Green functions, Segre numbers, and King’s formula

Mats Andersson[1]; Elizabeth Wulcan[1]

  • [1] Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 6, page 2639-2657
  • ISSN: 0373-0956

Abstract

top
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 ( d d c G ) k for k = 0 , 1 , 2 , ... , and prove that the Lelong numbers of the currents M k 𝒥 : = 1 Z ( d d 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.

How to cite

top

Andersson, Mats, and Wulcan, Elizabeth. "Green functions, Segre numbers, and King’s formula." Annales de l’institut Fourier 64.6 (2014): 2639-2657. <http://eudml.org/doc/275584>.

@article{Andersson2014,
abstract = {Let $\mathcal\{J\}$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log |f|+\mathcal\{O\}(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal\{J\}$. We give a meaning to the Monge-Ampère products $(dd^c G)^k$ for $k=0,1,2,\ldots $, and prove that the Lelong numbers of the currents $M_k^\{\mathcal\{J\}\}:=\{\bf 1\}_Z(dd^c G)^k$ 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 $M_k^\{\mathcal\{J\}\}$ satisfy a certain generalization of the classical King formula.},
affiliation = {Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN; Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN},
author = {Andersson, Mats, Wulcan, Elizabeth},
journal = {Annales de l’institut Fourier},
keywords = {Green function; Segre numbers; Monge-Ampère products; King’s formula; King's formula},
language = {eng},
number = {6},
pages = {2639-2657},
publisher = {Association des Annales de l’institut Fourier},
title = {Green functions, Segre numbers, and King’s formula},
url = {http://eudml.org/doc/275584},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Andersson, Mats
AU - Wulcan, Elizabeth
TI - Green functions, Segre numbers, and King’s formula
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 6
SP - 2639
EP - 2657
AB - Let $\mathcal{J}$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log |f|+\mathcal{O}(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal{J}$. We give a meaning to the Monge-Ampère products $(dd^c G)^k$ for $k=0,1,2,\ldots $, and prove that the Lelong numbers of the currents $M_k^{\mathcal{J}}:={\bf 1}_Z(dd^c G)^k$ 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 $M_k^{\mathcal{J}}$ satisfy a certain generalization of the classical King formula.
LA - eng
KW - Green function; Segre numbers; Monge-Ampère products; King’s formula; King's formula
UR - http://eudml.org/doc/275584
ER -

References

top
  1. R. Achilles, M. Manaresi, Multiplicities of bigraded And Intersection theory, Math. Ann. 309 (1997), 573-591 Zbl0894.14005MR1483824
  2. R. Achilles, S. Rams, Intersection numbers, Segre numbers and generalized Samuel multiplicities, Arch. Math. (Basel) 77 (2001), 391-398 Zbl1032.13012MR1858883
  3. M. Andersson, Residue currents of holomorphic sections and Lelong currents, Arkiv för matematik 43 (2005), 201-219 Zbl1103.32020MR2172988
  4. M. Andersson, H. Samuelsson Kalm, E. Wulcan, A. Yger, Segre numbers, a generalized King formula, and local intersections 
  5. E. Bedford, A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40 Zbl0547.32012MR674165
  6. E. Bedford, A. Taylor, Fine topology, Šilov boundary, and ( d d c ) n , J. Funct. Anal. 72 (1987), 225-251 Zbl0677.31005MR886812
  7. Z. Błocki, (2012) 
  8. S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes, Acta Math. 205 (2010), 199-262 Zbl1213.32025MR2746347
  9. J.-P. Demailly, Complex and Differential geometry Zbl1296.01027
  10. J.-P. Demailly, Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z. 194 (1987), 519-564 Zbl0595.32006MR881709
  11. J.-P. Demailly, Monge-Ampère Operators, Lelong Numbers, and Intersection Theory, Complex analysis and geometry (1993), 115-193, Plenum, New York Zbl0792.32006MR1211880
  12. J.-P. Demailly, H. H. Pham, A sharp lower bound for the log canonical threshold, Acta Math. 212 (2014), 1-9 Zbl1298.14006MR3179606
  13. W. Fulton, Intersection theory, (1998), Springer-Verlag, Berlin-Heidelberg Zbl0541.14005MR1644323
  14. T. Gaffney, R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geom. 8 (1999), 695-736 Zbl0971.13021MR1703611
  15. J. R. King, A residue formula for complex subvarieties, Proc. Carolina conf. on holomoprhic mappings and minimal surfaces (1970), 43-56, Univ. of North Carolina, Chapel Hill Zbl0224.32009MR273061
  16. R. Lazarsfeld, Positivity in Algebraic Geometry II. Positivity for vector bundles, and multiplier ideals, 49 (2004), Springer-Verlag, Berlin Zbl1093.14500MR2095472
  17. D. Massey, Lê cycles and hypersurface singularities, 1615 (1995), Springer-Verlag, Berlin Zbl0835.32002MR1441075
  18. David B. Massey, Numerical control over complex analytic singularities, Mem. Amer. Math. Soc. 163 (2003) Zbl1025.32010MR1962934
  19. A Rashkovskii, Multi-circled Singularities, Lelong Numbers, and Integrability Index, J. Geom. Anal. 23 (2013), 1976-1992 Zbl1282.32013MR3107686
  20. A. Rashkovskii, R. Sigurdsson, Green functions with singularities along complex spaces, Internat. J. Math. 16 (2005), 333-355 Zbl1085.32018MR2133260
  21. Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53-156 Zbl0289.32003MR352516
  22. H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans ℂ n , Bull. Soc. Math. France 100 (1972), 353-408 Zbl0246.32009MR352517
  23. J. Stückrad, W. Vogel, An algebraic approach to the intersection theory, Queen’s Papers in Pure and Appl. Math. 61 (1982), 1-32 Zbl0599.14003MR783085
  24. P. Tworzewski, Intersection theory in complex analytic geometry, Ann. Polon. Math. 62 (1995), 177-191 Zbl0911.32018MR1356791

NotesEmbed ?

top

You must be logged in to post comments.