On the monodromy of moduli spaces of sheaves on K3 surfaces

Markman, Eyal

doi:10.1090/S1056-3911-07-00457-2

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

M. F. Atiyah, $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR 0224083
Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112. MR 555258, DOI https://doi.org/10.1016/0001-8708%2880%2990043-2
W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR 216035, DOI https://doi.org/10.2307/1970457
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, DOI https://doi.org/10.1007/BF02392091
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
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
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
Arnaud Beauville, Some remarks on Kähler manifolds with $c_{1}=0$, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 1–26. MR 728605
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, DOI https://doi.org/10.1023/A%3A1002470302976
Tom Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115–133. MR 1629929, DOI https://doi.org/10.1515/crll.1998.046

Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Thesis, Cornell Univ., May 2000.

Wei-Liang Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32–67. MR 28057, DOI https://doi.org/10.2307/1969351

Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132

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

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

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

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, DOI https://doi.org/10.1016/S0012-365X%2898%2900141-1

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157

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

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).

Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870

Daniel Huybrechts and Paolo Stellari, Equivalences of twisted $K3$ surfaces, Math. Ann. 332 (2005), no. 4, 901–936. MR 2179782, DOI https://doi.org/10.1007/s00208-005-0662-2

Max Karoubi, $K$-theory, Springer-Verlag, Berlin-New York, 1978. An introduction; Grundlehren der Mathematischen Wissenschaften, Band 226. MR 0488029

Kaledin, D., Verbitsky, M.: Partial resolutions of Hilbert type, Dynkin diagrams and generalized Kummer varieties. Preprint, math.AG/9812078

Manfred Lehn and Christoph Sorger, The cup product of Hilbert schemes for $K3$ surfaces, Invent. Math. 152 (2003), no. 2, 305–329. MR 1974889, DOI https://doi.org/10.1007/s00222-002-0270-7

Lehn, M., Sorger, C.: Private communication of work in progress.

Eduard Looijenga and Valery A. Lunts, A Lie algebra attached to a projective variety, Invent. Math. 129 (1997), no. 2, 361–412. MR 1465328, DOI https://doi.org/10.1007/s002220050166

Eyal Markman, Brill-Noether duality for moduli spaces of sheaves on $K3$ surfaces, J. Algebraic Geom. 10 (2001), no. 4, 623–694. MR 1838974

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, DOI https://doi.org/10.1515/crll.2002.028

Markman, E.: On the monodromy of moduli spaces of sheaves on K$3$ surfaces II. Preprint, math.AG/0305043 v4.

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

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

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, DOI https://doi.org/10.1007/BF01389137

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

Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081

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, DOI https://doi.org/10.2969/aspm/01010515

Hiraku Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), no. 4, 671–721. MR 2023313, DOI https://doi.org/10.1007/s00208-003-0467-0

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, DOI https://doi.org/10.1090/conm/322/05680

Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344

Yoshinori Namikawa, Deformation theory of singular symplectic $n$-folds, Math. Ann. 319 (2001), no. 3, 597–623. MR 1819886, DOI https://doi.org/10.1007/PL00004451

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

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

O’Grady, K.: Involutions and linear systems on holomorphic symplectic manifolds. Geom. Funct. Anal. 15 (2005), 1223-1274. arXiv.org math.AG/0403519.

Keiji Oguiso, K3 surfaces via almost-primes, Math. Res. Lett. 9 (2002), no. 1, 47–63. MR 1892313, DOI https://doi.org/10.4310/MRL.2002.v9.n1.a4

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, DOI https://doi.org/10.1007/BF02399195

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, DOI https://doi.org/10.1070/IM2002v066n03ABEH000389

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

S. M. Salamon, On the cohomology of Kähler and hyper-Kähler manifolds, Topology 35 (1996), no. 1, 137–155. MR 1367278, DOI https://doi.org/10.1016/0040-9383%2895%2900006-2

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

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, DOI https://doi.org/10.1215/S0012-7094-01-10812-0

M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996), no. 4, 601–611. MR 1406664, DOI https://doi.org/10.1007/BF02247112

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

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

C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms. II, J. Reine Angew. Math. 213 (1963/64), 122–136. MR 155798, DOI https://doi.org/10.1515/crll.1964.213.122

K\B{o}ta Yoshioka, Some examples of Mukai’s reflections on $K3$ surfaces, J. Reine Angew. Math. 515 (1999), 97–123. MR 1717621, DOI https://doi.org/10.1515/crll.1999.080

Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K$3$ surfaces. math.AG/9907001

K\B{o}ta Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817–884. MR 1872531, DOI https://doi.org/10.1007/s002080100255

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
MR Author ID: 355854
Email: markman@math.umass.edu

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