## Arkiv för Matematik

### A residue criterion for strong holomorphicity

#### Abstract

We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A. Tsikh for the case of a reduced complete intersection.

#### Note

The author was partially supported by the Swedish Research Council.

#### Article information

Source
Ark. Mat., Volume 48, Number 1 (2010), 1-15.

Dates
Revised: 23 February 2009
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907095

Digital Object Identifier
doi:10.1007/s11512-009-0100-x

Mathematical Reviews number (MathSciNet)
MR2594583

Zentralblatt MATH identifier
1198.32001

Rights

#### Citation

Andersson, Mats. A residue criterion for strong holomorphicity. Ark. Mat. 48 (2010), no. 1, 1--15. doi:10.1007/s11512-009-0100-x. https://projecteuclid.org/euclid.afm/1485907095

#### References

• Altman, A. and Kleiman, S., Introduction to Grothendieck Duality Theory, Lecture Notes in Math. 146, Springer, Berlin–Heidelberg, 1970.
• Andersson, M., Coleff–Herrera currents, duality, and Noetherian operators, Preprint, 2009.
• Andersson, M., Uniqueness and factorization of Coleff–Herrera currents, to appear in Ann. Fac. Sci. Toulouse.
• Andersson, M. and Wulcan, E., Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup. 40 (2007), 985–1007.
• Andersson, M. and Wulcan, E., Decomposition of residue currents, to appear in J. Reine Angew. Math.
• Barlet, D., Le faisceau ωX sur un espace analytique X de dimension pure, in Fonctions de plusieurs variables complexes, III Sém. François Norguet, 1975–1977, Lecture Notes in Math. 670, pp. 187–204, Springer, Berlin–Heidelberg, 1978.
• Björk, J.-E., Residues and $\mathcal{D}$ -modules, in The Legacy of Niels Henrik Abel, pp. 605–651, Springer, Berlin, 2004.
• Coleff, N. R. and Herrera, M. E., Les courants résiduels associés àune forme méromorphe, Lecture Notes in Math. 663, Springer, Berlin–Heidelberg, 1978.
• Demailly, J.-P., Complex Analytic and Differential Geometry, In preparation.
• Dickenstein, A. and Sessa, C., Canonical representatives in moderate cohomology, Invent. Math. 80 (1985), 417–434.
• Eisenbud, D., Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Math. 150, Springer, New York, 1995.
• Henkin, G. and Passare, M., Abelian differentials on singular varieties and variations on a theorem of Lie–Griffiths, Invent. Math. 135 (1999), 297–328.
• Malgrange, B., Sur les fonctions différentiables et les ensembles analytiques, Bull. Soc. Math. France 91 (1963), 113–127.
• Passare, M., Residues, currents, and their relation to ideals of holomorphic functions, Math. Scand. 62 (1988), 75–152.
• Passare, M., Tsikh, A. and Yger, A., Residue currents of the Bochner–Martinelli type, Publ. Mat. 44 (2000), 85–117.
• Spallek, K., Über Singularitäten analytischer Mengen, Math. Ann. 172 (1967), 249–268.
• Tsikh, A., Multidimensional Residues and Their Applications, Nauka Sibirsk. Otdel., Novosibirsk, 1988 (Russian). English transl.: Transl. Math. Monographs 103, Amer. Math. Soc., Providence, RI, 1992.
• Wulcan, E., Products of residue currents of Cauchy–Fantappiè–Leray type, Ark. Mat. 45 (2007), 157–178.