Kyoto Journal of Mathematics

Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications

Kotaro Kawatani

Full-text: Open access


We show that some Gieseker stable sheaves on a projective K3 surface X are stable with respect to a stability condition of Bridgeland on the derived category of X if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion-free sheaves on a K3 surface with Picard number one is the moduli space of μ-stable locally free sheaves if the rank of the sheaves is not a square number.

Article information

Kyoto J. Math., Volume 53, Number 3 (2013), 597-627.

First available in Project Euclid: 19 August 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14J10: Families, moduli, classification: algebraic theory


Kawatani, Kotaro. Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications. Kyoto J. Math. 53 (2013), no. 3, 597--627. doi:10.1215/21562261-2265905.

Export citation


  • [1] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–345.
  • [2] T. Bridgeland, Stability conditions on $K3$ surfaces, Duke Math. J. 141 (2008), 241–291.
  • [3] D. Huybrechts, Fourier–Mukai Transformations in Algebraic Geometry, Oxford Math. Monogr. Oxford Univ. Press, Oxford, 2006.
  • [4] D. Huybrechts, Derived and abelian equivalence of $K3$ surfaces, J. Algebraic Geom. 17 (2008), 357–400.
  • [5] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects Math. E31, Vieweg, Braunschweig, 1997.
  • [6] D. Huybrechts, E. Macri, and P. Stellari, Stability conditions for generic $K3$ categories, Compos. Math. 144 (2008), 134–162.
  • [7] K. Kawatani, Stability conditions and $\mu$-stable sheaves on K3 surfaces with Picard number one, Osaka J. Math. 49 (2012), 1005–1034.
  • [8] K. Matsuki and R. Wentworth, Mumford–Thaddeus principles on the moduli space of vector bundles on an algebraic surface, Internat. J. Math. 8 (1997), 97–148.
  • [9] D. Orlov, Equivalences of derived categories and $K3$ surfaces, J. Math. Sci. 84 (1997), 1361–1381.
  • [10] Y. Toda, Moduli stacks and invariants of semistable objects on $K3$ surfaces, Adv. Math. 217 (2008), 2736–2781.
  • [11] K. Yoshioka, Stability and the Fourier–Mukai transform, II, Compos. Math. 145 (2009), 112–142.