Shifted symplectic structures
Publications Mathématiques de l'IHÉS, Tome 117 (2013), pp. 271-328.

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).

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     author = {Pantev, Tony and To\"en, Bertrand and Vaqui\'e, Michel and Vezzosi, Gabriele},
     title = {Shifted symplectic structures},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {271--328},
     publisher = {Springer-Verlag},
     volume = {117},
     year = {2013},
     doi = {10.1007/s10240-013-0054-1},
     zbl = {1328.14027},
     mrnumber = {3090262},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-013-0054-1/}
}
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Pantev, Tony; Toën, Bertrand; Vaquié, Michel; Vezzosi, Gabriele. Shifted symplectic structures. Publications Mathématiques de l'IHÉS, Tome 117 (2013), pp. 271-328. doi : 10.1007/s10240-013-0054-1. http://archive.numdam.org/articles/10.1007/s10240-013-0054-1/

[AKSZ] Alexandrov, M.; Kontsevich, M.; Schwarz, A.; Zaboronsky, O. The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A, Volume 12 (1997), pp. 1405-1429 | Article | MR 1432574 | Zbl 1073.81655

[Be] Behrend, K. Donaldson-Thomas type invariants via microlocal geometry, Ann. Math., Volume 170 (2009), pp. 1307-1338 | Article | MR 2600874 | Zbl 1191.14050

[Be-Fa] Behrend, K.; Fantechi, B. Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory, Volume 2 (2008), pp. 313-345 | Article | MR 2407118 | Zbl 1170.14004

[Ben-Nad] Ben-Zvi, D.; Nadler, D. Loop spaces and connections, J. Topol., Volume 5 (2012), pp. 377-430 | Article | MR 2928082 | Zbl 1246.14027

[Ber] Bergner, J. A survey of (∞,1)-categories, Towards Higher Categories (IMA Vol. Math. Appl.) (2010), pp. 69-83 | Article | MR 2664620 | Zbl 1200.18011

[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.

[Co] K. Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, preprint, October 2011, arXiv:1111.4234. | MR 3126501 | Zbl 1299.14013

[Co-Gw] K. Costello and O. Gwilliam, Factorization algebras in perturbative quantum field theory, preprint draft, 2011. | MR 3586504

[De-Ga] Demazure, M.; Gabriel, P. Introduction to Algebraic Geometry and Algebraic Groups, North Holland, Amsterdam, 1980 | MR 563524 | Zbl 0431.14015

[Fu] Fu, B. A survey on symplectic singularities and symplectic resolutions, Ann. Math. Blaise Pascal, Volume 13 (2006), pp. 209-236 | Article | MR 2275448 | Zbl 1116.14008

[Go] Goldman, W. The symplectic nature of fundamental groups of surfaces, Adv. Math., Volume 54 (1984), pp. 200-225 | Article | MR 762512 | Zbl 0574.32032

[Hu-Le] Huybrechts, D.; Lehn, M. The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, Vieweg, Braunschweig, 1997 | MR 1450870 | Zbl 0872.14002

[Il] Illusie, L. Complexe cotangent et déformations I, Lecture Notes in Mathematics, Springer, Berlin, 1971 | Article | MR 491680 | Zbl 0224.13014

[In] Inaba, M.-A. Smoothness of the moduli space of complexes of coherent sheaves on an Abelian or a projective K3 surface, Adv. Math., Volume 227 (2011), pp. 1399-1412 | Article | MR 2799799 | Zbl 1220.14010

[In-Iw-Sa] Inaba, M.-A.; Iwasaki, K.; Saito, M.-H. Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Publ. Res. Inst. Math. Sci., Volume 42 (2006), pp. 987-1089 | Article | MR 2289083 | Zbl 1127.34055

[Je] Jeffrey, L. Symplectic forms on moduli spaces of flat connections on 2-manifolds, Geometric Topology (Athens, GA, 1993) (AMS/IP Stud. Adv. Math.) (1997), pp. 268-281 | MR 1470732 | Zbl 0904.57009

[Kal] Kaledin, D. On crepant resolutions of symplectic quotient singularities, Sel. Math. New Ser., Volume 9 (2003), pp. 529-555 | Article | MR 2031751 | Zbl 1066.14003

[Kal-Le-So] Kaledin, D.; Lehn, M.; Sorger, Ch. Singular symplectic moduli spaces, Invent. Math., Volume 164 (2006), pp. 591-614 | Article | MR 2221132 | Zbl 1096.14037

[Ka] Kassel, C. Cyclic homology, comodules and mixed complexes, J. Algebra, Volume 107 (1987), pp. 195-216 | Article | MR 883882 | Zbl 0617.16015

[Ke-Lo] Keller, B.; Lowen, W. On Hochschild cohomology and Morita deformations, Int. Math. Res. Not., Volume 2009 (2009), pp. 3221-3235 | MR 2534996 | Zbl 1221.18014

[Ko-So] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, 0811.2435, 2008.

[Ku-Ma] Kuznetsov, A.; Markushevich, D. Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys., Volume 59 (2009), pp. 843-860 | Article | MR 2536849 | Zbl 1181.14049

[La] V. Lafforgue, Quelques calculs reliés à la correspondance de Langlands géométrique sur P 1, http://www.math.jussieu.fr/~vlafforg/.

[Lu1] Lurie, J. Higher Topos Theory, Annals of Mathematics Studies, Princeton University Press, Princeton, 2009 (xviii+925 pp) | MR 2522659 | Zbl 1175.18001

[Lu2] Lurie, J. Formal moduli problems, Proceedings of the International Congress of Mathematicians 2010 (2) (2010)

[Lu3] J. Lurie, DAG V, IX, http://www.math.harvard.edu/~lurie/.

[Lu5] J. Lurie, Higher algebra, http://www.math.harvard.edu/~lurie/.

[Mu] Mukai, S. Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., Volume 77 (1984), pp. 101-116 | Article | MR 751133 | Zbl 0565.14002

[Na] Y. Namikawa, Equivalence of symplectic singularities, 1102.0865, 2011. | MR 3079311 | Zbl 1277.32029

[Ne-McG] T. Nevins and K. McGerty, Derived equivalence for quantum symplectic resolutions, 1108.6267, 2011.

[Pa-Th] R. Pandharipande and R. Thomas, Almost closed 1-forms, 1204.3958, April 2012. | MR 3137857 | Zbl 1287.13009

[Pe] J. Pecharich, The derived Marsden-Weinstein quotient is symplectic, in preparation.

[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. | MR 3341464

[Si1] Simpson, C. Algebraic aspects of higher nonabelian Hodge theory, Motives, Polylogarithms and Hodge Theory, Part II (Int. Press Lect. Ser.) (2002), pp. 417-604 | MR 1978713 | Zbl 1051.14008

[Si2] Simpson, C. Geometricity of the Hodge filtration on the ∞-stack of perfect complexes over X DR , Mosc. Math. J., Volume 9 (2009), pp. 665-721 | MR 2562796 | Zbl 1189.14020

[Si3] Simpson, C. Homotopy Theory of Higher Categories, Cambridge University Press, Cambridge, 2011 | Article | MR 2883823

[To1] Toën, B. Derived Azumaya algebras and generators for twisted derived categories, Inv. Math., Volume 189 (2012), pp. 581-652 | Article | MR 2957304 | Zbl 06084026

[To2] Toën, B. Higher and derived stacks: a global overview, Algebraic Geometry—Seattle 2005. Part 1 (Proc. Symp. Pure Math.) (2009), pp. 435-487 | MR 2483943 | Zbl 1183.14001

[To3] Toën, B. Champs affines, Sel. Math. New Ser., Volume 12 (2006), pp. 39-135 | Article | MR 2244263 | Zbl 1108.14004

[To-Va] Toën, B.; Vaquié, M. Moduli of objects in dg-categories, Ann. Sci. Éc. Norm. Super., Volume 40 (2007), pp. 387-444 | Numdam | MR 2493386 | Zbl 1140.18005

[To-Va-Ve] B. Toën, M. Vaquié, and G. Vezzosi, Deformation theory of dg-categories revisited, in preparation.

[To-Ve-1] Toën, B.; Vezzosi, G. Homotopical Algebraic Geometry II: Geometric Stacks and Applications, Mem. Am. Math. Soc., 2008 (no. 902, x+224 pp) | MR 2394633 | Zbl 1145.14003

[To-Ve-2] B. Toën and G. Vezzosi, Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, 0903.3292, version of February 2011.

[To-Ve-3] Toën, B.; Vezzosi, G. Algèbres simpliciales S 1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs, Compos. Math., Volume 147 (2011), pp. 1979-2000 | Article | MR 2862069 | Zbl 1257.18014

[Ve] G. Vezzosi, Derived critical loci I—Basics, 1109.5213, 2011.

[Viz] Vizman, C. Induced differential forms on manifolds of functions, Arch. Math., Volume 47 (2011), pp. 201-215 | MR 2852381 | Zbl 1249.58005

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