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On the monodromy of moduli spaces of sheaves on \(K3\) surfaces. (English) Zbl 1185.14015

Summary: Let \( S\) be a \( K3\) surface and Aut \(D(S)\) the group of auto-equivalences of the derived category of \( S\). We construct a natural representation of 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 \( S^{[n]}\).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J28 \(K3\) surfaces and Enriques surfaces
14C05 Parametrization (Chow and Hilbert schemes)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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[1] M. F. Atiyah, \?-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967.
[2] Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50 – 112. , https://doi.org/10.1016/0001-8708(80)90043-2 Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme. II, Amer. J. Math. 101 (1979), no. 1, 10 – 41. · Zbl 0427.14016 · doi:10.2307/2373937
[3] W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442 – 528. · Zbl 0154.08602 · doi:10.2307/1970457
[4] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch and topological \? theory for singular varieties, Acta Math. 143 (1979), no. 3-4, 155 – 192. · Zbl 0474.14004 · doi:10.1007/BF02392091
[5] 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. · Zbl 1036.14016
[6] Arnaud Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755 – 782 (1984) (French). · Zbl 0537.53056
[7] 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). · Zbl 0880.14011
[8] Arnaud Beauville, Some remarks on Kähler manifolds with \?\(_{1}\)=0, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 1 – 26. · Zbl 0635.90058 · doi:10.1007/BF02592068
[9] 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. · Zbl 0994.18007 · doi:10.1023/A:1002470302976
[10] Tom Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115 – 133. · Zbl 0905.14020 · doi:10.1515/crll.1998.046
[11] Caldararu, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Thesis, Cornell Univ., May 2000.
[12] Wei-Liang Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32 – 67. · Zbl 0040.22901 · doi:10.2307/1969351
[13] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. · Zbl 0879.22001
[14] 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).
[15] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[16] 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. · Zbl 0484.14005
[17] Mark Haiman, \?,\?-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201 – 224. Selected papers in honor of Adriano Garsia (Taormina, 1994). · Zbl 1061.05509 · doi:10.1016/S0012-365X(98)00141-1
[18] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[19] 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 · Zbl 1070.14042
[20] 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). · Zbl 0953.53031
[21] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. · Zbl 0872.14002
[22] Daniel Huybrechts and Paolo Stellari, Equivalences of twisted \?3 surfaces, Math. Ann. 332 (2005), no. 4, 901 – 936. · Zbl 1092.14047 · doi:10.1007/s00208-005-0662-2
[23] Max Karoubi, \?-theory, Springer-Verlag, Berlin-New York, 1978. An introduction; Grundlehren der Mathematischen Wissenschaften, Band 226. · Zbl 0382.55002
[24] Kaledin, D., Verbitsky, M.: Partial resolutions of Hilbert type, Dynkin diagrams and generalized Kummer varieties. Preprint, math.AG/9812078
[25] Manfred Lehn and Christoph Sorger, The cup product of Hilbert schemes for \?3 surfaces, Invent. Math. 152 (2003), no. 2, 305 – 329. · Zbl 1035.14001 · doi:10.1007/s00222-002-0270-7
[26] Lehn, M., Sorger, C.: Private communication of work in progress. · Zbl 1035.14001
[27] Eduard Looijenga and Valery A. Lunts, A Lie algebra attached to a projective variety, Invent. Math. 129 (1997), no. 2, 361 – 412. · Zbl 0890.53030 · doi:10.1007/s002220050166
[28] Eyal Markman, Brill-Noether duality for moduli spaces of sheaves on \?3 surfaces, J. Algebraic Geom. 10 (2001), no. 4, 623 – 694. · Zbl 1074.14525
[29] Eyal Markman, Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. Reine Angew. Math. 544 (2002), 61 – 82. · Zbl 0988.14019 · doi:10.1515/crll.2002.028
[30] Markman, E.: On the monodromy of moduli spaces of sheaves on K\( 3\) surfaces II. Preprint, math.AG/0305043 v4.
[31] Markman, E.: Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces. Adv. in Math. 208 (2007), 622-646. · Zbl 1115.14036
[32] 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 · Zbl 1184.14074
[33] Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or \?3 surface, Invent. Math. 77 (1984), no. 1, 101 – 116. · Zbl 0565.14002 · doi:10.1007/BF01389137
[34] S. Mukai, On the moduli space of bundles on \?3 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. · Zbl 0674.14023
[35] Shigeru Mukai, Duality between \?(\?) and \?(\?) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153 – 175. · Zbl 0417.14036
[36] 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.
[37] Hiraku Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), no. 4, 671 – 721. · Zbl 1060.16017 · doi:10.1007/s00208-003-0467-0
[38] Hiraku Nakajima, Convolution on homology groups of moduli spaces of sheaves on \?3 surfaces, Vector bundles and representation theory (Columbia, MO, 2002) Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 75 – 87. · Zbl 1064.14043 · doi:10.1090/conm/322/05680
[39] Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. · Zbl 0949.14001
[40] Yoshinori Namikawa, Deformation theory of singular symplectic \?-folds, Math. Ann. 319 (2001), no. 3, 597 – 623. · Zbl 0989.53055 · doi:10.1007/PL00004451
[41] Nikulin, V. V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija, Vol. 14 (1980), No. 1. · Zbl 0427.10014
[42] Kieran G. O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a \?3 surface, J. Algebraic Geom. 6 (1997), no. 4, 599 – 644. · Zbl 0916.14018
[43] O’Grady, K.: Involutions and linear systems on holomorphic symplectic manifolds. Geom. Funct. Anal. 15 (2005), 1223-1274. arXiv.org math.AG/0403519. · Zbl 1093.53081
[44] Keiji Oguiso, K3 surfaces via almost-primes, Math. Res. Lett. 9 (2002), no. 1, 47 – 63. · Zbl 1043.14010 · doi:10.4310/MRL.2002.v9.n1.a4
[45] D. O. Orlov, Equivalences of derived categories and \?3 surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361 – 1381. Algebraic geometry, 7. · Zbl 0938.14019 · doi:10.1007/BF02399195
[46] 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. · Zbl 1031.18007 · doi:10.1070/IM2002v066n03ABEH000389
[47] C. Peters, Monodromy and Picard-Fuchs equations for families of \?3-surfaces and elliptic curves, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 583 – 607. · Zbl 0612.14006
[48] S. M. Salamon, On the cohomology of Kähler and hyper-Kähler manifolds, Topology 35 (1996), no. 1, 137 – 155. · Zbl 0854.58004 · doi:10.1016/0040-9383(95)00006-2
[49] 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. · Zbl 1017.14016
[50] Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37 – 108. · Zbl 1092.14025 · doi:10.1215/S0012-7094-01-10812-0
[51] M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996), no. 4, 601 – 611. · Zbl 0861.53069 · doi:10.1007/BF02247112
[52] 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. · Zbl 0926.32036
[53] 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. · Zbl 0844.14004
[54] C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms. II, J. Reine Angew. Math. 213 (1963/1964), 122 – 136. · Zbl 0135.08802 · doi:10.1515/crll.1964.213.122
[55] Kōta Yoshioka, Some examples of Mukai’s reflections on \?3 surfaces, J. Reine Angew. Math. 515 (1999), 97 – 123. · Zbl 0940.14026 · doi:10.1515/crll.1999.080
[56] Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K\( 3\) surfaces. math.AG/9907001
[57] Kōta Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817 – 884. · Zbl 1066.14013 · doi:10.1007/s002080100255
[58] Yoshioka, K.: A Note on Fourier-Mukai transform. Eprint arXiv:math.AG/0112267 v3. · Zbl 1177.14035
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