Skip to main content
SpringerLink
Log in

Non-commutative deformations of simple objects in a category of perverse coherent sheaves

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We define a category of perverse coherent sheaves as the abelian category corresponding to the category of modules under Bondal–Rickard equivalence which arises from a tilting bundle for a projective morphism. The purpose of this paper is to determine versal non-commutative deformations of simple collections in the categories of perverse coherent sheaves in some cases. In general we prove that the non-commutative structure algebra is recovered as the parameter algebra of the versal non-commutative deformation of the simple collection consisting of all simple objects over a closed point of the base space. In the case where the fiber dimensions are at most 1 and the structure sheaf is relatively acyclic, we determine the versal deformations of some partial simple collections consisting of vanishing simple objects. In particular it is proved that the parameter algebra of the versal non-commutative deformation is isomorphic to its opposite algebra in this case.

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. Bass, H.: Algebraic K-Theory. W. A. Benjamin, Inc., New York (1968)

    MATH  Google Scholar 

  2. Beilinson, A.A.: Coherent sheaves on \(\mathbf{P}^n\) and problems in linear algebra. Funkt. Anal. i Priloz. 12, 68–69 (1978)

    Google Scholar 

  3. Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque, vol. 100. Société Mathématique de France, Paris (1982). (in French)

    MATH  Google Scholar 

  4. Bodzenta, A., Bondal, A.: Flops and Spherical Functors. arXiv:1511.00665

  5. Bondal, A.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989) (Russian); translation in Math. USSR-Izv. 34 (1), 23–42 (1989)

  6. Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Bridgeland, T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Brown, K.A., Hajarnavis, C.R.: Homologically homogeneous rings. Trans. Am. Math. Soc. 281, 197–208 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Clemens, H.: The infinitesimal Abel–Jacobi mapping and moving the O(2)+O(-4) curve. Duke Math. J. 59(1), 233–240 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Donovan, W., Wemyss, M.: Noncommutative deformations and flops. Duke Math. J. 165(8), 1397–1474 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Donovan, W., Wemyss, M.: Twists and braids for general 3-fold flops. J. Eur. Math. Soc. 21(6), 1641–1701 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Donovan, W., Wemyss, M.: Contractions and deformations. Am. J. Math. 141(3), 563–592 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Donovan, W., Wemyss, M.: Noncommutative enhancements of contractions. Adv. Math. 344, 99–136 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19(2), 135–162 (1980)

    MathSciNet  MATH  Google Scholar 

  15. Hua, Z.: Contraction algebra and singularity of three-dimensional flopping contraction. Math. Z. 290, 431–443 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Hua, Z., Keller, B.: Cluster Categories and Rational Curves. arXiv:1810.00749

  17. Hua, Z., Toda, Y.: Contraction algebra and invariants of singularities. Int. Math. Res. Not. 2018(10), 3173–3198 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Hua, Z., Zhou, G.-S.: Noncommutative Mather–Yau theorem and ita applications to Calabi–Yau algebras and homological minimal model program. arXiv:1803.06128

  19. Kapranov, M.M.: On the derived category of coherent sheaves on Grassmann manifolds. Izv. Akad. Nauk SSSR Ser. Mat. 48(1), 192–202 (1984); Math. USSR-Izv. 24(1), 183–192 (1985)

  20. Kashiwara, M.: Faisceaux constructibles et systemes holonomes d’equations aux derivees partielles lineaires a points singuliers reguliers. Sem. Goulaouic-Schwartz, expose 19 (1979–80)

  21. Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: Algebraic Geometry Sendai 1985, Advanced Studies in Pure Mathematics, vol. 10, pp. 283–360. Kinokuniya and North-Holland (1987)

  22. Kawamata, Y.: D-equivalence and K-equivalence. J. Differ. Geom. 61(1), 147–171 (2002)

    MATH  Google Scholar 

  23. Kawamata, Y.: Birational geometry and derived categories. Surv. Differ. Geom. 22(1), 291–317 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Kawamata, Y.: On multi-pointed non-commutative deformations and Calabi–Yau threefolds. Compos. Math. 154, 1815–1842 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Mebkout, Z.: Sur le probleme de Riemann–Hilbert. Lecture Notes in Physics, vol. 126, pp. 99–110. Springer, Berlin (1980)

  26. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. (2) 39(3), 436–456 (1989)

    MathSciNet  MATH  Google Scholar 

  27. Toda, Y.: Moduli stacks of semistable sheaves and representations of Ext-quivers. Geom. Topol 22(5), 3083–3144 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004)

    MathSciNet  MATH  Google Scholar 

  29. Van den Bergh, M. Non-commutative Crepant Resolutions. The legacy of Niels Henrik Abel, pp. 749–770, Springer, Berlin (2004)

  30. Wemyss, M.: The \(GL(2,\mathbf{C})\) McKay correspondence. Math. Ann. 350(3), 631–659 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan

    Yujiro Kawamata

  2. Mathematics Division, National Center for Theoretical Sciences, National Taiwan University, Taipei, 106, Taiwan

    Yujiro Kawamata

  3. Morningside Center of Mathematics, Chinese Academy of Sciences, Haidian District, Beijing, 100190, China

    Yujiro Kawamata

  4. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Korea

    Yujiro Kawamata

Authors
  1. Yujiro Kawamata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yujiro Kawamata.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kawamata, Y. Non-commutative deformations of simple objects in a category of perverse coherent sheaves. Sel. Math. New Ser. 26, 43 (2020). https://doi.org/10.1007/s00029-020-00570-w

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s00029-020-00570-w

Keywords

Mathematics Subject Classification

Search

Navigation