Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities

  • Dmitri Orlov
Part of the Progress in Mathematics book series (PM, volume 270)


In this paper we establish an equivalence between the category of graded D-branes of type B in Landau–Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau–Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W = 0.

Key words

Triangulated categories of singularities derived categories of coherent sheaves branes Landau–Ginzburg models 


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  1. 1.
    M. Artin, J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 248–287.CrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. of Math. (2)167 (2008), no.3, 867–943.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Baer, Tilting sheaves in representation theory of algebras, Manuscripta Math. 60 (1988), no. 3, 323–247.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Beilinson, Coherent sheaves on \({\mathcal{P}}^{n}\) and problems in linear algebra, Funct. Anal. Appl. 12 (1978), no. 3, 68–69.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    P. Berthelot, A. Grothendieck, L. Illusie, Théorie des intersections et théoreme de Riemann-Roch, Lectere Notes in Mathematics, vol. 225, Springer, 1971.Google Scholar
  6. 6.
    A. Bondal, Representation of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR 53 (1989), no. 1, 25–44.MathSciNetGoogle Scholar
  7. 7.
    A. Bondal, M. Kapranov, Enhanced triangulated categories, Matem. Sb. 181 (1990), no. 5, 669–683.zbMATHGoogle Scholar
  8. 8.
    A. Bondal, D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint MPIM 95/15, 1995, arXiv:math.AG/9506012.Google Scholar
  9. 9.
    P. Gabriel, Des Catégories Abéliennes, Bull. Soc. Math. Fr. 90 (1962), 323–448.zbMATHMathSciNetGoogle Scholar
  10. 10.
    P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, New York, 1967.zbMATHGoogle Scholar
  11. 11.
    W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theorey of finite dimensional algebras, Singularities, representation of algebras, and vector bundles (Proc. Symp., Lambrecht/Pfalz/FRG 1985), Lect. Notes Math., vol. 1273, 1987, 265–297.CrossRefMathSciNetGoogle Scholar
  12. 12.
    S. Gelfand, Y. Manin, Homological algebra, algebra v, Encyclopaedia Math. Sci., vol. 38, Springer-Verlag, 1994.Google Scholar
  13. 13.
    D. Happel, On the derived categories of a finite-dimensional algebra, Comment. Math. Helv 62 (1987), 339–389.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer, 1966.Google Scholar
  15. 15.
    K. Hori, C. Vafa, Mirror Symmetry, arXiv:hep-th/0002222.Google Scholar
  16. 16.
    K. Hori, J. Walcher, F-term equation near Gepner points, arXiv: hep-th/0404196.Google Scholar
  17. 17.
    A. Kapustin, Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Physics, JHEP 12 (2003), no. 005, arXiv:hep-th/0210296.Google Scholar
  18. 18.
    Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model program, Adv. Stud. in Pure Math. 10 (1987), 283–360.MathSciNetGoogle Scholar
  19. 19.
    B. Keller, Derived categories and their uses, Handbook of Algebra, vol. 1, 671–701, North-Holland, Amsterdam, 1996.Google Scholar
  20. 20.
    M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of ICM, Zurich 1994 (Basel), Birkhauser, 1995, 120–139.Google Scholar
  21. 21.
    A. Neeman, Triangulated categories, Ann. of Math. Studies, vol. 148, Princeton University Press, 2001.Google Scholar
  22. 22.
    D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Trudy Steklov Math. Institute 246 (2004), 240–262.Google Scholar
  23. 23.
    D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Matem. Sbornik, (2006), 197(12):p.1827.zbMATHCrossRefGoogle Scholar
  24. 24.
    D. Quillen, Higher Algebraic K-theory I, Springer Lecture Notes in Math., vol. 341, Springer-Verlag, 1973.Google Scholar
  25. 25.
    J. P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire Dubreil–Pisot, vol. 23, Paris, 1958.Google Scholar
  26. 26.
    A. Takahashi, Matrix factorizations and representations of quivers 1, arXiv:math.AG/0506347.Google Scholar
  27. 27.
    J. L. Verdier, Categories derivées, SGA 4 1/2, Lecture Notes in Math., vol. 569, Springer-Verlag, 1977.Google Scholar
  28. 28.
    J. Walcher, Stability of Landau-Ginzburg branes, arXiv:hep-th/0412274.Google Scholar
  29. 29.
    A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), 41–84.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    A. Yekutieli, J. J. Zhang, Serre duality for noncommutative projective schemes, Proc. Amer. Math. Soc. 125 (1997), 697–707.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Algebra SectionSteklov Mathematical Institute RANMoscowRUSSIA

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