On the Kähler Rank of Compact Complex Surfaces

Matei Toma

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 2, page 243-260
  • ISSN: 0037-9484

Abstract

top
Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension ( 1 , 1 ) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions ( ℂ 2 , 0 ) → ( ℂ 2 , 0 ) .

How to cite

top

Toma, Matei. "On the Kähler Rank of Compact Complex Surfaces." Bulletin de la Société Mathématique de France 136.2 (2008): 243-260. <http://eudml.org/doc/272489>.

@article{Toma2008,
abstract = {Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension $(1,1)$ for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions $(\mathbb \{C\}^2,0)\rightarrow (\mathbb \{C\}^2,0)$.},
author = {Toma, Matei},
journal = {Bulletin de la Société Mathématique de France},
keywords = {compact complex surface; global spherical shell; closed positive current; iteration of polynomial maps},
language = {eng},
number = {2},
pages = {243-260},
publisher = {Société mathématique de France},
title = {On the Kähler Rank of Compact Complex Surfaces},
url = {http://eudml.org/doc/272489},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Toma, Matei
TI - On the Kähler Rank of Compact Complex Surfaces
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 2
SP - 243
EP - 260
AB - Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension $(1,1)$ for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions $(\mathbb {C}^2,0)\rightarrow (\mathbb {C}^2,0)$.
LA - eng
KW - compact complex surface; global spherical shell; closed positive current; iteration of polynomial maps
UR - http://eudml.org/doc/272489
ER -

References

top
  1. [1] W. P. Barth, K. Hulek, C. A. M. Peters & A. Van de Ven – Compact complex surfaces, second éd., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 4, Springer, 2004. Zbl0718.14023MR2030225
  2. [2] J.-P. Demailly – « Complex analytic and algebraic geometry », http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, 2007. 
  3. [3] G. Dloussky – « Structure des surfaces de Kato », Mém. Soc. Math. France (N.S.) (1984), p. 120. Zbl0543.32012MR763959
  4. [4] —, « Sur la classification des germes d’applications holomorphes contractantes », Math. Ann.280 (1988), p. 649–661. Zbl0677.32004MR939924
  5. [5] G. Dloussky & K. Oeljeklaus – « Vector fields and foliations on compact surfaces of class VII 0 », Ann. Inst. Fourier (Grenoble) 49 (1999), p. 1503–1545. Zbl0978.32021MR1723825
  6. [6] C. Favre – « Classification of 2-dimensional contracting rigid germs and Kato surfaces. I », J. Math. Pures Appl. (9) 79 (2000), p. 475–514. Zbl0983.32023MR1759437
  7. [7] H. Grauert & R. Remmert – « Plurisubharmonische Funktionen in komplexen Räumen », Math. Z.65 (1956), p. 175–194. Zbl0070.30403MR81960
  8. [8] R. Harvey & H. B. Lawson, Jr. – « An intrinsic characterization of Kähler manifolds », Invent. Math.74 (1983), p. 169–198. Zbl0553.32008MR723213
  9. [9] M. Kato – « Compact complex manifolds containing “global” spherical shells », Proc. Japan Acad.53 (1977), p. 15–16. Zbl0379.32023MR440076
  10. [10] A. Lamari – « Courants kählériens et surfaces compactes », Ann. Inst. Fourier (Grenoble) 49 (1999), p. 263–285. Zbl0926.32026MR1688140
  11. [11] —, « Le cône kählérien d’une surface », J. Math. Pures Appl. (9) 78 (1999), p. 249–263. Zbl0941.32007
  12. [12] M. Meo – « Image inverse d’un courant positif fermé par une application analytique surjective », C. R. Acad. Sci. Paris Sér. I Math.322 (1996), p. 1141–1144. Zbl0858.32012MR1396655
  13. [13] I. Nakamura – « On surfaces of class VII 0 with curves. II », Tohoku Math. J. (2) 42 (1990), p. 475–516. Zbl0732.14019MR1076173

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.