Let be a compact complex manifold and a -linear finite formal sum of divisors of . A theorem of Weil and Kodaira says that if is Kähler, then there is a closed logarithmic -form with residue divisor if and only if is homologous to zero in . We generalized their theorem to general compact complex manifolds. The necessary and sufficient condition is described by a new invariant called -flat class. In the second part of the paper, we classify all the pluriharmonic functions on a compact algebraic manifold with mild singularities.
Soit une variété complexe compacte et une somme formelle finie -linéaire des diviseurs de . Un théorème de Weil et Kodaira dit que si est kählerienne, alors il existe une forme logarithmique fermé avec un diviseur résiduel si et seulement si est homologue à zéro dans . Nous généralisons leur théorème aux variétès complexes compactes générales. La condition nécessaire et suffisante est décrite par un nouvel invariant appelé -flat class. Dans la deuxième partie de l’article, nous classons toutes les fonctions pluriharmoniques sur une variété algébrique compacte avec des singularités douces.
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Keywords: Residue, divisor, meromorphic 1-form, pluriharmonic functions
Mots-clés : Résidus, diviseur, 1-forme méromorphe, Fonction pluri harmonique
Fang, Hanlong 1
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@article{AIF_2021__71_5_1963_0,
author = {Fang, Hanlong},
title = {A geometric criterion for prescribing residues and some applications},
journal = {Annales de l'Institut Fourier},
pages = {1963--2018},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {71},
number = {5},
year = {2021},
doi = {10.5802/aif.3446},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3446/}
}
TY - JOUR AU - Fang, Hanlong TI - A geometric criterion for prescribing residues and some applications JO - Annales de l'Institut Fourier PY - 2021 SP - 1963 EP - 2018 VL - 71 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3446/ DO - 10.5802/aif.3446 LA - en ID - AIF_2021__71_5_1963_0 ER -
%0 Journal Article %A Fang, Hanlong %T A geometric criterion for prescribing residues and some applications %J Annales de l'Institut Fourier %D 2021 %P 1963-2018 %V 71 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3446/ %R 10.5802/aif.3446 %G en %F AIF_2021__71_5_1963_0
Fang, Hanlong. A geometric criterion for prescribing residues and some applications. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 1963-2018. doi : 10.5802/aif.3446. https://aif.centre-mersenne.org/articles/10.5802/aif.3446/
[1] Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer, 2004, xii+436 pages | DOI | MR
[2] Birational geometry of foliations, IMPA Monographs, 1, Springer, 2015, xiv+130 pages | DOI | MR
[3] Théorie de Hodge. III, Publ. Math., Inst. Hautes Étud. Sci. (1974) no. 44, pp. 5-77 | DOI | Numdam | MR | Zbl
[4] Complex analytic and differential geometry (open access book, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)
[5] Modern geometry—methods and applications. Part II The geometry and topology of manifolds, Graduate Texts in Mathematics, 104, Springer, 1985, xv+430 pages (translated from the Russian by Robert G. Burns) | DOI | MR
[6] Construct holomorphic invariants in Čech cohomology by a combinatorial formula (2018) | arXiv
[7] Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, 1994, xiv+813 pages (reprint of the 1978 original) | DOI | MR
[8] Differential topology, Graduate Texts in Mathematics, 33, Springer, 1976, x+221 pages | DOI | MR
[9] Integrals of the second kind on an algebraic variety, Ann. Math., Volume 62 (1955), pp. 56-91 | DOI | MR | Zbl
[10] Green’s forms and meromorphic functions on compact analytic varieties, Can. J. Math., Volume 3 (1951), pp. 108-128 | DOI | MR | Zbl
[11] On compact analytic surfaces. II, Ann. Math., Volume 77 (1963), pp. 563-626 | DOI | Zbl
[12] On compact analytic surfaces. III, Ann. Math., Volume 78 (1963), pp. 1-40 | DOI | MR | Zbl
[13] Stable bundles on Hopf manifolds (2004) | arXiv
[14] Complex parallelisable manifolds and their small deformations, J. Differ. Geom., Volume 10 (1975), pp. 85-112 | MR | Zbl
[15] On surfaces of class with curves, Invent. Math., Volume 78 (1984) no. 3, pp. 393-443 | DOI | MR | Zbl
[16] A short analytic proof of closedness of logarithmic forms, Kodai Math. J., Volume 18 (1995) no. 2, pp. 295-299 | DOI | MR | Zbl
[17] Fibrations, divisors and transcendental leaves, J. Algebr. Geom., Volume 15 (2006) no. 1, pp. 87-110 (with an appendix by Laurent Meersseman) | DOI | MR | Zbl
[18] Comparison of sheaf cohomology and singular cohomology (2016) | arXiv
[19] Topics in complex function theory. Vol. II Automorphic functions and abelian integrals, Wiley Classics Library, John Wiley & Sons, 1988, xii+193 pages (translated from the German by A. Shenitzer and M. Tretkoff, with a preface by Wilhelm Magnus) | MR
[20] Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, 439, Springer, 1975, xix+278 pages (notes written in collaboration with P. Cherenack) | DOI | MR
[21] Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, 2007, x+322 pages (translated from the French by Leila Schneps) | MR
[22] Sur la théorie des formes différentielles attachées à une variété analytique complexe, Comment. Math. Helv., Volume 20 (1947), pp. 110-116 | DOI | MR | Zbl
[23] On manifolds with trivial logarithmic tangent bundle, Osaka J. Math., Volume 41 (2004) no. 2, pp. 473-484 | MR | Zbl
[24] (Private communication, Piscataway, NJ, 2018)
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