Derived categories and birational geometry

Raphaël Rouquier

Séminaire Bourbaki (2004-2005)

  • Volume: 47, page 283-308
  • ISSN: 0303-1179

Abstract

top
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.

How to cite

top

Rouquier, Raphaël. "Catégories dérivées et géométrie birationnelle." Séminaire Bourbaki 47 (2004-2005): 283-308. <http://eudml.org/doc/252173>.

@article{Rouquier2004-2005,
abstract = {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.},
author = {Rouquier, Raphaël},
journal = {Séminaire Bourbaki},
keywords = {derived category; triangulated category; Calabi-Yau variety; flop},
language = {fre},
pages = {283-308},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Catégories dérivées et géométrie birationnelle},
url = {http://eudml.org/doc/252173},
volume = {47},
year = {2004-2005},
}

TY - JOUR
AU - Rouquier, Raphaël
TI - Catégories dérivées et géométrie birationnelle
JO - Séminaire Bourbaki
PY - 2004-2005
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 47
SP - 283
EP - 308
AB - 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.
LA - fre
KW - derived category; triangulated category; Calabi-Yau variety; flop
UR - http://eudml.org/doc/252173
ER -

References

top
  1. [1] D. Abramovich & J.C. Chen – “Computations with moduli of perverse point sheaves”, preprint arXiv : math.AG/0304353. 
  2. [2] —, “Flops, flips and perverse point sheaves on threefold stacks”, preprint arXiv : math.AG/0304354. 
  3. [3] P. Balmer – “Presheaves of triangulated categories and reconstruction of schemes”, Math. Ann.324 (2002), p. 557–580. Zbl1011.18007MR1938458
  4. [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. Zbl0545.14012MR863137
  5. [5] A. Bondal & M.M. Kapranov – “Representable functors, Serre functors, and mutations”, Math. USSR-Izv. 35 (1990), p. 519–541. Zbl0703.14011MR1039961
  6. [6] A. Bondal & D. Orlov – “Semiorthogonal decompositions for algebraic varieties”, preprint arXiv : alg-geom/9506012. 
  7. [7] —, “Reconstruction of a variety from the derived category and groups of autoequivalences”, Compositio Math.125 (2001), p. 327–344. Zbl0994.18007MR1818984
  8. [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. Zbl0996.18007MR1957019
  9. [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. Zbl1135.18302MR1996800
  10. [10] T. Bridgeland – “Fourier-Mukai transforms for elliptic surfaces”, J. reine angew. Math. 498 (1998), p. 115–133. Zbl0905.14020MR1629929
  11. [11] —, “Equivalences of triangulated categories and Fourier-Mukai transforms”, Bull. London Math. Soc.31 (1999), p. 25–34. Zbl0937.18012MR1651025
  12. [12] —, “Flops and derived categories”, Invent. Math.147 (2002), p. 613–632. Zbl1085.14017MR1893007
  13. [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. Zbl0966.14028MR1824990
  14. [14] T. Bridgeland & A. Maciocia – “Complex surfaces with equivalent derived categories”, Math. Z.236 (2001), p. 677–697. Zbl1081.14023MR1827500
  15. [15] —, “Fourier-Mukai transforms for K 3 and elliptic fibrations”, J. Algebraic Geom.11 (2002), p. 629–657. Zbl1066.14047MR1910263
  16. [16] A. Căldăraru – “Derived categories of sheaves : a skimming”, preprint arXiv : math.AG/0501094. MR2182889
  17. [17] —, “The Mukai pairing, I : the Hochschild structure”, preprint arXiv : math.AG/0308079(v2). 
  18. [18] —, “The Mukai pairing, II : the Hochschild-Kostant-Rosenberg isomorphism”, preprint arXiv : math.AG/0308080(v3). Zbl1098.14011
  19. [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. Zbl1090.14003MR1972146
  20. [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. Zbl1063.14065MR1929792
  21. [21] J. Chuang & R. Rouquier – “Derived equivalences for symmetric groups and 𝔰𝔩 2 -categorification”, preprint arXiv : math.RT/0407205. Zbl1144.20001MR2373155
  22. [22] H. Clemens, J. Kollár & S. Mori – Higher-dimensional complex geometry, Astérisque, vol. 166, Société Mathématique de France, Paris, 1988. Zbl0689.14016MR1004926
  23. [23] P. Gabriel – “Des catégories abéliennes”, Bull. Soc. math. France 90 (1962), p. 323–448. Zbl0201.35602
  24. [24] A.L. Gorodentsev & S.A. Kuleshov – “Helix theory”, Moscow Math. J. 4 (2004), p. 377–440, 535. Zbl1072.14020MR2108443
  25. [25] L. Hille & M. Van den Bergh – “Fourier-Mukai transforms”, preprint arXiv : math.AG/0402043(v2). MR2384610
  26. [26] D. Huybrechts – “Fourier-Mukai transforms in algebraic geometry”, livre en préparation. Zbl1095.14002MR2244106
  27. [27] M.M. Kapranov – “On the derived categories of coherent sheaves on some homogeneous spaces”, Invent. Math.92 (1988), p. 479–508. Zbl0651.18008MR939472
  28. [28] M. Kashiwara & P. Schapira – Sheaves on manifolds, Springer-Verlag, 1990. Zbl0709.18001MR1074006
  29. [29] Y. Kawamata – “Derived Categories of Toric Varieties”, preprint arXiv : math.AG/0503102. Zbl1159.14026MR2280493
  30. [30] —, “Derived equivalence for stratified Mukai flop on G ( 2 , 4 ) ”, preprint arXiv : math.AG/0503101. MR2282964
  31. [31] —, “ D -equivalence and K -equivalence”, J. Differential Geom.61 (2002), p. 147–171. MR1949787
  32. [32] —, “Francia’s flip and derived categories”, in Algebraic geometry, de Gruyter, 2002, p. 197–215. Zbl1092.14023MR1954065
  33. [33] —, “Equivalences of derived categories of sheaves on smooth stacks”, Amer. J. Math.126 (2004), p. 1057–1083. Zbl1076.14023MR2089082
  34. [34] A. King – “Tilting bundles on some rational surfaces”, preprint http://www.maths.bath.ac.uk/~masadk/papers/tilt.ps, 1997. 
  35. [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. Zbl0846.53021MR1403918
  36. [36] —, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys.66 (2003), p. 157–216. Zbl1058.53065MR2062626
  37. [37] A. Kuznetsov – “Derived category of V 12 Fano threefolds”, preprint arXiv : math.AG/0310008. Zbl1111.14038
  38. [38] —, “Derived category of a cubic threefold and the variety V 14 ”, Trudy Mat. Inst. Steklov.246 (2004), p. 183–207. MR2101293
  39. [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. Zbl0996.14011MR1886763
  40. [40] E. Markman – “Brill-Noether duality for moduli spaces of sheaves on K 3 surfaces”, J. Algebraic Geom.10 (2001), p. 623–694. Zbl1074.14525MR1838974
  41. [41] S. Mukai – “Duality between D ( X ) and D ( X ^ ) with its application to Picard sheaves”, Nagoya Math. J.81 (1981), p. 153–175. Zbl0417.14036MR607081
  42. [42] —, “On the moduli space of bundles on K 3 surfaces. I”, in Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res., 1987, p. 341–413. Zbl0674.14023MR893604
  43. [43] Y. Namikawa – “Mukai flops and derived categories”, J. reine angew. Math. 560 (2003), p. 65–76. Zbl1033.18008MR1992802
  44. [44] —, “Mukai flops and derived categories. II”, in Algebraic structures and moduli spaces, American Mathematical Society, 2004, p. 149–175. Zbl1086.14011MR2096144
  45. [45] D. Orlov – “Projective bundles, monoidal transformations, and derived categories of coherent sheaves”, Russian Acad. Sci. Izv. Math.41 (1993), p. 133–141. Zbl0798.14007MR1208153
  46. [46] —, “Equivalences of derived categories and K 3 surfaces”, J. Math. Sci.84 (1997), p. 1361–1381. Zbl0938.14019MR1465519
  47. [47] —, “Derived categories of coherent sheaves on abelian varieties and equivalences between them”, Izv. Math.66 (2002), p. 569–594. Zbl1031.18007MR1921811
  48. [48] —, “Derived categories of coherent sheaves and equivalences between them”, Russian Math. Surveys 58 (2003), no. 3, p. 511–591. Zbl1118.14021MR1998775
  49. [49] A. Polishchuk – Abelian varieties, theta functions and the Fourier transform, Cambridge University Press, 2003. Zbl1018.14016MR1987784
  50. [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. Zbl0996.14006MR1886756
  51. [51] R. Rouquier – “Dimensions of triangulated categories”, preprint arXiv : math.CT/0310134(v3). Zbl1165.18008MR2434186
  52. [52] —, “Catégories dérivées et géométrie algébrique”, notes d’exposés, http://www.math.jussieu.fr/~rouquier/preprints/luminy.dvi, janvier 2004. 
  53. [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. Zbl0855.14001MR1403970
  54. [54] R. Swan – “Hochschild cohomology of quasiprojective schemes”, J. Pure Appl. Algebra110 (1996), p. 57–80. Zbl0865.18010MR1390671
  55. [55] T. Tanisaki – “Hodge modules, equivariant K -theory and Hecke algebras”, Publ. RIMS, Kyoto Univ. 23 (1987), p. 841–879. Zbl0655.14004MR934674
  56. [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. Zbl0910.19002MR1367302
  57. [57] —, “Les K -groupes d’un schéma éclaté et une formule d’intersection excédentaire”, Invent. Math.112 (1993), p. 195–215. Zbl0816.19004MR1207482
  58. [58] —, “The classification of triangulated subcategories”, Compositio Math.105 (1997), p. 1–27. Zbl0873.18003MR1436741
  59. [59] B. Toen – “The homotopy category of dg-categories and derived Morita theory”, preprint arXiv : math.AG/0408337(v5). Zbl1118.18010
  60. [60] H. Uehara – “An example of Fourier-Mukai partners of minimal elliptic surfaces”, Math. Res. Lett.11 (2004), p. 371–375. Zbl1060.14055MR2067481
  61. [61] M. Van den Bergh – “Three-dimensional flops and noncommutative rings”, Duke Math. J.122 (2004), p. 423–455. Zbl1074.14013MR2057015
  62. [62] J. Wierzba – “Birational geometry of symplectic 4-folds”, preprint, http://www.dpmms.cam.ac.uk/~jw227/publications.html. 
  63. [63] J. Wierzba & J.A. Wiśniewski – “Small contractions of symplectic 4-folds”, Duke Math. J.120 (2003), p. 65–95. Zbl1036.14007MR2010734

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.