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
DOI: https://doi.org/10.1090/S1056-3911-07-00457-2
Published electronically: July 2, 2007
MathSciNet review: 2357680
Full-text PDF

Abstract | References | Additional Information

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]}$.


References [Enhancements On Off] (What's this?)

  • [At] Atiyah, M. F.: $ K$-theory. Lecture notes by D. W. Anderson. W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0224083 (36:7130)
  • [AK] Altman, A., Kleiman, S.: Compactifying the Picard scheme. Adv. in Math. 35, 50-112 (1980). MR 555258 (81f:14025a)
  • [BB] Baily, W. L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84 (1966), 442-528. MR 0216035 (35:6870)
  • [BFM] Baum, P., Fulton, W., MacPherson, R.: Riemann-Roch and topological $ K$-Theory for singular varieties. Acta Math. 143 (1979), no. 3-4, 155-192. MR 549773 (82c:14021)
  • [BHPV] Barth, W., Hulek, K., Peters, C., and Van de Ven, A.: Compact Complex Surfaces. Second edition, Springer-Verlag, 2004. MR 2030225 (2004m:14070)
  • [B1] Beauville, A. Variétés Kähleriennes dont la première classe de Chern est nulle. J. Diff. Geom. 18, (1983), 755-782. MR 730926 (86c:32030)
  • [B2] Beauville, A. Sur la cohomologie de certains espaces de modules de fibrés vectoriels. Geometry and analysis (Bombay, 1992), 37-40, Tata Inst. Fund. Res., Bombay, 1995. MR 1351502 (96f:14011)
  • [B3] Beauville, A. Some remarks on Kähler manifolds with $ c_1=0$. in Classification of algebraic and analytic manifolds (Katata, 1982), 1-26, Progr. Math., 39, Birkhäuser Boston, 1983. MR 728605 (86c:32031)
  • [BO] Bondal, A., Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math. 125 (2001), no. 3, 327-344. MR 1818984 (2001m:18014)
  • [Br] Bridgeland, T.: Fourier-Mukai transforms for elliptic surfaces. J. Reine Angew. Math. 498 (1998), 115-133. MR 1629929 (99f:14013)
  • [Ca] Caldararu, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Thesis, Cornell Univ., May 2000.
  • [C] Chow, W. L.: On the geometry of algebraic homogeneous spaces. Ann. of Math. (2) 50, (1949), 32-67. MR 0028057 (10:396d)
  • [CG] Chriss, N., Ginzburg, V.: Representation theory and complex geometry. Birkhäuser Boston, Boston, MA, 1997. MR 1433132 (98i:22021)
  • [De] Deligne, P.: 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, pp. 1-10, Lecture Notes in Math., 842, Springer, 1981. MR 636513 (83f:14026)
  • [Fu] Fulton, W.: Intersection Theory. Springer-Verlag, 1984. MR 732620 (85k:14004)
  • [FL] Fulton, W., Lazarsfeld, R.: Connectivity and its applications in algebraic geometry. Algebraic geometry (Chicago, Ill., 1980), pp. 26-92, Lecture Notes in Math., 862, Springer, 1981. MR 644817 (83i:14002)
  • [Hai] Haiman, M.: $ t,q$-Catalan numbers and the Hilbert scheme. Discrete Mathematics 193 (1998) 201-224. MR 1661369 (2000k:05264)
  • [Har] Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)
  • [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] Huybrechts, D, Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870 (98g:14012)
  • [HS] Huybrechts, D.; Stellari, P.: Equivalences of twisted $ K3$ surfaces. Math. Ann. 332 (2005), 901-936. arXiv:math.AG/0409030 v3 MR 2179782
  • [K] Karoubi, M.: $ K$-theory. An introduction. Springer-Verlag, 1978. MR 0488029 (58:7605)
  • [KV] Kaledin, D., Verbitsky, M.: Partial resolutions of Hilbert type, Dynkin diagrams and generalized Kummer varieties. Preprint, math.AG/9812078
  • [LS1] Lehn, M., Sorger, C.: The cup product of the Hilbert scheme for $ K3$ surfaces. Invent. Math. 152 (2003), no. 2, 305-329. MR 1974889 (2004a:14004)
  • [LS2] Lehn, M., Sorger, C.: Private communication of work in progress.
  • [LL] Looijenga, E., Lunts, V.: A Lie algebra attached to a projective variety. Invent. Math. 129 (1997), no. 2, 361-412. MR 1465328 (98m:14019)
  • [Ma1] Markman, E.: Brill-Noether duality for moduli spaces of sheaves on K$ 3$ surfaces J. of Alg. Geom. 10 (2001), no. 4, 623-694. MR 1838974 (2002d:14065)
  • [Ma2] Markman, E.: Generators of the cohomology ring of moduli spaces of sheaves on K$ 3$ and Abelian surfaces. Journal für die reine und angewandte Mathematik 544 (2002), 61-82. MR 1887889 (2003a:14011)
  • [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] Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K$ 3$ surface, Invent. Math. 77 (1984), 101-116. MR 751133 (85j:14016)
  • [Mu2] Mukai, S.: On the moduli space of bundles on K$ 3$ 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. MR 893604 (88i:14036)
  • [Mu3] Mukai, S.: Duality between $ D(X)$ and $ D(\hat X)$ with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153-175. MR 607081 (82f:14036)
  • [Mu4] Mukai, S.: Fourier functor and its application to the moduli of bundles on an Abelian variety. Adv. Studies in Pure Math. 10, 515-550 (1987). MR 946249 (89k:14026)
  • [Na1] Nakajima, H.: Reflection functors for quiver varieties and Weyl group actions. Math. Ann. 327 (2003), no. 4, 671-721. MR 2023313 (2004k:16036)
  • [Na2] Nakajima, H.: Convolution on homology groups of moduli spaces of sheaves on K$ 3$ surfaces. Vector bundles and representation theory (Columbia, MO, 2002), 75-87, Contemp. Math., 322, Amer. Math. Soc., Providence, RI, 2003. MR 1987740 (2004e:14068)
  • [Na3] Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University Lecture Series, 18. Amer. Math. Soc., Providence, RI, 1999. MR 1711344 (2001b:14007)
  • [Nam] Namikawa, Y.: Deformation theory of singular symplectic $ n$-folds. Math. Ann. 319 (2001), no. 3, 597-623. MR 1819886 (2002b:32025)
  • [Ni] Nikulin, V. V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija, Vol. 14 (1980), No. 1.
  • [OG1] O'Grady, K.: The weight-two Hodge structure of moduli spaces of sheaves on a K$ 3$ surface. J. Algebraic Geom. 6 (1997), no. 4, 599-644. MR 1487228 (2000a:14052)
  • [OG2] O'Grady, K.: Involutions and linear systems on holomorphic symplectic manifolds. Geom. Funct. Anal. 15 (2005), 1223-1274. arXiv.org math.AG/0403519.
  • [O] Oguiso, K.: K$ 3$ surfaces via almost-primes. Math. Res. Lett. 9 (2002), no. 1, 47-63. MR 1892313 (2002m:14031)
  • [Or1] Orlov, D. O.: Equivalences of derived categories and K$ 3$ surfaces. Algebraic geometry, 7. J. Math. Sci. (New York) 84 (1997), no. 5, 1361-1381. MR 1465519 (99a:14054)
  • [Or2] Orlov, D. O.: Derived categories of coherent sheaves on abelian varieties and equivalences between them. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 131-158; translation in Izv. Math. 66 (2002), no. 3, 569-594. MR 1921811 (2004b:14027)
  • [P] Peters, C.: Monodromy and Picard-Fuchs equations for families of $ K3$-surfaces and elliptic curves. Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 583-607. MR 875089 (88e:14009)
  • [Sa] Salamon, S. M.: On the cohomology of Kähler and hyper-Kähler manifolds. Topology 35 (1996), no. 1, 137-155. MR 1367278 (97f:32042)
  • [Sz] Szendröi, B.: Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry. Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), 317-337, NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, 2001. MR 1866907 (2003a:14059)
  • [ST] Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108 (2001), no. 1, 37-108. MR 1831820 (2002e:14030)
  • [Ve1] Verbitsky, M.: Cohomology of compact hyper-Kähler manifolds and its applications., Geom. Funct. Anal. 6 (1996), no. 4, 601-611. MR 1406664 (98a:53069)
  • [Ve2] Verbitsky, M.: Mirror symmetry for hyper-Kähler manifolds. Mirror symmetry, III (Montreal, PQ, 1995), 115-156, AMS/IP Stud. Adv. Math., 10, Amer. Math. Soc., Providence, RI, 1999. MR 1673084 (2000a:53084)
  • [Vi] Viehweg, E.: Quasi-projective Moduli for Polarized Manifolds. Springer-Verlag (1995). MR 1368632 (97j:14001)
  • [W] Wall, C. T. C.: On the orthogonal groups of unimodular quadratic forms. II. J. Reine Angew. Math. 213 (1963/1964), 122-136. MR 0155798 (27:5732)
  • [Y1] Yoshioka, K.: Some examples of Mukai reflections on K$ 3$ surfaces. J. Reine Angew. Math. 515 (1999), 97-123. MR 1717621 (2000h:14028)
  • [Y2] Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K$ 3$ surfaces. math.AG/9907001
  • [Y3] Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321 (2001), no. 4, 817-884. MR 1872531 (2002k:14020)
  • [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
Email: markman@math.umass.edu

DOI: https://doi.org/10.1090/S1056-3911-07-00457-2
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

American Mathematical Society