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Selecta Mathematica

, Volume 24, Issue 1, pp 85–143 | Cite as

Perverse schobers and birational geometry

  • Alexey Bondal
  • Mikhail Kapranov
  • Vadim Schechtman
Article
  • 99 Downloads

Abstract

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a “web of flops” provided by the Grothendieck resolution associated to a reductive Lie algebra \(\mathfrak {g}\) and study the corresponding schober-type diagram. For \(\mathfrak {g}={\mathfrak {s}\mathfrak {l}}_3\) we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

Mathematics Subject Classification

14E05 14F05 18D05 

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References

  1. 1.
    Babson, E., Gunnells, P.E., Scott, R.: A smooth space of tetrahedra. Adv. Math. 165, 285–312 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babson, E., Gunnells, P.E., Scott, R.: Geometry of the tetrahedron space. Adv. Math. 204, 176–203 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beilinson, A.: How to glue perverse sheaves. In: Manin, Y. I. (ed.) \(K\)-Theory, Arithmetic and Geometry (Moscow, 1984), Lecture Notes in Mathematics, vol. 1289, pp. 42–51. Springer, Berlin (1987)Google Scholar
  4. 4.
    Beilinson, A., Bernstein, I., Deligne, P.: Faisceaux Pervers. Astérisque 100, 5–171 (1982)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bezrukavnikov, R.: Noncommutative counterparts of the Springer resolution. arXiv: math/0604445
  6. 6.
    Bezrukavnikov, R.: Commutative and noncommutative symplectic resolutions and perverse sheaves. Lecture 18 May (2015)Google Scholar
  7. 7.
    Bezrukavnikov, R., Riche, S.: Affine braid group actions on derived categories of Springer resolutions. Ann. Sci. Éc. Norm. Supér. 45, 535–599 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bodzenta, A., Bondal, A.: Flops and spherical functors. arXiv:1511.00665
  9. 9.
    Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors and mutations. Math. USSR Izv. 35, 519–541 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bondal, A.I., Kapranov, M.M.: Enhanced triangulated categories. Math. USSR Sbornik 70, 93–107 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. arXiv: alg-geom/9506012
  12. 12.
    Bondal, A.I., Orlov, D.O.: Derived categories of coherent sheaves. Proceedings of the International Congress of Mathematicians, vol. II, pp. 47–56, (Beijing, 2002), Beijing, Higher Ed. Press (2002)Google Scholar
  13. 13.
    Bondal, A.I., van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3, 1–36 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bridgeland, T.: Flops and derived categories. Invent. Math. 146, 613–632 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brieskorn, E.: Die auflösung der rationalen singulariäten holomorpher abbildungen. Math. Ann. 178, 255–270 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brieskorn, E.: Singular elements of semi-simple algebraic groups. Actes du Congrés International des Mathématiciens, pp. 279–284 (Nice, 1970), Tome 2Google Scholar
  17. 17.
    Collino, A., Fulton, W.: Intersection rings of spaces of triangles. Mém. Soc. Math. France 38, 75–117 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birhkaüser, Boston (2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    Donovan, W., Wemyss, M.: Twists and braids for general 3-fold flops. arXiv:1504.05320
  20. 20.
    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272, 643–691 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dwyer, W.G., Hirschhorn, P.S., Kan, D.M., Smith, J.H.: Homotopy Limit Functors on Model Categories and Homotopical Categories. AMS Publication, Providence (2004)zbMATHGoogle Scholar
  22. 22.
    Dyckerhoff, T., Kapranov, M.: Triangulated surfaces in triangulated categories. arXiv:1306.2545
  23. 23.
    Dyckerhoff, T., Kapranov, M., Schechtman, V., Soibelman, Y.: Fukaya categoris with coefficients (in preparation) Google Scholar
  24. 24.
    Galligo, A., Granger, M., Maisonobe, P.: \(\cal{D}\)-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble 35, 1–48 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Harder, A., Katzarkov, L.: Perverse sheaves of categories and some applications. arXiv: 1708.01181
  26. 26.
    Hirschhorn, P.S.: Model Categories and Their Localizations. AMS Publication, Providence (2003)zbMATHGoogle Scholar
  27. 27.
    Hotta, R., Kashiwara, M.: The invariant holonomic system on a semisimple Lie algebra. Invent. Math. 75, 327–358 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Iliev, A., Manivel, L.: Severi varieties and their varieties of reductions. J. Reine Angew. Math. 585, 93–139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Iliev, A., Manivel, L.: Varieties of reductions for \(\mathfrak{g}\mathfrak{l}_{n}\). arXiv:math/0501329
  30. 30.
    van der Kallen, W., Magyar, P.: The space of triangles, vanishing theorems, and combinatorics. J. Algebra 222, 17–50 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kapranov, M., Schechtman, V.: Perverse sheaves over real hyperplane arrangements. Ann. Math. 183, 617–679 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Kapranov, M., Schechtman, V.: Perverse Schobers. arXiv:1411.2772
  33. 33.
    Kawamata, Y.: On the cone of divisors of Calabi–Yau fiber spaces. Int. J. Math. 8, 665–687 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kostant, B.: Lie group representations on polynomial rings. Bull. Am. Math. Soc. 69, 518–526 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kuznetsov, A.: Hyperplane sections and derived categories Russ. Math. Izv. 70, 447–547 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kuznetsov, A.: Homological projective duality. Publ. Math. Inst. Hautes Études Sci. 105, 157–220 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Namikawa, Y.: Mukai flops and derived categories. II. In: Algebraic Structures and Moduli Spaces, CRM Proceedings and Lecture Notes, vol. 38, pp. 149–175. AMS Publication, Providence (2004)Google Scholar
  38. 38.
    Pinkham, H.: Factorization of birational maps in dimension 3. In: Orlik, P. (ed.) Singularities, Proceedings of Symposia in Pure Mathematics, vol. 40, part 2, pp. 343–371. American Mathematical Society, Providence (1983)Google Scholar
  39. 39.
    Reid, M.: Young person’s guide to canonical singularities. In: Algebraic Geometry (Bowdoin, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46, Part 1, pp. 345–414. AMS, Providence (1987)Google Scholar
  40. 40.
    Riche, S.: Geometric braid group actions on derived categories of coherent sheaves (with a joint appendix with R. Bezrukavnikov). Represent. Theory 12, 131–169 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Roberts, J.: Old and New Results About the Triangle Varieties, vol. 1311, pp. 197–219. Springer, Berlin (1988)zbMATHGoogle Scholar
  42. 42.
    Schubert, H.: Anzahlgeometrische Behandlung des Dreiecks. Math. Ann. 17, 1213–1255 (1880)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Semple, J.G.: The triangle as a geometric variable. Mathematika 1, 80–88 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tabuada, G.: Théorie homotopique des DG-categories. arXiv:0710.4303
  45. 45.
    Toën, B.: The homotopy theory of DG-categories and derived Morita theory. Invent. Math. 167, 615–667 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122, 423–455 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alexey Bondal
    • 1
    • 2
    • 3
  • Mikhail Kapranov
    • 1
  • Vadim Schechtman
    • 4
  1. 1.Kavli Institute for Physics and Mathematics of the Universe (WPI)Kashiwa-shiJapan
  2. 2.Steklov Institute of MathematicsMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Université Paul SabatierInstitut de Mathématiques de ToulouseToulouseFrance

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