Duke Mathematical Journal

Derived equivalences of K3 surfaces and orientation

Daniel Huybrechts, Emanuele Macrì, and Paolo Stellari

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Every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms

Article information

Duke Math. J., Volume 149, Number 3 (2009), 461-507.

First available in Project Euclid: 24 August 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18E30: Derived categories, triangulated categories
Secondary: 14J28: $K3$ surfaces and Enriques surfaces


Huybrechts, Daniel; Macrì, Emanuele; Stellari, Paolo. Derived equivalences of K3 surfaces and orientation. Duke Math. J. 149 (2009), no. 3, 461--507. doi:10.1215/00127094-2009-043. https://projecteuclid.org/euclid.dmj/1251120010

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  • A. Beauville, J.-P. Bourguignon, and M. Demazure, Géométrie des surfaces K3: modules et périodes, Séminaires Palaiseau, Astérisque 126, Soc. Math. France, Montrouge, 1985.
  • C. Borcea, Diffeomorphisms of a K3 surface, Math. Ann. 275 (1986), 1--4.
  • T. Bridgeland, Stability conditions on K3 surfaces, Duke Math. J. 141 (2008), 241--291.
  • R.-O. Buchweitz and H. Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. 217 (2008), 205--242.
  • —, The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah Chern character, Adv. Math. 217 (2008), 243--281.
  • D. Calaque and M. Van Den Bergh, Hochschild cohomology and Atiyah classes, preprint,\arxiv0708.2725v4[math.KT]
  • A. CăLdăRaru, The Mukai pairing II: The Hochschild-Kostant-Rosenberg isomorphism, Adv. Math. 194 (2005), 34--66.
  • A. CăLdăRaru and S. Willerton, The Mukai pairing, I: A categorical approach, preprint,\arxiv0707.2052v1[math.AG]
  • V. Dolgushev, D. Tamarkin, and B. Tsygan, The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal, J. Noncommut. Geom. 1 (2007), 1--25.
  • —, Formality of the homotopy calculus algebra of Hochschild (co)chains, preprint,\arxiv0807.5117v1[math.KT]
  • S. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257--315.
  • A. Fujiki, ``On the de Rham cohomology group of a compact Kähler symplectic manifold'' in Algebraic Geometry (Sendai, Japan, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 105--165.
  • M. Gross, D. Huybrechts, and D. Joyce, Calabi-Yau manifolds and related geometries, Universitext, Springer, Berlin, 2003.
  • R. Hartshorne, Algebraic Geometry, Grad Texts in Math. 52, Springer, New York, 1977.
  • 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.
  • D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monogr., Oxford Univ. Press, New York, 2006.
  • D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects Math. E31, Vieweg, Braunschweig, 1997.
  • D. Huybrechts, E. Macr\`I, and P. Stellari, Stability conditions for generic K3 categories, Compos. Math. 144 (2008), 134--162.
  • —, Derived equivalences of K3 surfaces and orientation, preprint,\arxiv0710.1645v2[math.AG]
  • —, Formal deformations and their categorical general fibre, preprint,\arxiv0809.3201v1[math.AG]
  • D. Huybrechts and P. Stellari, Equivalences of twisted K3 surfaces, Math. Ann. 332 (2005), 901--936.
  • D. Huybrechts and R. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, to appear in Math. Ann., preprint,\arxiv0805.3527v1[math.AG]
  • L. Illusie, ``Grothendieck's existence theorem in formal geometry'' in Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, 2005, 179--233.
  • M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157--216.
  • M. Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), 175--206.
  • E. Macr\`I, M. Nieper-Wisskirchen, and P. Stellari, The module structure of Hochschild homology in some examples, C. R. Math. Acad. Sci. Paris 346 (2008), 863--866.
  • E. Macr\`I and P. Stellari, Infinitesimal derived Torelli theorem for K3 surfaces, with an appendix by S. Mehrotra, Int. Math. Res. Not. IMRN 2009, art. ID rnp 049.
  • N. Markarian, Poincaré-Birkhoff-Witt isomorphism, Hochschild homology and Riemann-Roch theorem, preprint, 2001.
  • 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.
  • D. Orlov, Equivalences of derived categories and K3 surfaces, J. Math. Sci. (N.Y.) 84 (1997), 1361--1381.
  • D. Ploog, Groups of autoequivalences of derived categories of smooth projective varieties, Ph.D. dissertation, Freie Universität, Berlin, 2005.
  • A. Ramadoss, The relative Riemann-Roch theorem from Hochschild homology, New York J. Math 14 (2008), 643--717.
  • B. SzendrőI, ``Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry'' in Applications of Algebraic Geometry to Coding Theory, Physics and Computation (Eilat, Israel, 2001), NATO Sci, Ser. II Math. Phys. Chem. 36, Kluwer, Dordrecht, 2001, 317--337.
  • Y. Toda, Deformations and Fourier-Mukai transforms, J. Differential Geom. 81 (2009), 197--224.
  • C. Weibel, Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 1655--1662.
  • A. Yekutieli, The continuous Hochschild cochain complex of a scheme, Canad. J. Math. 54 (2002), 1319--1337.