Skip to main content
SpringerLink
Log in

Perverse schobers and birational geometry

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babson, E., Gunnells, P.E., Scott, R.: A smooth space of tetrahedra. Adv. Math. 165, 285–312 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babson, E., Gunnells, P.E., Scott, R.: Geometry of the tetrahedron space. Adv. Math. 204, 176–203 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  4. Beilinson, A., Bernstein, I., Deligne, P.: Faisceaux Pervers. Astérisque 100, 5–171 (1982)

    MathSciNet  MATH  Google Scholar 

  5. Bezrukavnikov, R.: Noncommutative counterparts of the Springer resolution. arXiv: math/0604445

  6. Bezrukavnikov, R.: Commutative and noncommutative symplectic resolutions and perverse sheaves. Lecture 18 May (2015)

  7. Bezrukavnikov, R., Riche, S.: Affine braid group actions on derived categories of Springer resolutions. Ann. Sci. Éc. Norm. Supér. 45, 535–599 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bodzenta, A., Bondal, A.: Flops and spherical functors. arXiv:1511.00665

  9. Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors and mutations. Math. USSR Izv. 35, 519–541 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bondal, A.I., Kapranov, M.M.: Enhanced triangulated categories. Math. USSR Sbornik 70, 93–107 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. arXiv: alg-geom/9506012

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  14. Bridgeland, T.: Flops and derived categories. Invent. Math. 146, 613–632 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brieskorn, E.: Die auflösung der rationalen singulariäten holomorpher abbildungen. Math. Ann. 178, 255–270 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brieskorn, E.: Singular elements of semi-simple algebraic groups. Actes du Congrés International des Mathématiciens, pp. 279–284 (Nice, 1970), Tome 2

  17. Collino, A., Fulton, W.: Intersection rings of spaces of triangles. Mém. Soc. Math. France 38, 75–117 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birhkaüser, Boston (2010)

    Book  MATH  Google Scholar 

  19. Donovan, W., Wemyss, M.: Twists and braids for general 3-fold flops. arXiv:1504.05320

  20. Drinfeld, V.: DG quotients of DG categories. J. Algebra 272, 643–691 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  22. Dyckerhoff, T., Kapranov, M.: Triangulated surfaces in triangulated categories. arXiv:1306.2545

  23. Dyckerhoff, T., Kapranov, M., Schechtman, V., Soibelman, Y.: Fukaya categoris with coefficients (in preparation)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Harder, A., Katzarkov, L.: Perverse sheaves of categories and some applications. arXiv: 1708.01181

  26. Hirschhorn, P.S.: Model Categories and Their Localizations. AMS Publication, Providence (2003)

    MATH  Google Scholar 

  27. Hotta, R., Kashiwara, M.: The invariant holonomic system on a semisimple Lie algebra. Invent. Math. 75, 327–358 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Iliev, A., Manivel, L.: Severi varieties and their varieties of reductions. J. Reine Angew. Math. 585, 93–139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Iliev, A., Manivel, L.: Varieties of reductions for \(\mathfrak{g}\mathfrak{l}_{n}\). arXiv:math/0501329

  30. van der Kallen, W., Magyar, P.: The space of triangles, vanishing theorems, and combinatorics. J. Algebra 222, 17–50 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kapranov, M., Schechtman, V.: Perverse sheaves over real hyperplane arrangements. Ann. Math. 183, 617–679 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Kapranov, M., Schechtman, V.: Perverse Schobers. arXiv:1411.2772

  33. Kawamata, Y.: On the cone of divisors of Calabi–Yau fiber spaces. Int. J. Math. 8, 665–687 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kostant, B.: Lie group representations on polynomial rings. Bull. Am. Math. Soc. 69, 518–526 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kuznetsov, A.: Hyperplane sections and derived categories Russ. Math. Izv. 70, 447–547 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kuznetsov, A.: Homological projective duality. Publ. Math. Inst. Hautes Études Sci. 105, 157–220 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Roberts, J.: Old and New Results About the Triangle Varieties, vol. 1311, pp. 197–219. Springer, Berlin (1988)

    MATH  Google Scholar 

  42. Schubert, H.: Anzahlgeometrische Behandlung des Dreiecks. Math. Ann. 17, 1213–1255 (1880)

    MathSciNet  MATH  Google Scholar 

  43. Semple, J.G.: The triangle as a geometric variable. Mathematika 1, 80–88 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tabuada, G.: Théorie homotopique des DG-categories. arXiv:0710.4303

  45. Toën, B.: The homotopy theory of DG-categories and derived Morita theory. Invent. Math. 167, 615–667 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122, 423–455 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kavli Institute for Physics and Mathematics of the Universe (WPI), 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan

    Alexey Bondal & Mikhail Kapranov

  2. Steklov Institute of Mathematics, Moscow, Russia

    Alexey Bondal

  3. National Research University Higher School of Economics, Moscow, Russia

    Alexey Bondal

  4. Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 Route de Narbonne, 31062, Toulouse, France

    Vadim Schechtman

Authors
  1. Alexey Bondal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Kapranov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vadim Schechtman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vadim Schechtman.

Additional information

To Sasha Beilinson on his 60th birthday.

‘A travers le brouillard, il contemplait des clochers, des édifices, dont il ne savait pas les noms.

Flaubert, L’Éducation sentimentale.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bondal, A., Kapranov, M. & Schechtman, V. Perverse schobers and birational geometry. Sel. Math. New Ser. 24, 85–143 (2018). https://doi.org/10.1007/s00029-018-0395-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-018-0395-1

Mathematics Subject Classification

Search

Navigation