Duke Mathematical Journal

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

Arend Bayer and Tom Bridgeland

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

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1473854468

Digital Object Identifier
doi:10.1215/00127094-3674332

Subjects
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

Keywords
derived category autoequivalences stability conditions K3 surfaces mirror symmetry

Citation

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. https://projecteuclid.org/euclid.dmj/1473854468.


Export citation

References

  • [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].