Kyoto Journal of Mathematics

Perverse coherent sheaves and Fourier–Mukai transforms on surfaces, II

Kōta Yoshioka

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on 2 -dimensional moduli spaces of stable sheaves on K3 and elliptic surfaces. Then we show that perverse coherent sheaves appear in the theory of Fourier–Mukai transforms. As an application, we generalize the Fourier–Mukai duality for K3 surfaces to our situation.

Article information

Kyoto J. Math., Volume 55, Number 2 (2015), 365-459.

Received: 4 September 2012
Revised: 4 April 2014
Accepted: 4 April 2014
First available in Project Euclid: 11 June 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}


Yoshioka, Kōta. Perverse coherent sheaves and Fourier–Mukai transforms on surfaces, II. Kyoto J. Math. 55 (2015), no. 2, 365--459. doi:10.1215/21562261-2871785.

Export citation


  • [Br1] T. Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115–133.
  • [Br2] T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), 25–34.
  • [Br3] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613–632.
  • [Br4] T. Bridgeland, Stability conditions on K3 surfaces, Duke Math. J. 141 (2008), 241–291.
  • [E] H. Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362 (1985), 63–71.
  • [H] D. Huybrechts, Derived and abelian equivalence of K3 surfaces, J. Algebraic Geom. 17 (2008), 375–400.
  • [In] M.-a. Inaba, Moduli of stable objects in a triangulated category, J. Math. Soc. Japan 62 (2010), 395–429.
  • [K] V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
  • [MYY] H. Minamide, S. Yanagida, and K. Yoshioka, Fourier-Mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions, preprint, arXiv:1106.5217v2 [math.AG].
  • [Mu1] S. Mukai, Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175.
  • [Mu2] S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), 101–116.
  • [Mu3] S. Mukai, “On the moduli space of bundles on K3 surfaces, I” in Vector Bundles on Algebraic Varieties (Bombay 1984), Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Bombay, 1987, 341–413.
  • [MFK] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
  • [OY] N. Onishi and K. Yoshioka, Singularities on the $2$-dimensional moduli spaces of stable sheaves on K3 surfaces, Internat. J. Math. 14 (2003), 837–864.
  • [O] D. O. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (N. Y.) 84 (1997), 1361–1381.
  • [Se] C. S. Seshadri, Geometric reductivity over arbitrary base, Adv. Math. 26 (1977), 225–274.
  • [S] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, Publ. Math. Inst. Hautes Études Sci. 79 (1994), 47–129.
  • [VB] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), 423–455.
  • [Y1] K. Yoshioka, Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface, Internat. J. Math. 7 (1996), 411–431.
  • [Y2] K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), 817–884.
  • [Y3] K. Yoshioka, Twisted stability and Fourier-Mukai transform, I, Compos. Math. 138 (2003), 261–288.
  • [Y4] K. Yoshioka, Twisted stability and Fourier-Mukai transform, II, Manuscripta Math. 110 (2003), 433–465.
  • [Y5] K. Yoshioka, “Moduli spaces of twisted sheaves on a projective variety” in Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, 2006, 1–30.
  • [Y6] K. Yoshioka, Stability and the Fourier-Mukai transform, II, Compos. Math. 145 (2009), 112–142.
  • [Y7] K. Yoshioka, “An action of a Lie algebra on the homology groups of moduli spaces of stable sheaves” in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), Adv. Stud. Pure Math. 58, Math. Soc. Japan, 2010, 403–459.
  • [Y8] K. Yoshioka, Perverse coherent sheaves and Fourier-Mukai transforms on surfaces, I, Kyoto J. Math. 53 (2013), 261–344.