Catégories dérivées et géométrie birationnelle [d'après Bondal, Orlov, Bridgeland, Kawamata...]

Rouquier, Raphaël

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 946, pp. 283-307.

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.

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.

Classification : 14Exx, 14Jxx, 18Exx
Mot clés : catégorie dérivée, catégorie triangulée, variété de Calabi-Yau, flop
Keywords: derived category, triangulated category, Calabi-Yau variety, flop

[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

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

[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 | Zbl

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

[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. | DOI | MR | Zbl

[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 | Zbl

[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. | DOI | MR | Zbl

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

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

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

[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. | DOI | MR | Zbl

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

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

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

[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

[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. | DOI | MR | Zbl

[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 | Zbl

[21] J. Chuang & R. Rouquier - “Derived equivalences for symmetric groups and ${\mathrm{𝔰𝔩}}_2$-categorification”, preprint arXiv : math.RT/0407205. | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

[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. | EuDML | Numdam | MR | Zbl

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

[41] S. Mukai - “Duality between $D\left(X\right)$ and $D\left(\widehat{X}\right)$ with its application to Picard sheaves”, Nagoya Math. J. 81 (1981), p. 153-175. | DOI | MR | Zbl

[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 | Zbl

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

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

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

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

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

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

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

[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. | EuDML | Numdam | MR | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

[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. | DOI | MR | Zbl