Deformation quantization modules - Tome (2012) no. 345

Deformation quantization modules

Kashiwara, Masaki ; Schapira, Pierre

Astérisque, no. 345 (2012) , 159 p.

@book{AST_2012__345__R1_0,
     author = {Kashiwara, Masaki and Schapira, Pierre},
     title = {Deformation quantization modules},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {345},
     year = {2012},
     mrnumber = {3012169},
     zbl = {1260.32001},
     language = {en},
     url = {http://www.numdam.org/item/AST_2012__345__R1_0/}
}

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%A Kashiwara, Masaki
%A Schapira, Pierre
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%S Astérisque
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Kashiwara, Masaki; Schapira, Pierre. Deformation quantization modules. Astérisque, no. 345 (2012), 159 p. http://numdam.org/item/AST_2012__345__R1_0/

Bibliographie
Cité par

[1] M. Artin, A. Grothendieck J.-L. Verdier (eds.) - Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Math., vol. 270, Springer, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4). Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. | Zbl

[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz et D. Sternheimer - "Deformation theory and quantization. I. Deformations of symplectic structures", Ann. Physics 111 (1978), p. 61-110. | MR | Zbl | DOI

[3] O. Ben-Bassat, J. Block et T. Pantev - "Non-commutative tori and Fourier-Mukai duality", Compos. Math. 143 (2007), p. 423-475. | MR | Zbl | DOI

[4] F. A. Berezin - "Quantization", Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), p. 1116-1175. | MR | Zbl

[5] F. A. Berezin, "General concept of quantization", Comm. Math. Phys. 40 (1975), p. 153-174. | MR | Zbl | DOI

[6] M. Van Den Bergh - "A relation between Hochschild homology and cohomology for Gorenstein rings", Proc. Amer. Math. Soc. 126 (1998), p. 1345-1348. | MR | Zbl | DOI

[7] M. Van Den Bergh - "On global deformation quantization in the algebraic case", J. Algebra 315 (2007), p. 326-395. | MR | Zbl | DOI

[8] R. Bezrukavnikov et D. Kaledin - "Fedosov quantization in algebraic context", Mosc. Math. J. 4 (2004), p. 559-592, 782. | MR | Zbl | DOI

[9] L. Boutet De Monvel - "Complex star algebras", Math. Phys. Anal. Geom. 2 (1999), p. 113-139. | MR | Zbl | DOI

[10] L. Breen - "On the classification of $2$ -gerbes and $2$ -stacks", Astérisque 225 (1994). | MR | Zbl | Numdam

[11] P. Bressler, A. Gorokhovsky, R. Nest et B. Tsygan - "Algebraic index theorem for symplectic deformations of gerbes", in Noncommutative geometry and global analysis, Contemp. Math., vol. 546, Amer. Math. Soc., 2011, p. 23-38. | MR | Zbl | DOI

[12] P. Bressler, A. Gorokhovsky, R. Nest et B. L. Tsygan - "Deformation quantization of gerbes", Adv. Math. 214 (2007), p. 230-266. | MR | Zbl | DOI

[13] P. Bressler, A. Gorokhovsky, R. Nest et B. L. Tsygan, "Chern character for twisted complexes", in Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, 2008, p. 309-324. | MR | Zbl

[14] P. Bressler, R. Nest et B. L. Tsygan - "Riemann-Roch theorems via deformation quantization. I, II", Adv. Math. 167 (2002), p. 1-25, 26-73. | MR | Zbl | DOI

[15] J.-L. Brylinski et E. Getzler - "The homology of algebras of pseudodifferential symbols and the noncommutative residue", $K$ -Theory 1 (1987), p. 385-403. | MR | Zbl | DOI

[16] D. Calaque et G. Halbout - "Weak quantization of Poisson structures", J. Geom. Phys. 61 (2011), p. 1401-1414. | MR | Zbl | DOI

[17] A. Căldăraru - "The Mukai pairing. II. The Hochschild-Kostant-Rosenberg isomorphism", Adv. Math. 194 (2005), p. 34-66. | MR | Zbl | DOI

[18] A. Căldăraru et S. Willerton - "The Mukai pairing. I. A categorical approach", New York J. Math. 16 (2010), p. 61-98. | MR | Zbl | EuDML

[19] H.-Y. Chen - "GAGA for $D Q$ -algebroids", Rend. Semin. Mat. Univ. Padova 123 (2010), p. 211-231. | MR | Zbl | EuDML | Numdam | DOI

[20] A. D'Agnolo, S. Guillermou et P. Schapira - "Regular holonomic $𝒟 [[ℏ]]$ -modules", Publ. Res. Inst. Math. Sci. 47 (2011), p. 221-255. | MR | Zbl

[21] A. D'Agnolo et P. Polesello - "Deformation quantization of complex involutive submanifolds", in Noncommutative geometry and physics, World Sci. Publ., Hackensack, NJ, 2005, p. 127-137. | MR

[22] A. D'Agnolo et P. Schapira - "Quantization of complex Lagrangian submanifolds", Adv. Math. 213 (2007), p. 358-379. | MR | Zbl | DOI

[23] V. Dolgushev - "The Van den Bergh duality and the modular symmetry of a Poisson variety", Selecta Math. (N.S.) 14 (2009), p. 199-228. | MR | Zbl | DOI

[24] V. Dolgushev et V. N. Rubtsov - "An algebraic index theorem for Poisson manifolds", J. reine angew. Math. 633 (2009), p. 77-113. | MR | Zbl

[25] B. Fedosov - Deformation quantization and index theory, Mathematical Topics, vol. 9, Akademie Verlag, 1996. | MR | Zbl

[26] B. L. Feigin, A. Losev et B. Shoikhet - "Riemann-Roch-Hirzebruch theorem and topological quantum mechanics", preprint arXiv:math.QA/0401400.

[27] B. L. Feigin et B. L. Tsygan - "Riemann-Roch theorem and Lie algebra cohomology. I", in Proceedings of the Winter School on Geometry and Physics (Srní, 1988), vol. 21, 1989, p. 15-52. | MR | Zbl | EuDML

[28] O. Gabber - "The integrability of the characteristic variety", Amer. J. Math. 103 (1981), p. 445-468. | MR | Zbl | DOI

[29] J. Giraud - Cohomologie non abélienne, Die Grund. Math. Wiss., vol. 179, Springer, 1971. | MR | Zbl

[30] R. Godement - Topologie algébrique et théorie des faisceaux, Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, 1958. | MR | Zbl

[31] H. Grauert - "Ein Theorem der analytischen Garbentheorie und die Modulräume complexer Strukturen", Publ. IHÉS 5 (1960). | MR | EuDML | Numdam

[32] J. Grivaux - "On a conjecture of Kashiwara relating Chern and Euler classes of $𝒪$ -modules-modules", preprint arXiv:0910.5384. | MR | Zbl

[33] A. Grothendieck - "Éléments de Géométrie Algébrique III, Étude cohomologique des faisceaux cohérents I", Publ. IHÉS 11 (1961). | MR | Numdam

[34] C. Houzel - "Espaces analytiques relatifs et théorème de finitude", Math. Ann. 205 (1973), p. 13-54. | MR | Zbl | EuDML | DOI

[35] D. Huybrechts - Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 2006. | Zbl | MR

[36] M. Kashiwara - "On the maximally overdetermined system of linear differential equations. I", Publ. Res. Inst. Math. Sci. 10 (1974/75), p. 563-579. | MR | Zbl | DOI

[37] M. Kashiwara, "Quantization of contact manifolds", Publ. Res. Inst. Math. Sci. 32 (1996), p. 1-7. | MR | Zbl | DOI

[38] M. Kashiwara, $D$ -modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217, Amer. Math. Soc., 2003. | MR | Zbl

[39] M. Kashiwara, Letter to Pierre Schapira, 18/11/1991.

[40] M. Kashiwara et T. Ōshima - "Systems of differential equations with regular singularities and their boundary value problems", Ann. Math. 106 (1977), p. 145-200. | MR | Zbl | DOI

[41] M. Kashiwara et P. Schapira - Sheaves on manifolds, Grund. Math. Wiss., vol. 292, Springer, 1990. | MR | Zbl

[42] M. Kashiwara et P. Schapira, "Moderate and formal cohomology associated with constructible sheaves", Mém. Soc. Math. France (N.S.) 64 (1996). | MR | Zbl | EuDML | Numdam

[43] M. Kashiwara et P. Schapira, Categories and sheaves, Grund. Math. Wiss., vol. 332, Springer, 2006. | MR | Zbl

[44] M. Kashiwara et P. Schapira, "Constructibility and duality for simple holonomic modules on complex symplectic manifolds", Amer. J. Math. 130 (2008), p. 207-237. | MR | Zbl | DOI

[45] M. Kashiwara et P. Schapira, "Modules over deformation quantization algebroids : an overview", Lett. Math. Phys. 88 (2009), p. 79-99. | MR | Zbl | DOI

[46] M. Kashiwara et P. Schapira,"Deformation quantization modules", preprint arXiv: 1003.3304. | MR | Zbl | Numdam

[47] M. Kontsevich - "Deformation quantization of algebraic varieties", Lett. Math. Phys. 56 (2001), p. 271-294. | MR | Zbl | DOI

[48] M. Kontsevich, "Deformation quantization of Poisson manifolds", Lett. Math. Phys. 66 (2003), p. 157-216. | MR | Zbl | DOI

[49] G. Laumon - "Sur la catégorie dérivée des $𝒟$ -modules filtrés", in Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, 1983, p. 151-237. | MR | Zbl | DOI

[50] N. Markarian - "The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem", J. Lond. Math. Soc. 79 (2009), p. 129-143. | MR | Zbl | DOI

[51] J. P. May - "The additivity of traces in triangulated categories", Adv. Math. 163 (2001), p. 34-73. | MR | Zbl | DOI

[52] P. Polesello - "Classification of deformation quantization algebroids on complex symplectic manifolds", Publ. Res. Inst. Math. Sci. 44 (2008), p. 725-748. | MR | Zbl | DOI

[53] P. Polesello et P. Schapira - "Stacks of quantization-deformation modules on complex symplectic manifolds", Int. Math. Res. Not. 2004 (2004), p. 2637-2664. | MR | Zbl | DOI

[54] D. Popescu - "On a question of Quillen", Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 45(93) (2002), p. 209-212. | MR | Zbl

[55] A. C. Ramadoss - "The relative Riemann-Roch theorem from Hochschild homology", New York J. Math. 14 (2008), p. 643-717. | MR | Zbl | EuDML

[56] M. Sato, T. Kawai et M. Kashiwara - "Microfunctions and pseudo-differential equations", in Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971 ; dedicated to the memory of André Martineau), Springer, 1973, p. 265-529. Lecture Notes in Math., Vol. 287. | MR | Zbl

[57] P. Schapira - Microdifferential systems in the complex domain, Grund. Math. Wiss., vol. 269, Springer, 1985. | MR | Zbl

[58] P. Schapira et J.-P. Schneiders - "Elliptic pairs. II. Euler class and relative index theorem", Astérisque 224 (1994), p. 61-98. | MR | Zbl | Numdam

[59] D. Shklyarov - "Hirzebruch-Riemann-Roch theorem for DG algebras", preprint arXiv:math/0710.1937. | Zbl

[60] R. G. Swan - "Néron-Popescu desingularization", in Algebra and geometry (Taipei, 1995), Lect. Algebra Geom., vol. 2, Int. Press, Cambridge, MA, 1998, p. 135-192. | MR | Zbl

[61] M. Wodzicki - "Cyclic homology of pseudodifferential operators and noncommutative Euler class", C. R. Acad. Sci. Paris Ser. I Math. 306 (1988), p. 321-325. | MR | Zbl

[62] A. Yekutieli - "The continuous Hochschild cochain complex of a scheme", Canad. J. Math. 54 (2002), p. 1319-1337. | MR | Zbl | DOI

[63] A. Yekutieli, "Deformation quantization in algebraic geometry", Adv. Math. 198 (2005), p. 383-432 | MR | Zbl | DOI

A. Yekutieli, "Deformation quantization in algebraic geometry", erratum : Adv. Math. 217 (2008), 2897-2906. | MR | Zbl

[64] A. Yekutieli, "Twisted deformation quantization of algebraic varieties", preprint arXiv:math.AG/0905.0488. | MR | Zbl | DOI