Functional Analysis and Its Applications

, Volume 50, Issue 1, pp 1–16 | Cite as

Differential Forms on Quasihomogeneous Noncomplete Intersections

  • A. G. Aleksandrov


In this article, we discuss a few simple methods for computing the Poincaré series of modules of differential forms given on quasihomogeneous noncomplete intersections of various types. Among them are curves associated with a semigroup, bouquets of such curves, affine cones over rational or elliptic curves, and normal determinantal and toric varieties, including some types of quotient singularities, as well as cones over the Veronese embedding of projective spaces or over the Segre embedding of products of projective spaces, rigid singularities, fans, etc. In many cases, correct formulas can be derived without resorting to analysis of complicated resolvents or using computer systems of algebraic calculations. The obtained results allow us to compute the basic invariants of singularities in an explicit form by means of elementary operations on rational functions.

Key words

differential forms Poincaré complex de Rham complex graded singularities determinantal singularities rigid singularities fans 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. G. Aleksandrov, “The index of differential forms on complete intersections,” Funkts. Anal. Prilozhen., 49:1 (2015), 1–17; English transl.: Functional Anal. Appl., 49:1 (2015), 1–14.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. G. Aleksandrov, “L’indice des champs de vecteurs sur les courbes de Cohen–Macaulay,” Bull. Sci. Math., 137:6 (2013), 791–804.MathSciNetCrossRefGoogle Scholar
  3. [3]
    A. G. Aleksandrov, “L’indice topologique des champs de vecteurs sur les intersections complètes quasi-homogènes (à la mémoire de Henri Poincaré),” C. R. Acad. Sci., Paris. Sér. I. Math., 350:19–20 (2012), 911–916.CrossRefGoogle Scholar
  4. [4]
    A. G. Aleksandrov, “A de Rham complex of nonisolated singularities,” Funkts. Anal. Prilozhen., 22:2 (1988), 59–60; English transl.: Functional Anal. Appl., 22:2 (1988), 131–133.MathSciNetMATHGoogle Scholar
  5. [5]
    A. G. Aleksandrov, “Cohomology of a quasihomogeneous complete intersection,” Izv. Akad. Nauk SSSR, Ser. Mat., 49:3 (1985), 467–510; English transl.: Math. USSR Izv., 26:3 (1986), 437–477.MathSciNetGoogle Scholar
  6. [6]
    A. G. Aleksandrov, “The de Rham complex of quasihomogeneous complete intersection,” Funkts. Anal. Prilozhen., 17:1 (1983), 63–64; English transl.: Functional Anal. Appl., 17:1 (1983), 48–49.MathSciNetMATHGoogle Scholar
  7. [7]
    A. G. Aleksandrov, “On deformations of one-dimensional singularities with the invariants c = δ+1,” Uspekhi Mat. Nauk, 33:3 (1978), 157–158; English transl.: Russian Math. Surveys, 33:3 (1978), 139–140.MathSciNetGoogle Scholar
  8. [8]
    A. G. Aleksandrov, “Duality, derivations and deformations of zero-dimensional singularities,” in: Zero-dimensional schemes, de Gruyter, Berlin, 1994, 11–31.Google Scholar
  9. [9]
    S. M. Gusein-Zade and W. Ebeling, “On indices of 1-forms on determinantal singularities,” Trudy Mat. Inst. Steklov., 267 (2009), 119–131; English transl.: Proc. Steklov Inst. Math., 267 (2009), 113–124.MathSciNetMATHGoogle Scholar
  10. [10]
    V. I. Danilov, “The geometry of toric varieties,” Uspekhi Mat. Nauk, 33:2(200) (1978), 85–134; English transl.: Russian Math. Surveys, 33:2 (1978), 97–154.MathSciNetMATHGoogle Scholar
  11. [11]
    W. Ebeling, S. M. Gusein-Zade, and J. Seade, “Homological index for 1-forms and a Milnor number for isolated singularities,” Internat. J. Math., 15:9 (2004), 895–905.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    A. R. Grandjean and M. J. Vale, “Almost smooth algebras,” Cahiers Topologie Géom. Différentielle Catég., 32:2 (1991), 131–138.MathSciNetMATHGoogle Scholar
  13. [13]
    C. Huneke, “The Koszul homology of an ideal,” Adv. in Math., 56:3 (1985), 295–318.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    S. L. Kleiman and J. Landolfi, “Geometry and deformations of special Schubert varieties,” in: Algebraic Geometry (F. Oort, ed.), Oslo, 1970, Wolters-Noordhoff Publ., Groningen, 1972, 97–124.Google Scholar
  15. [15]
    S. Lichtenbaum and M. Schlessinger, “The cotangent complex of a morphism,” Trans. Amer. Math. Soc., 128:1 (1967), 41–70.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    T. Matsuoka, “On almost complete intersections,” Manuscripta Math., 21:3 (1977), 329–340.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    I. Naruki, “Some remarks on isolated singularities and their application to algebraic manifolds,” Publ. RIMS, 13:1 (1977), 17–46.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J. J. Nuñno-Ballesteros, B. Orréfice-Okamoto, and J. N. Tomazella, “The vanishing Euler characteristic of an isolated determinantal singularity,” Israel J. Math., 197:1 (2013), 475–495.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    R. Pellikaan, “Finite determinacy of functions with non-isolated singularities,” Proc. London Math. Soc. (3), 57:2 (1988), 357–382.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    H. C. Pinkham, “Deformations of algebraic varieties with Gmaction,” in: Astérisque, vol. 20, Société Mathématique de France, Paris, 1974.Google Scholar
  21. [21]
    O. Riemenschneider, “Deformationen von Quotientensingularit¨aten (nach zyklischen Gruppen),” Math. Ann., 209:2 (1974), 211–248.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D. S. Rim, “Torsion differentials and deformation,” Trans. Amer. Math. Soc., 169:442 (1972), 257–278.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    D. S. Rim and M. A. Vitulli, “Weierstrass points and monomial curves,” J. Algebra, 48:2 (1977), 454–476.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    M. Schlessinger, “Rigidity of quotient singularities.,” Invent. Math., 14:1 (1971), 17–26.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    M. Schlessinger, “On rigid singularities,” Rice Univ. Studies, 59:1 (1973), 147–162.MathSciNetMATHGoogle Scholar
  26. [26]
    P. Seibt, “Differential filtrations and symbolic powers of regular primes,” Math. Z., 166:2 (1979), 159–164.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    P. Seibt, “Infnitesimal extensions of commutative algebras,” J. Pure Appl. Algebra, 16:2 (1980), 197–206.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    A. Simis, B. Ulrich, and W. V. Vasconselos, “Tangent algebras,” Trans. Amer. Math. Soc., 364:2 (2012), 571–594.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Control SciencesRussian Academy of ScienceSt. PetersburgRussia

Personalised recommendations