[Classes caractéristiques de fibrés plats en géométrie complexe]
Sur une variété kählérienne compacte , tout fibré plat semi-simple admet une métrique harmonique. On peut grâce à elle définir certaines classes caractéristiques du fibré plat, dans la cohomologie de . Nous montrons que ces classes de cohomologie proviennent d’un espace de dimension infinie construit à partir de groupes de lacets, cet espace étant un analogue des domaines de périodes de la théorie de Hodge.
On a compact Kähler manifold , any semisimple flat bundle carries a harmonic metric. It can be used to define some characteristic classes of the flat bundle, in the cohomology of . We show that these cohomology classes come from an infinite-dimensional space, constructed with loop groups, an analogue of the period domains used in Hodge theory.
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3255
Keywords: harmonic bundles, non-abelian Hodge theory, flat bundles, loop groups, period domains
Mot clés : fibrés harmoniques, théorie de Hodge non-abélienne, fibrés plats, groupes de lacets, domaines de périodes
@article{AIF_2019__69_2_729_0,
author = {Daniel, Jeremy},
title = {On somes characteristic classes of flat bundles in complex geometry},
journal = {Annales de l'Institut Fourier},
pages = {729--751},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {69},
number = {2},
year = {2019},
doi = {10.5802/aif.3255},
zbl = {07067416},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3255/}
}
TY - JOUR AU - Daniel, Jeremy TI - On somes characteristic classes of flat bundles in complex geometry JO - Annales de l'Institut Fourier PY - 2019 SP - 729 EP - 751 VL - 69 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3255/ DO - 10.5802/aif.3255 LA - en ID - AIF_2019__69_2_729_0 ER -
%0 Journal Article %A Daniel, Jeremy %T On somes characteristic classes of flat bundles in complex geometry %J Annales de l'Institut Fourier %D 2019 %P 729-751 %V 69 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3255/ %R 10.5802/aif.3255 %G en %F AIF_2019__69_2_729_0
Daniel, Jeremy. On somes characteristic classes of flat bundles in complex geometry. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 729-751. doi : 10.5802/aif.3255. http://www.numdam.org/articles/10.5802/aif.3255/
[1] Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, 44, American Mathematical Society, 1996, xi+140 pages | Zbl
[2] Flat vector bundles, direct images and higher real analytic torsion, J. Am. Math. Soc., Volume 8 (1995) no. 2, pp. 291-363 | DOI | MR | Zbl
[3] Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. Math., Volume 57 (1953), pp. 115-207 | DOI | Zbl
[4] Sur l’homologie et la cohomologies des groupes de Lie, Am. J. Math., Volume 76 (1954) no. 2, pp. 273-342 | DOI | MR | Zbl
[5] Some Remarks on the Paper of Callias, Commun. Math. Phys., Volume 62 (1978), pp. 235-245 | DOI | Zbl
[6] Characteristic Forms and Geometric Invariants, Ann. Math., Volume 99 (1974) no. 1, pp. 48-69 | MR
[7] Differential characters and geometric invariants, Geometry and topology (Lecture Notes in Mathematics), Volume 1167, Springer, 1985 | DOI | MR | Zbl
[8] Loop Hodge structures and harmonic bundles, Algebr. Geom., Volume 4 (2017) no. 5, pp. 603-643 | MR | Zbl
[9] Torsion Invariants for Families, From Probability to Geometry (II). Volume in honor of the 60th birthday of Jean-Michel Bismut (Astérisque), Volume 328, Société Mathématique de France, 2009, pp. 161-206 | MR | Zbl
[10] Periods of integrals on algebraic manifolds, III (Some global differential-geometric properties of the period mapping), Publ. Math., Inst. Hautes Étud. Sci., Volume 38 (1970), pp. 125-180 | DOI | Zbl
[11] Locally homogeneous complex manifolds, Acta Math., Volume 123 (1969), pp. 253-302 | DOI | MR | Zbl
[12] Characteristic invariants of foliated bundles, Manuscr. Math., Volume 11 (1974), pp. 51-89 | DOI | MR | Zbl
[13] Fibrés de Higgs et systèmes locaux, Séminaire Bourbaki, Vol. 1990/91 (Astérisque), Volume 201-203, Société Mathématique de France, 1990–1991, pp. 221-268 | Zbl
[14] A Basic Course in Algebraic Topology, Graduate Texts in Mathematics, 127, Springer, 1991, xiv+428 pages | MR | Zbl
[15] Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press, 1974, vii+331 pages | Zbl
[16] Loop Groups, Oxford Mathematical Monographs, Oxford Science Publications, 1988, viii+318 pages | Zbl
[17] All regulators of flat bundles are torsion, Ann. Math., Volume 141 (1995) no. 2, pp. 373-386 | MR | Zbl
[18] The structures of Kähler groups, I: Second cohomology, Motives, polylogarithms and Hodge theory. Part II: Hodge theory (International Press Lecture Series), Volume 3, International Press (2002), pp. 717-730 | MR | Zbl
[19] Higgs bundles and local systems, Publ. Math., Inst. Hautes Étud. Sci., Volume 75 (1992), pp. 5-95 | DOI | Zbl
[20] Mixed twistor structures (1997) (https://arxiv.org/abs/alg-geom/9705006)
[21] Constructing smooth manifolds of loop spaces, Proc. Lond. Math. Soc., Volume 99 (2009) no. 1, pp. 195-216 | DOI | MR | Zbl
[22] Continuous cohomology of groups and classifying spaces, Bull. Am. Math. Soc., Volume 84 (1978) no. 4, pp. 513-530 | DOI | MR | Zbl
Cité par Sources :
