Green functions, Segre numbers, and King’s formula
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2639-2657.

Let 𝒥 be a coherent ideal sheaf on a complex manifold X with zero set Z, and let G be a plurisubharmonic function such that G=log|f|+𝒪(1) locally at Z, where f is a tuple of holomorphic functions that defines 𝒥. We give a meaning to the Monge-Ampère products (dd c G) k for k=0,1,2,..., and prove that the Lelong numbers of the currents M k 𝒥 :=1 Z (dd c G) k at x coincide with the so-called Segre numbers of J at x, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that M k 𝒥 satisfy a certain generalization of the classical King formula.

Soit 𝒥 un faisceau cohérent d’ideaux sur un variété complexe lisse X, et soit Z la variété de 𝒥. Soit G une fonction plurisousharmonique telle que G=log|f|+𝒪(1) localement sur Z, où f est un n-uple de fonctions holomorphes qui définit 𝒥. Nous donnons un sens au produit de Monge-Ampère (dd c G) k pour k=0,1,2,..., et nous montrons que les nombres de Lelong des courants M k 𝒥 :=1 Z (dd c G) k en x coïncident avec les nombres de Segre de 𝒥 en x, introduits indépendemment par Tworzewski, Gaffney-Gassler et Achilles-Manaresi. Plus généralement, nous montrons que les M k 𝒥 satisfont une certaine généralisation de la formule de King.

DOI: 10.5802/aif.2922
Classification: 32U35, 32U25, 32U40, 32B30, 14B05
Keywords: Green function, Segre numbers, Monge-Ampère products, King’s formula
Mot clés : Fonctions de Green, nombres de Segre, produits de Monge-Ampère, formule de King
Andersson, Mats 1; Wulcan, Elizabeth 1

1 Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN
     author = {Andersson, Mats and Wulcan, Elizabeth},
     title = {Green functions, {Segre} numbers,  and {King{\textquoteright}s} formula},
     journal = {Annales de l'Institut Fourier},
     pages = {2639--2657},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     doi = {10.5802/aif.2922},
     mrnumber = {3331176},
     zbl = {06387349},
     language = {en},
     url = {}
AU  - Andersson, Mats
AU  - Wulcan, Elizabeth
TI  - Green functions, Segre numbers,  and King’s formula
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 2639
EP  - 2657
VL  - 64
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  -
DO  - 10.5802/aif.2922
LA  - en
ID  - AIF_2014__64_6_2639_0
ER  - 
%0 Journal Article
%A Andersson, Mats
%A Wulcan, Elizabeth
%T Green functions, Segre numbers,  and King’s formula
%J Annales de l'Institut Fourier
%D 2014
%P 2639-2657
%V 64
%N 6
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.2922
%G en
%F AIF_2014__64_6_2639_0
Andersson, Mats; Wulcan, Elizabeth. Green functions, Segre numbers,  and King’s formula. Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2639-2657. doi : 10.5802/aif.2922.

[1] Achilles, R.; Manaresi, M. Multiplicities of bigraded And Intersection theory, Math. Ann., Volume 309 (1997), pp. 573-591 | DOI | MR | Zbl

[2] Achilles, R.; Rams, S. Intersection numbers, Segre numbers and generalized Samuel multiplicities, Arch. Math. (Basel), Volume 77 (2001), pp. 391-398 | DOI | MR | Zbl

[3] Andersson, M. Residue currents of holomorphic sections and Lelong currents, Arkiv för matematik, Volume 43 (2005), pp. 201-219 | DOI | MR | Zbl

[4] Andersson, M.; Samuelsson Kalm, H.; Wulcan, E.; Yger, A. Segre numbers, a generalized King formula, and local intersections (arXiv:1009.2458v3)

[5] Bedford, E.; Taylor, A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982) no. 1–2, pp. 1-40 | DOI | MR | Zbl

[6] Bedford, E.; Taylor, A. Fine topology, Šilov boundary, and (dd c ) n , J. Funct. Anal., Volume 72 (1987) no. 2, pp. 225-251 | DOI | MR | Zbl

[7] Błocki, Z., 2012 (Personal communication)

[8] Boucksom, S.; Eyssidieux, P.; Guedj, V.; Zeriahi, A. Monge-Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010), pp. 199-262 | DOI | MR | Zbl

[9] Demailly, J.-P. Complex and Differential geometry (available at

[10] Demailly, J.-P. Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z., Volume 194 (1987) no. 4, pp. 519-564 | DOI | MR | Zbl

[11] Demailly, J.-P. Monge-Ampère Operators, Lelong Numbers, and Intersection Theory, Complex analysis and geometry (Univ. Ser. Math.), Plenum, New York, 1993, pp. 115-193 | MR | Zbl

[12] Demailly, J.-P.; Pham, H. H. A sharp lower bound for the log canonical threshold, Acta Math., Volume 212 (2014), pp. 1-9 | DOI | MR

[13] Fulton, W. Intersection theory, Springer-Verlag, Berlin-Heidelberg, 1998 | MR | Zbl

[14] Gaffney, T.; Gassler, R. Segre numbers and hypersurface singularities, J. Algebraic Geom., Volume 8 (1999), pp. 695-736 | MR | Zbl

[15] King, J. R. A residue formula for complex subvarieties, Proc. Carolina conf. on holomoprhic mappings and minimal surfaces, Univ. of North Carolina, Chapel Hill, 1970, pp. 43-56 | MR | Zbl

[16] Lazarsfeld, R. Positivity in Algebraic Geometry II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 49, Springer-Verlag, Berlin, 2004 | MR | Zbl

[17] Massey, D. Lê cycles and hypersurface singularities, Lecture Notes in Mathematics, 1615, Springer-Verlag, Berlin, 1995, pp. xii+131 | MR | Zbl

[18] Massey, David B. Numerical control over complex analytic singularities, Mem. Amer. Math. Soc., Volume 163 (2003) no. 778, pp. xii+268 | DOI | MR | Zbl

[19] Rashkovskii, A Multi-circled Singularities, Lelong Numbers, and Integrability Index, J. Geom. Anal., Volume 23 (2013), pp. 1976-1992 | DOI | MR | Zbl

[20] Rashkovskii, A.; Sigurdsson, R. Green functions with singularities along complex spaces, Internat. J. Math., Volume 16 (2005), pp. 333-355 | DOI | MR | Zbl

[21] Siu, Y. T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., Volume 27 (1974), pp. 53-156 | DOI | MR | Zbl

[22] Skoda, H. Sous-ensembles analytiques d’ordre fini ou infini dans n , Bull. Soc. Math. France, Volume 100 (1972), pp. 353-408 | Numdam | MR | Zbl

[23] Stückrad, J.; Vogel, W. An algebraic approach to the intersection theory, Queen’s Papers in Pure and Appl. Math., Volume 61 (1982), pp. 1-32 | MR | Zbl

[24] Tworzewski, P. Intersection theory in complex analytic geometry, Ann. Polon. Math., Volume 62 (1995), pp. 177-191 | MR | Zbl

Cited by Sources: