Abstract
A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, Vol. I. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1985
Beauville, A.: Some remarks on Kähler manifolds with c 1 = 0. Classification of algebraic and analytic manifolds (Katata, 1982), 1–26, Progr. Math., 39, Birkhäuser, Boston, MA, 1983
Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983)
Beauville, A.: Systèmes hamiltoniens complètement intégrables associés aux surfaces K3. Problems in the theory of surfaces and their classification (Cortona, 1988), 25–31, Sympos. Math., XXXII, Academic Press, London, 1991
Bogomolov, F.A.: Hamiltonian Kähler manifolds. Sov. Math., Dokl. 19, 1462–1465 (1978)
Boucksom, S.: Le cone kählérien d'une variété hyperkählérienne. C. R. Acad. Sci. Paris, Ser. I Math. 333, 935–938 (2001)
Caldararu, A.: Derived Categories of Twisted Sheaves on Calabi-Yau Manifolds. Ph.D. Thesis, Cornell University, 2000
Caldararu, A.: Nonfine moduli spaces of sheaves on K3 surfaces. Int. Math. Res. Not. 2002, 1027–1056 (2002)
Fujiki, A.: On primitively symplectic compact Kahler V-manifolds of dimension four. In: Classification of algebraic and analytic manifolds, Proc. Symp., Katata/Jap. 1982, Prog. Math. 39, 71–250 (1983)
Griffiths, P.A., Harris, J.: Principles of Algebraic Geometry. John Wiley & Sons, New York, 1978
Gulbrandsen, M.G.: Lagrangian fibrations on generalized Kummer varieties. math.AG/0510145
Hartshorne, R.: Residues and Duality. Lecture Notes in Math., No. 20, Springer, 1966
Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52, Springer, 1977
Hassett, B., Tschinkel, Yu.: Abelian fibrations and rational points on symmetric products. Internat. J. Math. 11, 1163–1176 (2000)
Hassett, B., Tschinkel, Yu.: Rational curves on holomorphic symplectic fourfolds. Geom. Funct. Anal. 11, 1201–1228 (2001)
Huybrechts, D.: Birational symplectic manifolds and their deformations. J. Differ. Geom. 45, 488–513 (1997)
Huybrechts, D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135, 63–113 (1999); Erratum, ibid, 152, 209–212 (2003)
Huybrechts, D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326, 499–513 (2003)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Math., Vol. E 31, Friedr. Vieweg & Sohn, Braunschweig (1997)
Huybrechts, D., Stellari, P.: Equivalences of twisted K3 surfaces. Math. Annalen 332, 901–936 (2005)
Iliev, A., Ranestad, K.: The abelian fibration on the Hilbert cube of a K3 surface of genus 9. e-print math.AG/0507016
Kovács, S.J.: The cone of curves of a K3 surface. Math. Ann. 300, 681–691 (1994)
Lazarsfeld, R.: Brill-Noether-Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)
Le Potier, J.: Sur l'espace de modules des fibrés de Yang et Mills. Mathematics and physics (Paris, 1979/1982), 65–137, Progr. Math., 37, Birkhäuser, Boston, 1983
Le Potier, J.: Faisceaux semi-stables et systèmes cohérents. Vector bundles in algebraic geometry (Durham, 1993), Ed. N. Hitchin et al., 179–239, London Math. Soc. Lecture Note Ser., 208, Cambridge Univ. Press, Cambridge, 1995
Looijenga, E., Peters, C.: Torelli theorems for Kähler K3 surfaces. Compositio Math. 42, 145–186 (1980/81)
Maclane, S.: Homology. Springer, Berlin, 1963
Markman, E.: Brill-Noether duality for moduli spaces of sheaves on K3 surfaces. J. Algebraic Geometry 10, 623–694 (2002)
Matsushita, D.: On fibre space structures of a projective irreducible symplectic manifold. Topology 38, 79–83 (1999); Addendum, ibid, 40, 431–432 (2001)
Matsushita, D.: Higher direct images of dualizing sheaves of Lagrangian fibrations. Amer. J. Math. 127, 243–259 (2005)
Morrison, D.: The geometry of K3 surfaces. Lectures delivered at the Scuola Matematica Interuniversitaria, Cortona, 1988
Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77, 101–116 (1984)
Mukai, S.: On the moduli space of bundles on K3 surfaces. I. Vector bundles on algebraic varieties (Bombay, 1984), 341–413, Tata Inst. Fund. Res. Stud. Math., 11, Bombay, 1987
Mukai, S.: Duality of polarized K3 surfaces. In: New trends in algebraic geometry (Warwick, 1996), 311–326, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999
O'Grady, K.G.: The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface. J. Algebr. Geom. 6, 599–644 (1997)
O'Grady, K. G.: Involutions and linear systems on holomorphic symplectic manifolds. math.AG/0403519
Sawon, J.: Abelian fibred holomorphic symplectic manifolds. Turk. J. Math. 27, 197–230 (2003)
Sawon, J.: Lagrangian fibrations on Hilbert schemes of points on K3 surfaces. e-print math.AG/0509224
Saint-Donat, B.: Projective models of K-3 surfaces. Amer. J. Math. 96, 602–639 (1974)
Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. I.H.E.S. 79, 47–129 (1994)
Tyurin, A.N.: Special 0-cycles on a polarized surface of type K3. (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 51, 131–151 (1987)
Tyurin, A.N.: Cycles, curves and vector bundles on an algebraic surface. Duke Math. J. 54, 1–26 (1987)
Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321, 817–884 (2001)
Yoshioka, K.: Moduli spaces of twisted sheaves on a projective variety. math.AG/0411538
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Markushevich, D. Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces. manuscripta math. 120, 131–150 (2006). https://doi.org/10.1007/s00229-006-0631-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-006-0631-4