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Publications mathématiques de l'IHÉS

, Volume 117, Issue 1, pp 271–328 | Cite as

Shifted symplectic structures

  • Tony Pantev
  • Bertrand Toën
  • Michel Vaquié
  • Gabriele Vezzosi
Article

Abstract

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see Toën and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toën in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (nd)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π 1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E,T X), for E an elliptic curve and T X is the total space of the cotangent bundle of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π 1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).

Keywords

Modulus Space Symplectic Form Symplectic Structure Mixed Complex Chern Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AKSZ]
    M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A, 12 (1997), 1405–1429. MathSciNetCrossRefMATHGoogle Scholar
  2. [Be]
    K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. Math., 170 (2009), 1307–1338. MathSciNetCrossRefMATHGoogle Scholar
  3. [Be-Fa]
    K. Behrend and B. Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory, 2 (2008), 313–345. MathSciNetCrossRefMATHGoogle Scholar
  4. [Ben-Nad]
    D. Ben-Zvi and D. Nadler, Loop spaces and connections, J. Topol., 5 (2012), 377–430. MathSciNetCrossRefMATHGoogle Scholar
  5. [Ber]
    J. Bergner, A survey of (∞,1)-categories, in Towards Higher Categories, IMA Vol. Math. Appl., vol. 152, pp. 69–83, Springer, New York, 2010. CrossRefGoogle Scholar
  6. [Br-Bu-Du-Jo]
    C. Brav, V. Bussi, D. Dupont, and D. Joyce, Shifted symplectic structures on derived schemes and critical loci, preprint, May 2012. Google Scholar
  7. [Co]
    K. Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, preprint, October 2011, arXiv:1111.4234.
  8. [Co-Gw]
    K. Costello and O. Gwilliam, Factorization algebras in perturbative quantum field theory, preprint draft, 2011. Google Scholar
  9. [De-Ga]
    M. Demazure and P. Gabriel, Introduction to Algebraic Geometry and Algebraic Groups, North Holland, Amsterdam, 1980. MATHGoogle Scholar
  10. [Fu]
    B. Fu, A survey on symplectic singularities and symplectic resolutions, Ann. Math. Blaise Pascal, 13 (2006), 209–236. MathSciNetCrossRefMATHGoogle Scholar
  11. [Go]
    W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), 200–225. CrossRefMATHGoogle Scholar
  12. [Hu-Le]
    D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, vol. 31, Vieweg, Braunschweig, 1997. MATHGoogle Scholar
  13. [Il]
    L. Illusie, Complexe cotangent et déformations I, Lecture Notes in Mathematics, vol. 239, Springer, Berlin, 1971. CrossRefMATHGoogle Scholar
  14. [In]
    M.-A. Inaba, Smoothness of the moduli space of complexes of coherent sheaves on an Abelian or a projective K3 surface, Adv. Math., 227 (2011), 1399–1412. MathSciNetCrossRefMATHGoogle Scholar
  15. [In-Iw-Sa]
    M.-A. Inaba, K. Iwasaki, and M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Publ. Res. Inst. Math. Sci., 42 (2006), 987–1089. MathSciNetCrossRefMATHGoogle Scholar
  16. [Je]
    L. Jeffrey, Symplectic forms on moduli spaces of flat connections on 2-manifolds, in Geometric Topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2.1, pp. 268–281, Am. Math. Soc., Providence, 1997. Google Scholar
  17. [Kal]
    D. Kaledin, On crepant resolutions of symplectic quotient singularities, Sel. Math. New Ser., 9 (2003), 529–555. MathSciNetCrossRefMATHGoogle Scholar
  18. [Kal-Le-So]
    D. Kaledin, M. Lehn, and Ch. Sorger, Singular symplectic moduli spaces, Invent. Math., 164 (2006), 591–614. MathSciNetCrossRefMATHGoogle Scholar
  19. [Ka]
    C. Kassel, Cyclic homology, comodules and mixed complexes, J. Algebra, 107 (1987), 195–216. MathSciNetCrossRefMATHGoogle Scholar
  20. [Ke-Lo]
    B. Keller and W. Lowen, On Hochschild cohomology and Morita deformations, Int. Math. Res. Not., 2009 (2009), 3221–3235. MathSciNetMATHGoogle Scholar
  21. [Ko-So]
    M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, 0811.2435, 2008.
  22. [Ku-Ma]
    A. Kuznetsov and D. Markushevich, Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys., 59 (2009), 843–860. MathSciNetCrossRefMATHGoogle Scholar
  23. [La]
    V. Lafforgue, Quelques calculs reliés à la correspondance de Langlands géométrique sur P 1, http://www.math.jussieu.fr/~vlafforg/.
  24. [Lu1]
    J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, 2009, xviii+925 pp. MATHGoogle Scholar
  25. [Lu2]
    J. Lurie, Formal moduli problems, in R. Bhatia, A. Pal, G. Rangarajan, V. Srinivas, and M. Vanninathan (eds.), Proceedings of the International Congress of Mathematicians 2010, Vol. 2, World Scientific, Singapore, 2010. Google Scholar
  26. [Lu3]
    J. Lurie, DAG V, IX, http://www.math.harvard.edu/~lurie/.
  27. [Lu5]
    J. Lurie, Higher algebra, http://www.math.harvard.edu/~lurie/.
  28. [Mu]
    S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., 77 (1984), 101–116. MathSciNetCrossRefMATHGoogle Scholar
  29. [Na]
    Y. Namikawa, Equivalence of symplectic singularities, 1102.0865, 2011.
  30. [Ne-McG]
    T. Nevins and K. McGerty, Derived equivalence for quantum symplectic resolutions, 1108.6267, 2011.
  31. [Pa-Th]
    R. Pandharipande and R. Thomas, Almost closed 1-forms, 1204.3958, April 2012.
  32. [Pe]
    J. Pecharich, The derived Marsden-Weinstein quotient is symplectic, in preparation. Google Scholar
  33. [Sch-To-Ve]
    T. Schürg, B. Toën, and G. Vezzosi, Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes, J. Reine Angew. Math., to appear. Google Scholar
  34. [Si1]
    C. Simpson, Algebraic aspects of higher nonabelian Hodge theory, in Motives, Polylogarithms and Hodge Theory, Part II (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, pp. 417–604, International Press, Somerville, 2002. Google Scholar
  35. [Si2]
    C. Simpson, Geometricity of the Hodge filtration on the ∞-stack of perfect complexes over X DR, Mosc. Math. J. 9 (2009), 665–721. MathSciNetMATHGoogle Scholar
  36. [Si3]
    C. Simpson, Homotopy Theory of Higher Categories, Cambridge University Press, Cambridge, 2011. CrossRefGoogle Scholar
  37. [To1]
    B. Toën, Derived Azumaya algebras and generators for twisted derived categories, Inv. Math., 189 (2012), 581–652. CrossRefMATHGoogle Scholar
  38. [To2]
    B. Toën, Higher and derived stacks: a global overview, in Algebraic Geometry—Seattle 2005. Part 1, Proc. Symp. Pure Math., vol. 80, pp. 435–487, Am. Math. Soc., Providence, 2009. Google Scholar
  39. [To3]
    B. Toën, Champs affines, Sel. Math. New Ser., 12 (2006), 39–135. CrossRefMATHGoogle Scholar
  40. [To-Va]
    B. Toën and M. Vaquié, Moduli of objects in dg-categories, Ann. Sci. Éc. Norm. Super. 40 (2007), 387–444. MATHGoogle Scholar
  41. [To-Va-Ve]
    B. Toën, M. Vaquié, and G. Vezzosi, Deformation theory of dg-categories revisited, in preparation. Google Scholar
  42. [To-Ve-1]
    B. Toën and G. Vezzosi, Homotopical Algebraic Geometry II: Geometric Stacks and Applications, Mem. Am. Math. Soc., vol. 193, 2008, no. 902, x+224 pp. Google Scholar
  43. [To-Ve-2]
    B. Toën and G. Vezzosi, Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, 0903.3292, version of February 2011.
  44. [To-Ve-3]
    B. Toën and G. Vezzosi, Algèbres simpliciales S 1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs, Compos. Math., 147 (2011), 1979–2000. MathSciNetCrossRefMATHGoogle Scholar
  45. [Ve]
    G. Vezzosi, Derived critical loci I—Basics, 1109.5213, 2011.
  46. [Viz]
    C. Vizman, Induced differential forms on manifolds of functions, Arch. Math., 47 (2011), 201–215. MathSciNetMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tony Pantev
    • 1
  • Bertrand Toën
    • 2
  • Michel Vaquié
    • 3
  • Gabriele Vezzosi
    • 4
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Institut de Mathématiques et de Modélisations de MontpellierUniversité de Montpellier2MontpellierFrance
  3. 3.Institut de Math. de ToulouseUniversité Paul SabatierToulouseFrance
  4. 4.Institut de Mathématiques de JussieuUniversité Paris DiderotParisFrance

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