifolds: Basic results. Invent. Math. 135 (1999), no. 1, 63–113 and Erratum: Invent. Math. 152 (2003), no. 1, 209–212.

    Article  MATH  MathSciNet  Google Scholar 

  27. Huybrechts, D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326 (2003), no. 3, 499–513.

    MATH  MathSciNet  Google Scholar 

  28. Huybrechts, D.: Compact hyperkähier manifolds. Calabi-Yau manifolds and related geometries (Nordfjordeid, 2001), 161–225, Universitext, Springer, Berlin, 2003.

    Google Scholar 

  29. Huybrechts, D.: Finiteness results for compact hyperkähler manifolds. J. Reine Angew. Math. 558 (2003), 15–22.

    Article  MATH  MathSciNet  Google Scholar 

  30. Huybrechts, D.: The global Torelli theorem: classical, derived, twisted. Algebraic geometry-Seattle 2005. Part 1, 235–258, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., 2009.

    Google Scholar 

  31. Huybrechts, D.: A global Torelli Theorem for hyperkähler manifolds, after M. Verbitsky. Séminaire Bourbaki, preprint 2010.

    Google Scholar 

  32. Iliev, A., Ranestad, K.: K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds. Trans. Amer. Math. Soc. 353 (2001), no. 4, 1455–1468

    Article  MATH  MathSciNet  Google Scholar 

  33. Kawamata, Y.: On the cone of divisors of Caiab-Yau fiber spaces. Internat. J. Math. 8 (1997), no. 5, 665–687.

    Article  MATH  MathSciNet  Google Scholar 

  34. Kovács, S.: The cone of curves of a K3 surface. Math. Ann. 300 (1994), no. 4, 681–691.

    Article  MATH  MathSciNet  Google Scholar 

  35. Li, J.: Algebraic geometric interpretation of Donaldson’s polynomial invariants of algebraic surfaces. J. Diff. Geom. 37 (1993), 417–466.

    MATH  Google Scholar 

  36. Looijenga, E., Peters, C.: Torelli theorems for Kähler K3 surfaces. Compositio Math. 42 (1980/81), no. 2, 145–186.

    MathSciNet  Google Scholar 

  37. Markman, E.: Brill-Noether duality for moduli spaces of sheaves on K3 surfaces. J. Algebraic Geom. 10 (2001), no. 4, 623–694.

    MATH  MathSciNet  Google Scholar 

  38. Markman, E.: On the monodromy of moduli spaces of sheaves on K3 surfaces. J. Algebraic Geom. 17 (2008), 29–99.

    MATH  MathSciNet  Google Scholar 

  39. Markman, E.: Generators of the cohomoiogy ring of moduli spaces of sheaves on symplectic surfaces. Journal für die reine und angewandte Mathematik 544 (2002), 61–82.

    Article  MATH  MathSciNet  Google Scholar 

  40. Markman, E.: Integral generators for the cohomoiogy ring of moduli spaces of sheaves over Poisson surfaces. Adv. in Math. 208 (2007) 622–646.

    Article  MATH  MathSciNet  Google Scholar 

  41. Markman, E.: Integral constraints on the monodromy group of the hyperkähler resolution ofa symmetric product of a K3 surface. Internat. J. of Math. 21, (2010), no. 2, 169–223.

    Article  MATH  MathSciNet  Google Scholar 

  42. Markman, E.: Modular Galois covers associated to symplectic resolutions of singularities. J. Reine Angew. Math. 644 (2010), 189–220.

    Article  MATH  MathSciNet  Google Scholar 

  43. Markman, E.: Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections. Preprint arXiv:0912.4981 v1.

    Google Scholar 

  44. Markushevich, D.: Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces. Manuscripta Math. 120 (2006), no. 2, 131–150.

    Article  MATH  MathSciNet  Google Scholar 

  45. Matsushita, D.: On deformations of Lagrangian fibrations. Preprint arXiv:0903.2098.

    Google Scholar 

  46. Matsushita, D.: Higher direct images of dualizing sheaves of Lagrangian fibrations. Amer. J. Math. 127 (2005), no. 2, 243–259.

    Article  MATH  MathSciNet  Google Scholar 

  47. Morrison, D. R.: Compactifications of moduli spaces inspired by mirror symmetry. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). Astérisque No. 218 (1993), 243–271.

    MathSciNet  Google Scholar 

  48. Morrison, D. R.: Beyond the Kähier cone. Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., 9, Bar-Ilan Univ., 1996, 361–376.

    MathSciNet  Google Scholar 

  49. Mukai, S.: On the moduli space of bundles on K3 surfaces I, Vector bundles on algebraic varieties, Proc. Bombay Conference, 1984, Tata Institute of Fundamental Research Studies, no. 11, Oxford University Press, 1987, pp. 341–413.

    Google Scholar 

  50. Mukai, S.: New developments in Fano manifold theory related to the vector bundle method and moduli problems. Sugaku 47 (1995), no. 2, 125–144. English translation: Sugaku Expositions 15 (2002), no. 2, 125–150.

    MathSciNet  Google Scholar 

  51. Namikawa, Yukihiko: Periods of Enriques surfaces. Math. Ann. 270 (1985), No. 2, 201–222.

    Article  MATH  MathSciNet  Google Scholar 

  52. Namikawa, Yoshinori: Deformation theory of singular symplectic n-folds. Math. Ann. 319 (2001), no. 3, 597–623.

    Article  MATH  MathSciNet  Google Scholar 

  53. Namikawa, Yoshinori: Counter-example to global Torelli problem for irreducible symplectic manifolds. Math. Ann. 324 (2002), no. 4, 841–845.

    Article  MATH  MathSciNet  Google Scholar 

  54. Namikawa, Yoshinori: Poisson deformations of affine symplectic varieties II. Preprint arXiv:0902.2832, to appear in Nagata memorial issue of Kyoto J. of Math.

    Google Scholar 

  55. Nikulin, V. V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija, Vol. 14 (1980), No. 1, 103–167.

    Article  MATH  Google Scholar 

  56. O’Grady, K.: The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface. J. Algebraic Geom. 6 (1997), no. 4, 599–644.

    MATH  MathSciNet  Google Scholar 

  57. O’Grady, K.: Desingularized moduli spaces of sheaves on a K3. J. Reine Angew. Math. 512 (1999), 49–117.

    Article  MATH  MathSciNet  Google Scholar 

  58. O’Grady, K.: A new six-dimensional irreducible symplectic variety. J. Algebraic Geom. 12 (2003), no. 3, 435–505.

    MATH  MathSciNet  Google Scholar 

  59. O’Grady, K.: Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics. Duke Math. J. 134 (2006), no. 1, 99–137.

    Article  MATH  MathSciNet  Google Scholar 

  60. O’Grady, K.: Higher-dimensional analogues of K3 surfaces. Preprint arXiv:1005.3131.

    Google Scholar 

  61. Oguiso, K.: K3 surfaces via almost-primes. Math. Res. Lett. 9 (2002), no. 1, 47–63.

    MATH  MathSciNet  Google Scholar 

  62. Orlov, D.: Equivalences of derived categories and K3 surfaces. Algebraic geometry, 7. J. Math. Sci. (New York) 84 (1997), no. 5, 1361–1381.

    MATH  Google Scholar 

  63. Ploog, D.: Equivariant autoequivalences for finite group actions. Adv. Math. 216 (2007), no. 1, 62–74.

    Article  MATH  MathSciNet  Google Scholar 

  64. Prendergast-Smith, A.: The cone conjecture for abelian varieties. Electronic preprint arXiv:1008.4467 (2010).

    Google Scholar 

  65. Piatetski-Shapiro, I. I., Shafarevich, I. R.: Torelli’s theorem for algebraic surfaces of type K3. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 1971 530–572.

    MATH  Google Scholar 

  66. Rapagnetta, A.: On the Beauville form of the known irreducible symplectic varieties. Math. Ann. 340 (2008), no. 1, 77–95.

    Article  MathSciNet  Google Scholar 

  67. Satake, I.: Algebraic structures of symmetric domains. Kano Memorial Lectures, 4. Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980.

    Google Scholar 

  68. Sawon, J.: Lagrangian fibrations on Hilbert schemes of points on K3 surfaces. J. Algebraic Geom. 16 (2007), no. 3, 477–497.

    MATH  MathSciNet  Google Scholar 

  69. Sterk, H.: Finiteness results for algebraic K3 surfaces. Math. Z. 189, 507–513 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  70. Sullivan, D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331 (1978).

    Google Scholar 

  71. Totaro, B.: The cone conjecture for Calabi-Yau pairs in dimension 2. Duke Math. J. 154 (2010), no. 2, 241–263

    Article  MATH  MathSciNet  Google Scholar 

  72. Verbitsky, M.: Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6 (1996), no. 4, 601–611.

    Article  MATH  MathSciNet  Google Scholar 

  73. Verbitsky, M.: A global Torelli theorem for hyperkähler manifolds. Preprint arXiv:0908.4121 v6.

    Google Scholar 

  74. Verbitsky, M.: HyperKähler SYZ conjecture and semipositive line bundles. Geom. Funct. Anal. 19 (2010), no. 5, 1481–1493.

    Article  MATH  MathSciNet  Google Scholar 

  75. Viehweg, E.: Quasi-projective moduli for polarized manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 30. Springer, Berlin, 1995.

    Google Scholar 

  76. Voisin, C.: Hodge Theory and Complex Algebraic Geometry I. Cambridge studies in advanced mathematics 76, Cambridge Univ. Press (2002).

    Google Scholar 

  77. Vinberg, E. B., Shvartsman, O. V.: Discrete groups of motions of spaces of constant curvature. Geometry, II, 139–248, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993.

    Google Scholar 

  78. Yoshioka, K.: Some examples of Mukai’s reflections on K3 surfaces. J. Reine Angew. Math. 515 (1999), 97–123.

    Article  MATH  MathSciNet  Google Scholar 

  79. Yoshioka, K. : Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321 (2001), no. 4, 817–884.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Eyal Markman

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  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

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Markman, E. (2011). A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_15

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