Abstract
We consider base spaces of Lagrangian fibrations from singular symplectic varieties. After defining cohomologically irreducible symplectic varieties, we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space. We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.
Similar content being viewed by others
References
Bayer A, Macr È. MMP for moduli of sheaves on K3s via wall-crossing: Nef and movable cones, Lagrangian fibrations. ArXiv:1301. 6968, 2013
Beauville A. Symplectic singularities. Invent Math, 2000, 139: 541–549
Greb D, Kebekus S, Kovács S J, et al. Differential forms on log canonical spaces. Publ Math Inst Hautes Études Sci, 2011, 114: 87–169
Greb D, Kebekus S, Peternell T. Singular spaces with trivial canonical class. ArXiv:1110. 5250, 2011
Griffits F, Šmid V. Locally homogeneous complex manifolds. Uspehi Mat Nauk, 1971, 26: 117–168
Hwang J-M. Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1. Ann Inst Fourier, 2007, 57: 815–823
Hwang J M. Base manifolds for fibrations of projective irreducible symplectic manifolds. Invent Math, 2008, 174: 625–644
Kaledin D. Symplectic singularities from the Poisson point of view. J Reine Angew Math, 2006, 600: 135–156
Kaledin D, Lehn M, Sorger C. Singular symplectic moduli spaces. Invent Math, 2006, 164: 591–614
Kebekus S, Solá Conde L. Existence of rational curves on algebraic varieties, minimal rational tangents, and applications. In: Global Aspects of Complex Geometry. Berlin: Springer, 2006, 359–416
Kollár J. Higher direct images of dualizing sheaves, II. Ann Math, 1986, 124: 171–202
Kollár J. Rational Curves on Algebraic Varieties. Berlin: Springer-Verlag, 1996
Markman E. Lagrangian fibrations of holomorphic-symplectic varieties of K3∧[n]-type. ArXiv:1301. 6584, 2013
Matsushita D. On isotropic divisors on irreducible symplectic manifolds. ArXiv:1310. 0896, 2013
Matsushita D. On fibre space structures of a projective irreducible symplectic manifold. Topology, 1999, 38: 79–83
Matsushita D. Addendum: On fibre space structures of a projective irreducible symplectic manifold. Topology, 2001, 40: 431–432
Matsushita D. Higher direct images of dualizing sheaves of Lagrangian fibrations. Amer J Math, 2005, 127: 243–259
Namikawa Y. A note on symplectic singularities. ArXiv: 0101028, 2001
Namikawa Y. Extension of 2-forms and symplectic varieties. J Reine Angew Math, 2001, 539: 123–147
Schmid W. Variation of Hodge structure: the singularities of the period mapping. Invent Math, 1973, 22: 211–319
Takayama S. Simple connectedness of weak Fano varieties. J Algebraic Geom, 2000, 9: 403–407
Yoshioka K. Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface. ArXiv:1206. 4838, 2012
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Matsushita, D. On base manifolds of Lagrangian fibrations. Sci. China Math. 58, 531–542 (2015). https://doi.org/10.1007/s11425-014-4927-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4927-7