Catégories dérivées et géométrie birationnelle  [ Derived categories and birational geometry ]
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 946, p. 283-307

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance and of minimization of the derived category have appeared, inspired by Kontsevich's homological mirror symmetry conjecture and Mori's minimal model program. We present the main conjectures and their proofs in dimension 3 and for particular classes of flops.

À l’origine conçue comme un outil technique, la catégorie dérivée des faisceaux cohérents d’une variété algébrique est apparue lors de ces dix dernières années comme un invariant important dans l’étude birationnelle des variétés algébriques. Des problèmes d’invariance birationnelle et de minimisation de la catégorie dérivée sont apparus, inspirés par la conjecture homologique de symétrie miroir de Kontsevich et le programme de Mori de modèles minimaux pour les variétés algébriques. Nous présenterons les conjectures générales et leur preuve en dimension 3 et pour des flops particuliers.

Classification:  14Exx,  14Jxx,  18Exx
Keywords: derived category, triangulated category, Calabi-Yau variety, flop
@incollection{SB_2004-2005__47__283_0,
     author = {Rouquier, Rapha\"el},
     title = {Cat\'egories d\'eriv\'ees et g\'eom\'etrie birationnelle},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:946},
     pages = {283-307},
     zbl = {1123.14009},
     mrnumber = {2296422},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__283_0}
}
Rouquier, Raphaël. Catégories dérivées et géométrie birationnelle, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 946, pp. 283-307. http://www.numdam.org/item/SB_2004-2005__47__283_0/

[1] D. Abramovich & J. C. Chen - “Computations with moduli of perverse point sheaves”, preprint arXiv : math.AG/0304353.

[2] -, “Flops, flips and perverse point sheaves on threefold stacks”, preprint arXiv : math.AG/0304354. | Zbl 1087.14010

[3] P. Balmer - “Presheaves of triangulated categories and reconstruction of schemes”, Math. Ann. 324 (2002), p. 557-580. | Article | MR 1938458 | Zbl 1011.18007

[4] A. A. Beilinson - “The derived category of coherent sheaves on P n , Selecta Math. Soviet. 3 (1983/84), p. 233-237, ou (en russe) Funktsional. Anal. i Prilozhen. 12 (1978), p. 68-69. | MR 863137 | Zbl 0545.14012

[5] A. Bondal & M. M. Kapranov - “Representable functors, Serre functors, and mutations”, Math. USSR-Izv. 35 (1990), p. 519-541. | MR 1039961 | Zbl 0703.14011

[6] A. Bondal & D. Orlov - “Semiorthogonal decompositions for algebraic varieties”, preprint arXiv : alg-geom/9506012.

[7] -, “Reconstruction of a variety from the derived category and groups of autoequivalences”, Compositio Math. 125 (2001), p. 327-344. | Article | MR 1818984 | Zbl 0994.18007

[8] -, “Derived categories of coherent sheaves”, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, 2002, p. 47-56. | MR 1957019 | Zbl 0996.18007

[9] A. Bondal & M. Van Den Bergh - “Generators and representability of functors in commutative and noncommutative geometry”, Moscow Math. J. 3 (2003), p. 1-36. | Article | MR 1996800 | Zbl 1135.18302

[10] T. Bridgeland - “Fourier-Mukai transforms for elliptic surfaces”, J. reine angew. Math. 498 (1998), p. 115-133. | MR 1629929 | Zbl 0905.14020

[11] -, “Equivalences of triangulated categories and Fourier-Mukai transforms”, Bull. London Math. Soc. 31 (1999), p. 25-34. | Article | MR 1651025 | Zbl 0937.18012

[12] -, “Flops and derived categories”, Invent. Math. 147 (2002), p. 613-632. | Article | MR 1893007 | Zbl 1085.14017

[13] T. Bridgeland, A. King & M. Reid - “The McKay correspondence as an equivalence of derived categories”, J. Amer. Math. Soc. 14 (2001), p. 535-554. | Article | MR 1824990 | Zbl 0966.14028

[14] T. Bridgeland & A. Maciocia - “Complex surfaces with equivalent derived categories”, Math. Z. 236 (2001), p. 677-697. | Article | MR 1827500 | Zbl 1081.14023

[15] -, “Fourier-Mukai transforms for K3 and elliptic fibrations”, J. Algebraic Geom. 11 (2002), p. 629-657. | Article | MR 1910263 | Zbl 1066.14047

[16] A. Căldăraru - “Derived categories of sheaves : a skimming”, preprint arXiv : math.AG/0501094. | Article | MR 2182889

[17] -, “The Mukai pairing, I : the Hochschild structure”, preprint arXiv : math.AG/0308079(v2).

[18] -, “The Mukai pairing, II : the Hochschild-Kostant-Rosenberg isomorphism”, preprint arXiv : math.AG/0308080(v3). | Zbl 1098.14011

[19] J.-C. Chen - “Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities”, J. Differential Geom. 61 (2002), p. 227-261. | Article | MR 1972146 | Zbl 1090.14003

[20] K. Cho, Y. Miyaoka & N. I. Shepherd-Barron - “Characterizations of projective space and applications to complex symplectic manifolds”, in Higher dimensional birational geometry (Kyoto, 1997), Math. Soc. Japan, 2002, p. 1-88. | MR 1929792 | Zbl 1063.14065

[21] J. Chuang & R. Rouquier - “Derived equivalences for symmetric groups and 𝔰𝔩 2 -categorification”, preprint arXiv : math.RT/0407205. | MR 2373155 | Zbl 1144.20001

[22] H. Clemens, J. Kollár & S. Mori - Higher-dimensional complex geometry, Astérisque, vol. 166, Société Mathématique de France, Paris, 1988. | Numdam | MR 1004926 | Zbl 0689.14016

[23] P. Gabriel - “Des catégories abéliennes”, Bull. Soc. math. France 90 (1962), p. 323-448. | Numdam | Zbl 0201.35602

[24] A. L. Gorodentsev & S. A. Kuleshov - “Helix theory”, Moscow Math. J. 4 (2004), p. 377-440, 535. | MR 2108443 | Zbl 1072.14020

[25] L. Hille & M. Van Den Bergh - “Fourier-Mukai transforms”, preprint arXiv : math.AG/0402043(v2). | Article | MR 2384610

[26] D. Huybrechts - “Fourier-Mukai transforms in algebraic geometry”, livre en préparation. | MR 2244106 | Zbl 1095.14002

[27] M. M. Kapranov - “On the derived categories of coherent sheaves on some homogeneous spaces”, Invent. Math. 92 (1988), p. 479-508. | Article | MR 939472 | Zbl 0651.18008

[28] M. Kashiwara & P. Schapira - Sheaves on manifolds, Springer-Verlag, 1990. | Article | MR 1074006 | Zbl 0709.18001

[29] Y. Kawamata - “Derived Categories of Toric Varieties”, preprint arXiv : math.AG/0503102. | Article | MR 2280493 | Zbl 1159.14026

[30] -, “Derived equivalence for stratified Mukai flop on G(2,4), preprint arXiv : math.AG/0503101. | MR 2282964 | Zbl 1137.14305

[31] -, D-equivalence and K-equivalence”, J. Differential Geom. 61 (2002), p. 147-171. | Article | MR 1949787 | Zbl 1056.14021

[32] -, “Francia's flip and derived categories”, in Algebraic geometry, de Gruyter, 2002, p. 197-215. | MR 1954065 | Zbl 1092.14023

[33] -, “Equivalences of derived categories of sheaves on smooth stacks”, Amer. J. Math. 126 (2004), p. 1057-1083. | Article | MR 2089082 | Zbl 1076.14023

[34] A. King - “Tilting bundles on some rational surfaces”, preprint http://www.maths.bath.ac.uk/~masadk/papers/tilt.ps, 1997.

[35] M. Kontsevich - “Homological algebra of mirror symmetry”, in Proceedings of the International Congress of Mathematicians, Vol. 1 (Zürich, 1994), Birkhäuser, 1995, p. 120-139. | MR 1403918 | Zbl 0846.53021

[36] -, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys. 66 (2003), p. 157-216. | Article | MR 2062626 | Zbl 1058.53065

[37] A. Kuznetsov - “Derived category of V 12 Fano threefolds”, preprint arXiv : math.AG/0310008. | Zbl 1111.14038

[38] -, “Derived category of a cubic threefold and the variety V 14 , Trudy Mat. Inst. Steklov. 246 (2004), p. 183-207. | MR 2101293

[39] E. Looijenga - “Motivic measures”, in Séminaire Bourbaki (1999/2000), Astérisque, vol. 276, Société Mathématique de France, Paris, 2002, Exp. no 874, p. 267-297. | Numdam | MR 1886763 | Zbl 0996.14011

[40] E. Markman - “Brill-Noether duality for moduli spaces of sheaves on K3 surfaces”, J. Algebraic Geom. 10 (2001), p. 623-694. | MR 1838974 | Zbl 1074.14525

[41] S. Mukai - “Duality between D(X) and D(X ^) with its application to Picard sheaves”, Nagoya Math. J. 81 (1981), p. 153-175. | Article | MR 607081 | Zbl 0417.14036

[42] -, “On the moduli space of bundles on K3 surfaces. I”, in Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res., 1987, p. 341-413. | MR 893604 | Zbl 0674.14023

[43] Y. Namikawa - “Mukai flops and derived categories”, J. reine angew. Math. 560 (2003), p. 65-76. | MR 1992802 | Zbl 1033.18008

[44] -, “Mukai flops and derived categories. II”, in Algebraic structures and moduli spaces, American Mathematical Society, 2004, p. 149-175. | MR 2096144 | Zbl 1086.14011

[45] D. Orlov - “Projective bundles, monoidal transformations, and derived categories of coherent sheaves”, Russian Acad. Sci. Izv. Math. 41 (1993), p. 133-141. | MR 1208153 | Zbl 0798.14007

[46] -, “Equivalences of derived categories and K3 surfaces”, J. Math. Sci. 84 (1997), p. 1361-1381. | Article | MR 1465519 | Zbl 0938.14019

[47] -, “Derived categories of coherent sheaves on abelian varieties and equivalences between them”, Izv. Math. 66 (2002), p. 569-594. | Article | MR 1921811 | Zbl 1031.18007

[48] -, “Derived categories of coherent sheaves and equivalences between them”, Russian Math. Surveys 58 (2003), no. 3, p. 511-591. | MR 1998775 | Zbl 1118.14021

[49] A. Polishchuk - Abelian varieties, theta functions and the Fourier transform, Cambridge University Press, 2003. | Article | MR 1987784 | Zbl 1018.14016

[50] M. Reid - “La correspondance de McKay”, in Séminaire Bourbaki (1999/2000), Astérisque, vol. 276, Société Mathématique de France, Paris, 2002, Exp. no 867, p. 53-72. | Numdam | MR 1886756 | Zbl 0996.14006

[51] R. Rouquier - “Dimensions of triangulated categories”, preprint arXiv : math.CT/0310134(v3). | Article | MR 2434186 | Zbl 1165.18008

[52] -, “Catégories dérivées et géométrie algébrique”, notes d'exposés, http://www.math.jussieu.fr/~rouquier/preprints/luminy.dvi, janvier 2004.

[53] A. Rudakov - “Rigid and exceptional vector bundles and sheaves on a Fano variety”, in Proceedings of the International Congress of Mathematicians, Vol. 1 (Zürich, 1994), Birkhäuser, 1995, p. 697-705. | MR 1403970 | Zbl 0855.14001

[54] R. Swan - “Hochschild cohomology of quasiprojective schemes”, J. Pure Appl. Algebra 110 (1996), p. 57-80. | Article | MR 1390671 | Zbl 0865.18010

[55] T. Tanisaki - “Hodge modules, equivariant K-theory and Hecke algebras”, Publ. RIMS, Kyoto Univ. 23 (1987), p. 841-879. | MR 934674 | Zbl 0655.14004

[56] R. W. Thomason - “Les K-groupes d’un fibré projectif”, in Algebraic K-theory and algebraic topology (Lake Louise, 1991), Kluwer, 1993, p. 243-248. | MR 1367302 | Zbl 0910.19002

[57] -, “Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire”, Invent. Math. 112 (1993), p. 195-215. | Article | MR 1207482 | Zbl 0816.19004

[58] -, “The classification of triangulated subcategories”, Compositio Math. 105 (1997), p. 1-27. | Article | MR 1436741 | Zbl 0873.18003

[59] B. Toen - “The homotopy category of dg-categories and derived Morita theory”, preprint arXiv : math.AG/0408337(v5). | Zbl 1118.18010

[60] H. Uehara - “An example of Fourier-Mukai partners of minimal elliptic surfaces”, Math. Res. Lett. 11 (2004), p. 371-375. | Article | MR 2067481 | Zbl 1060.14055

[61] M. Van Den Bergh - “Three-dimensional flops and noncommutative rings”, Duke Math. J. 122 (2004), p. 423-455. | Article | MR 2057015 | Zbl 1074.14013

[62] J. Wierzba - “Birational geometry of symplectic 4-folds”, preprint, http://www.dpmms.cam.ac.uk/~jw227/publications.html.

[63] J. Wierzba & J. A. Wiśniewski - “Small contractions of symplectic 4-folds”, Duke Math. J. 120 (2003), p. 65-95. | Article | MR 2010734 | Zbl 1036.14007