On Bochner-Martinelli residue currents and their annihilator ideals
[Sur les courants résiduels de type Bochner-Martinelli et leurs idéaux annihilateurs]
Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2119-2142.

Nous étudions le courant résiduel de type Bochner-Martinelli R f associé à un m-uple de germes de fonctions holomorphes f=(f 1 ,,f m ) définies à l’origine 0C n dont l’ensemble des zéros communs se réduit à 0. Nos résultats principaux sont : une description géométrique de R f en terme des valuations de Rees associées à l’idéal (f) engendré par f et la caractérisation du cas où l’idéal annihilateur de R f est égal à (f).

We study the residue current R f of Bochner-Martinelli type associated with a tuple f=(f 1 ,,f m ) of holomorphic germs at 0C n , whose common zero set equals the origin. Our main results are a geometric description of R f in terms of the Rees valuations associated with the ideal (f) generated by f and a characterization of when the annihilator ideal of R f equals (f).

DOI : https://doi.org/10.5802/aif.2485
Classification : 32A26,  32A27,  32S45
Mots clés : courant résiduel, idéal annihilateur, valuation de Rees
@article{AIF_2009__59_6_2119_0,
     author = {Jonsson, Mattias and Wulcan, Elizabeth},
     title = {On Bochner-Martinelli residue currents and their annihilator ideals},
     journal = {Annales de l'Institut Fourier},
     pages = {2119--2142},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     doi = {10.5802/aif.2485},
     mrnumber = {2640915},
     zbl = {1189.32003},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2485/}
}
Jonsson, Mattias; Wulcan, Elizabeth. On Bochner-Martinelli residue currents and their annihilator ideals. Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2119-2142. doi : 10.5802/aif.2485. http://www.numdam.org/articles/10.5802/aif.2485/

[1] Andersson, M. Uniqueness and factorization of Coleff-Herrera currents, Preprint, to appear in Ann. Fac. Sci. Toulouse Math

[2] Andersson, M. Residue currents and ideals of holomorphic functions, Bull. Sci. Math., Volume 128 (2004) no. 6, pp. 481-512 | Article | MR 2074610 | Zbl 1086.32005

[3] Andersson, M. Residues of holomorphic sections and Lelong currents, Ark. Mat., Volume 43 (2005) no. 2, pp. 201-219 | Article | MR 2172988 | Zbl 1103.32020

[4] Andersson, M.; Götmark, E. Explicit representation of membership of polynomial ideals, Preprint, Göteborg, available at arXiv:0806.2592

[5] Andersson, M.; Samuelsson, H.; Sznajdman, J. On the Briancon-Skoda theorem on a singular variety, Preprint, to appear in Ann. Inst. Fourier

[6] Andersson, M.; Wulcan, E. Decomposition of residue currents, to appear in Journal für die reine und angewandte Mathematik, available at arXiv:0710.2016

[7] Andersson, M.; Wulcan, E. Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup., Volume 40 (2007) no. 6, pp. 985-1007 | Numdam | MR 2419855 | Zbl 1143.32003

[8] Berenstein, C. A.; Gay, R.; Vidras, A.; Yger, A. Residue currents and Bezout identities, Progress in Mathematics, 114, Birkhäuser Verlag, 1993 | MR 1249478 | Zbl 0802.32001

[9] Berenstein, C. A.; Yger, A. Analytic residue theory in the non-complete intersection case, J. Reine Angew. Math., Volume 527 (2000), pp. 203-235 | Article | MR 1794023 | Zbl 0960.32004

[10] Björk, J-E. Residues and 𝒟 -modules, The legacy of Niels Henrik Abel (2004), pp. 605-651 | MR 2077588 | Zbl 1069.32001

[11] Briançon, J.; Skoda, H. Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de n , C. R. Acad. Sci. Paris Sér. A, Volume 278 (1974), pp. 949-951 | MR 340642 | Zbl 0307.32007

[12] Bruns, W.; Herzog, J. Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993 | MR 1251956 | Zbl 0788.13005

[13] Coleff, N.; Herrera, M. Les courants résiduels associcés à une forme méromorphe, Lecture Notes in Mathematics, 633, Springer Verlag, Berlin, 1978 | MR 492769 | Zbl 0371.32007

[14] Dickenstein, A.; Sessa, C. Canonical representatives in moderate cohomology, Invent. Math., Volume 80 (1985), pp. 417-434 | Article | MR 791667 | Zbl 0556.32005

[15] Hartshorne, R. Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer Verlag, Berlin, 1977 | MR 463157 | Zbl 0367.14001

[16] Heinzer, W.; Ratliff, L. J.; Shah, K. On the irreducible components of an ideal, Comm. Algebra, Volume 25 (1997) no. 5, pp. 1609-1634 | Article | MR 1444023 | Zbl 0872.13002

[17] Hickel, M. Une note à propos du Jacobien de n fonctions holomorphes à l’origine de n , Preprint, 2007 | Zbl 1154.32010

[18] Huneke, C.; Swanson, I. Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006 | MR 2266432 | Zbl 1117.13001

[19] Lazarsfeld, R. Positivity in algebraic geometry. I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, 49, Springer-Verlag, Berlin, 2004 | MR 2095471 | Zbl 1093.14500

[20] Passare, M. Residues, currents, and their relation to ideals of holomorphic functions, Math. Scand., Volume 62 (1988) no. 1, pp. 75-152 | MR 961584 | Zbl 0633.32005

[21] Passare, M.; Tsikh, A.; Yger, A. Residue currents of the Bochner-Martinelli type, Publ. Mat., Volume 44 (2000), pp. 85-117 | MR 1775747 | Zbl 0964.32003

[22] Teissier, B. Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981), Lecture Notes in Mathematics, 961, Springer-Verlag, Berlin, 1982 | MR 708342 | Zbl 0585.14008

[23] Vidras, A..; Yger, A. On some generalizations of Jacobi’s residue formula, Ann. Sci. École Norm. Sup., Volume 34 (2001) no. 1, pp. 131-157 | Numdam | MR 1833092 | Zbl 0991.32003

[24] Wulcan, E. Residue currents constructed from resolutions of monomial ideals, To appear in Math. Z.  available at arXiv:math/0702847

[25] Wulcan, E. Residue currents of monomial ideals, Indiana Univ. Math. J., Volume 56 (2007) no. 1, pp. 365-388 | Article | MR 2305939 | Zbl 1120.32004