Duke Mathematical Journal

Derived automorphism groups of K3 surfaces of Picard rank $1$

Arend Bayer and Tom Bridgeland

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We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank $1$. We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible.

Article information

Duke Math. J. Volume 166, Number 1 (2017), 75-124.

Received: 14 May 2014
Revised: 5 February 2016
First available in Project Euclid: 14 September 2016

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Digital Object Identifier

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14J33: Mirror symmetry [See also 11G42, 53D37] 18E30: Derived categories, triangulated categories

derived category autoequivalences stability conditions K3 surfaces mirror symmetry


Bayer, Arend; Bridgeland, Tom. Derived automorphism groups of K3 surfaces of Picard rank 1 . Duke Math. J. 166 (2017), no. 1, 75--124. doi:10.1215/00127094-3674332. http://projecteuclid.org/euclid.dmj/1473854468.

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  • [1] D. Allcock, Completions, branched covers, Artin groups and singularity theory, Duke Math. J. 162 (2013), 2645–2689.
  • [2] A. Bayer and E. Macrî, The space of stability conditions on the local projective plane, Duke Math. J. 160 (2011), 263–322.
  • [3] A. Bayer and E. Macrî, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math. 198 (2014), 505–590.
  • [4] A. Bondal and D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math. 125 (2001), 327–344.
  • [5] C. Brav and H. Thomas, Braid groups and Kleinian singularities, Math. Ann. 351 (2011), 1005–1017.
  • [6] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–345.
  • [7] T. Bridgeland, Stability conditions on $K3$ surfaces, Duke Math. J. 141 (2008), 241–291.
  • [8] T. Bridgeland, “Spaces of stability conditions” in Algebraic Geometry—Seattle 2005, Part 1, Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, 2009, 1–21.
  • [9] N. Broomhead and D. Ploog, Autoequivalences of toric surfaces, Proc. Amer. Math. Soc. 142 (2014), 1133–1146.
  • [10] I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), 2599–2630.
  • [11] H. Hartmann, Cusps of the Kähler moduli space and stability conditions on K3 surfaces, Math. Ann. 354 (2012), 1–42.
  • [12] S. Hosono, B. H. Lian, K. Oguiso, and S.-T. Yau, Autoequivalences of derived category of a $K3$ surface and monodromy transformations, J. Algebraic Geom. 13 (2004), 513–545.
  • [13] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2006.
  • [14] D. Huybrechts, Stability conditions via spherical objects, Math. Z. 271 (2012), 1253–1270.
  • [15] D. Huybrechts, E. Macrî, and P. Stellari, Stability conditions for generic $K3$ categories, Compos. Math. 144 (2008), 134–162.
  • [16] D. Huybrechts, E. Macrî, and P. Stellari, Derived equivalences of $K3$ surfaces and orientation, Duke Math. J. 149 (2009), 461–507.
  • [17] A. Ishii, K. Ueda, and H. Uehara, Stability conditions on $A_{n}$-singularities, J. Differential Geom. 84 (2010), 87–126.
  • [18] K. Kawatani, A hyperbolic metric and stability conditions on K3 surfaces with $\rho=1$, preprint, arXiv:1204.1128v3 [math.AG].
  • [19] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435v1 [math.AG].
  • [20] S. A. Kuleshov, A theorem on the existence of exceptional bundles on surfaces of type $K3$ (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53, no. 2 (1989), 363–378; English translation in Math. USSR-Izv. 34 (1990), 373–388.
  • [21] E. Macrî and P. Stellari, Infinitesimal derived Torelli theorem for $K3$ surfaces, with an appendix by S. Mehrotra, Int. Math. Res. Not. IMRN 2009, no. 17, 3190–3220.
  • [22] 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.
  • [23] D. O. Orlov, Equivalences of derived categories and $K3$ surfaces, J. Math. Sci. (N.Y.) 84 (1997), 1361–1381.
  • [24] D. O. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 66, no. 3 (2002), 131–158; English translation in Izv. Math. 66, no. 3 (2002), 569–594.
  • [25] D. Ploog, Groups of autoequivalences of derived categories of smooth projective varieties, Ph.D. dissertation, Freie Universität Berlin, Berlin, 2005.
  • [26] Y. Qiu, Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math. 269 (2015), 220–264.
  • [27] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37–108.
  • [28] T. Sutherland, The modular curve as the space of stability conditions of a CY3 algebra, preprint, arXiv:1111.4184v1 [math.AG].
  • [29] B. Szendrői, “Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry” in Applications of Algebraic Geometry to Coding Theory, Physics and Computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem. 36, Kluwer, Dordrecht, 2001, 317–337.
  • [30] K. Yoshioka, Irreducibility of moduli spaces of vector bundles on K3 surfaces, preprint, arXiv:math/9907001 [math.AG].