Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On the monodromy of moduli spaces of sheaves on K3 surfaces

Author: Eyal Markman
Journal: J. Algebraic Geom. 17 (2008), 29-99
Published electronically: July 2, 2007
MathSciNet review: 2357680
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Abstract: Let $ S$ be a $ K3$ surface and $ \operatorname{Aut}D(S)$ the group of auto-equivalences of the derived category of $ S$. We construct a natural representation of $ \operatorname{Aut}D(S)$ on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on $ S$. The main result of this paper is the precise relation of this action with the monodromy of the Hilbert schemes $ S^{[n]}$ of points on the surface. A formula is provided for the monodromy representation in terms of the Chern character of the universal sheaf. Isometries of the second cohomology of $ S^{[n]}$ are lifted, via this formula, to monodromy operators of the whole cohomology ring of $ S^{[n]}$.

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  • [At] M. F. Atiyah, 𝐾-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0224083
  • [AK] Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112. MR 555258, 10.1016/0001-8708(80)90043-2
  • [BB] W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR 0216035
  • [BFM] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch and topological 𝐾 theory for singular varieties, Acta Math. 143 (1979), no. 3-4, 155–192. MR 549773, 10.1007/BF02392091
  • [BHPV] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
  • [B1] Arnaud Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782 (1984) (French). MR 730926
  • [B2] Arnaud Beauville, Sur la cohomologie de certains espaces de modules de fibrés vectoriels, Geometry and analysis (Bombay, 1992) Tata Inst. Fund. Res., Bombay, 1995, pp. 37–40 (French). MR 1351502
  • [B3] Arnaud Beauville, Some remarks on Kähler manifolds with 𝑐₁=0, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 1–26. MR 728605, 10.1007/BF02592068
  • [BO] Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327–344. MR 1818984, 10.1023/A:1002470302976
  • [Br] Tom Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115–133. MR 1629929, 10.1515/crll.1998.046
  • [Ca] Caldararu, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Thesis, Cornell Univ., May 2000.
  • [C] Wei-Liang Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32–67. MR 0028057
  • [CG] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
  • [De] Pierre Deligne, Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 1–10 (French). MR 636513
  • [Fu] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
  • [FL] William Fulton and Robert Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR 644817
  • [Hai] Mark Haiman, 𝑡,𝑞-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201–224. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661369, 10.1016/S0012-365X(98)00141-1
  • [Har] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [HLOY] Hosono, S., Lian, B., Oguiso, K., Yau, S-T.: Autoequivelences of derived category of a K3 surface and monodromy transformations. J. Algebraic Geom. 13 (2004), 513-545. math.AG/0201047
  • [Hu] Huybrechts, D.: Compact Hyper-Kähler Manifolds: Basic results. Invent. Math. 135 (1999), no. 1, 63-113 and Erratum in Invent. Math. 152, 209-212 (2003).
  • [HL] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870
  • [HS] Daniel Huybrechts and Paolo Stellari, Equivalences of twisted 𝐾3 surfaces, Math. Ann. 332 (2005), no. 4, 901–936. MR 2179782, 10.1007/s00208-005-0662-2
  • [K] Max Karoubi, 𝐾-theory, Springer-Verlag, Berlin-New York, 1978. An introduction; Grundlehren der Mathematischen Wissenschaften, Band 226. MR 0488029
  • [KV] Kaledin, D., Verbitsky, M.: Partial resolutions of Hilbert type, Dynkin diagrams and generalized Kummer varieties. Preprint, math.AG/9812078
  • [LS1] Manfred Lehn and Christoph Sorger, The cup product of Hilbert schemes for 𝐾3 surfaces, Invent. Math. 152 (2003), no. 2, 305–329. MR 1974889, 10.1007/s00222-002-0270-7
  • [LS2] Lehn, M., Sorger, C.: Private communication of work in progress.
  • [LL] Eduard Looijenga and Valery A. Lunts, A Lie algebra attached to a projective variety, Invent. Math. 129 (1997), no. 2, 361–412. MR 1465328, 10.1007/s002220050166
  • [Ma1] Eyal Markman, Brill-Noether duality for moduli spaces of sheaves on 𝐾3 surfaces, J. Algebraic Geom. 10 (2001), no. 4, 623–694. MR 1838974
  • [Ma2] Eyal Markman, Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. Reine Angew. Math. 544 (2002), 61–82. MR 1887889, 10.1515/crll.2002.028
  • [Ma3] Markman, E.: On the monodromy of moduli spaces of sheaves on K$ 3$ surfaces II. Preprint, math.AG/0305043 v4.
  • [Ma4] Markman, E.: Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces. Adv. in Math. 208 (2007), 622-646.
  • [Ma5] Markman, E.: Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a $ K3$ surface. Preprint, arXiv:math.AG/0601304 v1
  • [Mu1] Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or 𝐾3 surface, Invent. Math. 77 (1984), no. 1, 101–116. MR 751133, 10.1007/BF01389137
  • [Mu2] S. Mukai, On the moduli space of bundles on 𝐾3 surfaces. I, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 341–413. MR 893604
  • [Mu3] Shigeru Mukai, Duality between 𝐷(𝑋) and 𝐷(𝑋) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081
  • [Mu4] Shigeru Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 515–550. MR 946249
  • [Na1] Hiraku Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), no. 4, 671–721. MR 2023313, 10.1007/s00208-003-0467-0
  • [Na2] Hiraku Nakajima, Convolution on homology groups of moduli spaces of sheaves on 𝐾3 surfaces, Vector bundles and representation theory (Columbia, MO, 2002) Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 75–87. MR 1987740, 10.1090/conm/322/05680
  • [Na3] Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344
  • [Nam] Yoshinori Namikawa, Deformation theory of singular symplectic 𝑛-folds, Math. Ann. 319 (2001), no. 3, 597–623. MR 1819886, 10.1007/PL00004451
  • [Ni] Nikulin, V. V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija, Vol. 14 (1980), No. 1.
  • [OG1] Kieran G. O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a 𝐾3 surface, J. Algebraic Geom. 6 (1997), no. 4, 599–644. MR 1487228
  • [OG2] O'Grady, K.: Involutions and linear systems on holomorphic symplectic manifolds. Geom. Funct. Anal. 15 (2005), 1223-1274. math.AG/0403519.
  • [O] Keiji Oguiso, K3 surfaces via almost-primes, Math. Res. Lett. 9 (2002), no. 1, 47–63. MR 1892313, 10.4310/MRL.2002.v9.n1.a4
  • [Or1] D. O. Orlov, Equivalences of derived categories and 𝐾3 surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361–1381. Algebraic geometry, 7. MR 1465519, 10.1007/BF02399195
  • [Or2] D. O. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 131–158 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 3, 569–594. MR 1921811, 10.1070/IM2002v066n03ABEH000389
  • [P] C. Peters, Monodromy and Picard-Fuchs equations for families of 𝐾3-surfaces and elliptic curves, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 583–607. MR 875089
  • [Sa] S. M. Salamon, On the cohomology of Kähler and hyper-Kähler manifolds, Topology 35 (1996), no. 1, 137–155. MR 1367278, 10.1016/0040-9383(95)00006-2
  • [Sz] B. Szendrői, Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 317–337. MR 1866907
  • [ST] Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, 10.1215/S0012-7094-01-10812-0
  • [Ve1] M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996), no. 4, 601–611. MR 1406664, 10.1007/BF02247112
  • [Ve2] Misha Verbitsky, Mirror symmetry for hyper-Kähler manifolds, Mirror symmetry, III (Montreal, PQ, 1995) AMS/IP Stud. Adv. Math., vol. 10, Amer. Math. Soc., Providence, RI, 1999, pp. 115–156. MR 1673084
  • [Vi] Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
  • [W] C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms. II, J. Reine Angew. Math. 213 (1963/1964), 122–136. MR 0155798
  • [Y1] Kōta Yoshioka, Some examples of Mukai’s reflections on 𝐾3 surfaces, J. Reine Angew. Math. 515 (1999), 97–123. MR 1717621, 10.1515/crll.1999.080
  • [Y2] Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K$ 3$ surfaces. math.AG/9907001
  • [Y3] Kōta Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817–884. MR 1872531, 10.1007/s002080100255
  • [Y4] Yoshioka, K.: A Note on Fourier-Mukai transform. Eprint arXiv:math.AG/0112267 v3.

Additional Information

Eyal Markman
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003

Received by editor(s): December 5, 2005
Published electronically: July 2, 2007
Additional Notes: The author was partially supported by NSF grant number DMS-9802532

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