A comparison formula for residue currents

  • Richard Lärkäng

Abstract

Given two ideals $\mathcal {I}$ and $\mathcal {J}$ of holomorphic functions such that $\mathcal {I} \subseteq \mathcal {J}$, we describe a comparison formula relating the Andersson-Wulcan currents of $\mathcal {I}$ and $\mathcal {J}$. More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes.

One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an analytic proof by means of residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the Jacobian ideal of a holomorphic mapping.

References

Andersson, M., Residue currents and ideals of holomorphic functions, Bull. Sci. Math. 128 (2004), no. 6, 481–512. https://doi.org/10.1016/j.bulsci.2004.03.003

Andersson, M., Integral representation with weights. II. Division and interpolation, Math. Z. 254 (2006), no. 2, 315–332. https://doi.org/10.1007/s00209-006-0949-3

Andersson, M., Uniqueness and factorization of Coleff-Herrera currents, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 4, 651–661.

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

Andersson, M., Coleff-Herrera currents, duality, and Noetherian operators, Bull. Soc. Math. France 139 (2011), no. 4, 535–554. https://doi.org/10.24033/bsmf.2618

Andersson, M. and Samuelsson, H., Weighted Koppelman formulas and the $\overline \partial $-equation on an analytic space, J. Funct. Anal. 261 (2011), no. 3, 777–802. https://doi.org/10.1016/j.jfa.2011.02.018

Andersson, M. and Samuelsson, H., A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas, Invent. Math. 190 (2012), no. 2, 261–297. https://doi.org/10.1007/s00222-012-0380-9

Andersson, M., Samuelsson, H., and Sznajdman, J., On the Briançon-Skoda theorem on a singular variety, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 417–432.

Andersson, M. and Wulcan, E., Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 6, 985–1007. https://doi.org/10.1016/j.ansens.2007.11.001

Andersson, M. and Wulcan, E., Decomposition of residue currents, J. Reine Angew. Math. 638 (2010), 103–118. https://doi.org/10.1515/CRELLE.2010.004

Andersson, M. and Wulcan, E., Global effective versions of the Briançon-Skoda-Huneke theorem, Invent. Math. 200 (2015), no. 2, 607–651. https://doi.org/10.1007/s00222-014-0544-x

Andersson, M. and Wulcan, E., Direct images of semi-meromorphic currents, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 2, 875–900.

Björk, J.-E., $\mathcal D$-modules and residue currents on complex manifolds, preprint, 1996.

Björk, J.-E., Residues and $\mathcal D$-modules, in “The legacy of Niels Henrik Abel: Papers from the Abel Bicentennial Conference held at the University of Oslo, Oslo, June 3–8, 2002” (Laudal, O. A. and Piene, R., eds.), Springer, Berlin, 2004, pp. 605--651. https://doi.org/10.1007/978-3-642-18908-1

Coleff, N. R. and Herrera, M. E., Les courants résiduels associés à une forme méromorphe, Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978.

Dickenstein, A. and Sessa, C., Canonical representatives in moderate cohomology, Invent. Math. 80 (1985), no. 3, 417–434. https://doi.org/10.1007/BF01388723

Dickenstein, A. and Sessa, C., Résidus de formes méromorphes et cohomologie modérée, in “Géométrie complexe (Paris, 1992)'', Actualités Sci. Indust., vol. 1438, Hermann, Paris, 1996, pp. 35--59. https://doi.org/10.1016/j.brainres.2011.12.039

Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-5350-1

Griffiths, P. and Harris, J., Principles of algebraic geometry, Wiley-Interscience, New York, 1978.

Hickel, M., Une note à propos du jacobien de $n$ fonctions holomorphes à l'origine de $\Bbb C^n$, Ann. Polon. Math. 94 (2008), no. 3, 245–264. https://doi.org/10.4064/ap94-3-4

Jonsson, M. and Wulcan, E., On Bochner-Martinelli residue currents and their annihilator ideals, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2119–2142.

Lärkäng, R., Residue currents associated with weakly holomorphic functions, Ark. Mat. 50 (2012), no. 1, 135–164. https://doi.org/10.1007/s11512-010-0141-1

Lärkäng, R., On the duality theorem on an analytic variety, Math. Ann. 355 (2013), no. 1, 215–234. https://doi.org/10.1007/s00208-012-0782-4

Lärkäng, R., Explicit versions of the local duality theorem in $\C ^n$, preprint arXiv:1510.01965 [math.CV], 2015.

Lärkäng, R., Residue currents with prescribed annihilator ideals on singular varieties, Math. Z. 279 (2015), no. 1-2, 333–358. https://doi.org/10.1007/s00209-014-1371-x

Lärkäng, R. and Wulcan, E., Computing residue currents of monomial ideals using comparison formulas, Bull. Sci. Math. 138 (2014), no. 3, 376–392. https://doi.org/10.1016/j.bulsci.2013.06.003

Lärkäng, R. and Wulcan, E., Residue currents and fundamental cycles, Indiana Univ. Math. J. 67 (2018), no. 3, 1085–1114. https://doi.org/10.1512/iumj.2018.67.7285

Lejeune-Jalabert, M., Liaison et résidu, in “Algebraic geometry (La Rábida, 1981)'', Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 233--240. https://doi.org/10.1007/BFb0071285

Lundqvist, J., A local Grothendieck duality theorem for Cohen-Macaulay ideals, Math. Scand. 111 (2012), no. 1, 42–52. https://doi.org/10.7146/math.scand.a-15212

Lundqvist, J., A local duality principle for ideals of pure dimension, preprint arXiv:1306.6252 [math.CV], 2013.

Passare, M., Residues, currents, and their relation to ideals of holomorphic functions, Math. Scand. 62 (1988), no. 1, 75–152. https://doi.org/10.7146/math.scand.a-12211

Passare, M., Tsikh, A., and Yger, A., Residue currents of the Bochner-Martinelli type, Publ. Mat. 44 (2000), no. 1, 85–117. https://doi.org/10.5565/PUBLMAT_44100_02

Sznajdman, J., A Briançon-Skoda-type result for a non-reduced analytic space, J. Reine Angew. Math. 742 (2018), 1–16. https://doi.org/10.1515/crelle-2015-0099

Tong, Y. L. L., Integral representation formulae and Grothendieck residue symbol, Amer. J. Math. 95 (1973), 904–917. https://doi.org/10.2307/2373701

Tsikh, A. K., Multidimensional residues and their applications, Translations of Mathematical Monographs, vol. 103, American Mathematical Society, Providence, RI, 1992.

Vasconcelos, W. V., The top of a system of equations, Bol. Soc. Mat. Mexicana (2) 37 (1992), no. 1-2, 549–556.

Wiebe, H., Über homologische Invarianten lokaler Ringe, Math. Ann. 179 (1969), 257–274. https://doi.org/10.1007/BF01350771

Wulcan, E., Products of residue currents of Cauchy-Fantappiè-Leray type, Ark. Mat. 45 (2007), no. 1, 157–178. https://doi.org/10.1007/s11512-006-0036-3
Published
2019-08-29
How to Cite
Lärkäng, R. (2019). A comparison formula for residue currents. MATHEMATICA SCANDINAVICA, 125(1), 39-66. https://doi.org/10.7146/math.scand.a-113032
Section
Articles