Abstract
We study the multiple residue of logarithmic differential forms with poles along a reducible divisor and compute the kernel and the image of the multiple residue map. As an application we describe the weight filtration on the logarithmic de Rham complex for divisors whose irreducible components are given locally by a regular sequence of holomorphic functions. In particular, this allows us to compute the mixed Hodge structure on the cohomology of the complement of divisors of certain types without the use of theorems on resolution of singularities and the standard reduction to the case of normal crossings.
Similar content being viewed by others
References
A. G. Aleksandrov, “Non-isolated hypersurface singularities,” in: Theory of Singularities and Its Applications, Advances in Soviet Mathematics, vol. 1, Amer. Math. Soc., Providence, RI, 1990, 211–246.
A. G. Aleksandrov and A. K. Tsikh, “Théorie des résidus de Leray et formes de Barlet sur une intersection complète singulière,” C. R. Acad. Sci. Paris, Ser. I, 333:11 (2001), 973–978.
A. G. Aleksandrov and A. K. Tsikh, “Multi-logarithmic differential forms on complete intersections,” J. Siberian Federal University. Math. & Phys., 2 (2008), 105–124.
D. Barlet, Le faisceau ω ⊙X sur un espace analytique X de dimension pure, Lecture Notes in Math., vol. 670, Springer-Verlag, Berlin, 1978.
D. A. Buchsbaum and D. S. Rim, “A generalized Koszul complex. II. Depth and multiplicity,” Trans. Amer. Math. Soc., 111 (1964), 197–224.
P. Deliau]gne, “Théorie de Hodge II,” Inst. Hautes Études Sci. Publ. Math., 40 (1971), 5–57.
Ph. Griffiths and J. Harris, Principles of Algebraic Geometry, vol. 2, Wiley-Interscience, New York, 1978.
M. P. Holland and D. Mond, “Logarithmic differential forms and the cohomology of the complement of a divisor,” Math. Scand., 83:2 (1998), 235–254.
M. Kersken, “Reguläre Differentialformen,” Manuscripta Math., 46:1 (1984), 1–25.
E. Kunz, “Holomorphe Differentialformen auf algebraischen Varietäten mit Singularitäten. I,” Manuscripta Math., 15 (1975), 91–108.
K. Morita, “On the basis of twisted de Rham cohomology,” Hokkaido Math. J., 27:3 (1998), 567–603.
K. Saito, “On a generalization of de Rham lemma,” Ann. Inst. Fourier (Grenoble), 26:2 (1976), 165–170.
K. Saito, “Theory of logarithmic differential forms and logarithmic vector fields,” J. Fac. Sci. Univ. Tokyo, ser. IA, 27:2 (1980), 265–291.
J. H. M. Steenbrink, “Mixed Hodge structure on the vanishing cohomology,” in: Real and Complex Singularities (Proc. Nordic Summer School, Symp. Math., Oslo, 1976), Sijthoff and Noordhoff Publ., Alphen aan den Rijn, 1977.
H. Wiebe, “Über homologische Invarianten lokaler Ringe,” Math. Ann., 179:4 (1969), 257–274.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 47, No. 4, pp. 1–17, 2013
Original Russian Text Copyright © by A. G. Aleksandrov
Rights and permissions
About this article
Cite this article
Aleksandrov, A.G. The multiple residue and the weight filtration on the logarithmic de Rham complex. Funct Anal Its Appl 47, 247–260 (2013). https://doi.org/10.1007/s10688-013-0032-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-013-0032-x