Abstract
In the article we introduce the notion of logarithmic differential forms with poles along a Cartier divisor given on a variety with singularities, discuss some properties of such forms, and describe highly efficient methods for computing the Poincaré series and generators of modules of logarithmic differential forms in various situations. We also examine several concrete examples by applying these methods to the study of divisors on varieties with singularities of many types, including quasi-homogeneous complete intersections, normal, determinantal, and rigid varieties, and so on.
Similar content being viewed by others
References
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.
A. G. Aleksandrov, “Nonisolated Saito singularities,” Mat. Sb. (N.S.), 137(179):4 (1988), 554–567; English transl.: Math. USSR Sb., 65:2 (1990), 561–574.
A. G. Aleksandrov, Nonisolated Hypersurface Singularities, Adv. Soviet Math., vol. 1, Amer. Math. Soc., Providence, RI, 1990.
A. G. Aleksandrov, “Logarithmic differential forms, torsion differentials and residue,” Complex Var. Theory Appl., 50:7–11 (2005), 777–802.
A. G. Aleksandrov, “Residues of logarithmic differential forms in complex analysis and geometry,” Anal. Theory Appl., 30:1 (2014), 34–50.
A. G. Aleksandrov, “Differential forms on quasihomogeneous noncomplete intersections,” Funkts. Anal. Prilozhen., 50:1 (2016), 1–19; English transl.: Functional Anal. Appl., 50:1 (2016), 1–16.
N. Bourbaki, Algèbre commutative, Hermann, Paris, 1961.
G. de Rham, “Sur la division de formes et de courants par une forme linéaire,” Comment. Math. Helvet., 28 (1954), 346–352.
V. Grandjean, “Coherent vector fields and logarithmic stratification,” in: Real and Complex Singularities, Proceedings of the 5th Workshop, S˜ao Carlos, Brazil, July 27–31, 1998, Chapman Hall/CRC Res. Notes Math., vol. 412, Chapman & Hall/CRC, Boca Raton, FL, 2000, 46–60.
G.–M. Greuel, “Der Gauß–Manin–Zusammenhang isolierter Singularitäten von vollständigen Durchscnitten,” Math. Ann., 214:1 (1975), 235–266.
D. Mumford, Lectures on Curves on an Algebraic Surface, Princeton University Press, Princeton, New Jersey, 1966.
I. Naruki, “Some remarks on isolated singularities and their application to algebraic manifolds,” Publ. Res. Inst. Math. Sci., 13:1 (1977), 17–46.
H. C. Pinkham, Deformations of Algebraic Varieties with Gm-Action, Astérisque, vol. 20, Société Mathématique de France, Paris, 1974.
D. S. Rim, “Torsion differentials and deformations,” Trans. Amer. Math. Soc., 169:442 (1972), 257–278.
K. Saito, “Theory of logarithmic differential forms and logarithmic vector fields,” J. Fac. Sci. Univ. Tokyo, sect. IA, 27:2 (1980), 265–291.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 51, No. 4, pp. 3–15, 2017
Original Russian Text Copyright © by A. G. Aleksandrov
Rights and permissions
About this article
Cite this article
Aleksandrov, A.G. Logarithmic differential forms on varieties with singularities. Funct Anal Its Appl 51, 245–254 (2017). https://doi.org/10.1007/s10688-017-0190-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-017-0190-3