On somes characteristic classes of flat bundles in complex geometry
Annales de l'Institut Fourier, to appear, 23 p.

On a compact Kähler manifold X, 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 X. 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.

Sur une variété kählérienne compacte X, 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 X. 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.

Received : 2017-09-06
Revised : 2018-02-06
Accepted : 2018-03-13
Classification:  57R20,  22E67,  58A14
Keywords: harmonic bundles, non-abelian Hodge theory, flat bundles, loop groups, period domains
@unpublished{AIF_0__0_0_A19_0,
     author = {Daniel, Jeremy},
     title = {On somes characteristic classes of flat bundles in complex geometry},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Daniel, Jeremy. On somes characteristic classes of flat bundles in complex geometry. Annales de l'Institut Fourier, to appear, 23 p.

[1] Amorós, Jaum; Burger, Marc; Corlette, A.; Kotschick, Dieter; Toledo, Domingo Fundamental groups of compact Kähler manifolds, American Mathematical Society, Mathematical Surveys and Monographs, Tome 44 (1996), xi+140 pages | Zbl 0849.32006

[2] Bismut, Jean-Michel; Lott, John Flat vector bundles, direct images and higher real analytic torsion, J. Am. Math. Soc., Tome 8 (1995) no. 2, pp. 291-363 | Zbl 0837.58028

[3] Borel, Armand Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. Math., Tome 57 (1953), pp. 115-207

[4] Borel, Armand Sur l’homologie et la cohomologies des groupes de Lie, Am. J. Math., Tome 76 (1954) no. 2, pp. 273-342

[5] Bott, Raoul; Seeley, Robert Some Remarks on the Paper of Callias, Commun. Math. Phys., Tome 62 (1978), pp. 235-245

[6] Cheeger, Jeff; Simons, James Characteristic Forms and Geometric Invariants, Ann. Math., Tome 99 (1974) no. 1, pp. 48-69

[7] Cheeger, Jeff; Simons, James Differential characters and geometric invariants, Geometry and topology, Springer (Lecture Notes in Mathematics) Tome 1167 (1985) | Zbl 0621.57010

[8] Daniel, Jeremy Loop Hodge structures and harmonic bundles, Algebr. Geom., Tome 4 (2017) no. 5, pp. 603-643 | Zbl 1388.32008

[9] Goette, Sebastian Torsion Invariants for Families, From Probability to Geometry (II). Volume in honor of the 60th birthday of Jean-Michel Bismut, Société Mathématique de France (Astérisque) Tome 328 (2009), pp. 161-206 | Zbl 1247.58019

[10] Griffiths, Phillip Periods of integrals on algebraic manifolds, III (Some global differential-geometric properties of the period mapping), Publ. Math., Inst. Hautes Étud. Sci., Tome 38 (1970), pp. 125-180 | Zbl 0212.53503

[11] Griffiths, Phillip; Schmid, Wilfried Locally homogeneous complex manifolds, Acta Math., Tome 123 (1969), pp. 253-302 | Zbl 0209.25701

[12] Kamber, Franz; Tondeur, Philippe Characteristic invariants of foliated bundles, Manuscr. Math., Tome 11 (1974), pp. 51-89 | Zbl 0267.57012

[13] Le Potier, Joseph Fibrés de Higgs et systèmes locaux, Séminaire Bourbaki, Vol. 1990/91, Société Mathématique de France (Astérisque) Tome 201-203 (1990–1991), pp. 221-268 | Zbl 0762.14011

[14] Massey, William S. A Basic Course in Algebraic Topology, Springer, Graduate Texts in Mathematics, Tome 127 (1991), xiv+428 pages | Zbl 0725.55001

[15] Milnor, John W.; Stasheff, James D. Characteristic classes, Princeton University Press, Annals of Mathematics Studies, Tome 76 (1974), vii+331 pages | Zbl 0298.57008

[16] Pressley, Andrew; Segal, Graeme Loop Groups, Oxford Science Publications, Oxford Mathematical Monographs (1988), viii+318 pages | Zbl 0638.22009

[17] Reznikov, Alexander All regulators of flat bundles are torsion, Ann. Math., Tome 141 (1995) no. 2, pp. 373-386 | Zbl 0865.14003

[18] Reznikov, Alexander The structures of Kähler groups, I: Second cohomology, Motives, polylogarithms and Hodge theory. Part II: Hodge theory, International Press (International Press Lecture Series) Tome 3 (2002), pp. 717-730 | Zbl 1048.32008

[19] Simpson, Carlos Higgs bundles and local systems, Publ. Math., Inst. Hautes Étud. Sci., Tome 75 (1992), pp. 5-95 | Zbl 0814.32003

[20] Simpson, Carlos Mixed twistor structures (1997) (https://arxiv.org/abs/alg-geom/9705006 )

[21] Stacey, Andrew Constructing smooth manifolds of loop spaces, Proc. Lond. Math. Soc., Tome 99 (2009) no. 1, pp. 195-216 | Zbl 1185.58004

[22] Stasheff, James D. Continuous cohomology of groups and classifying spaces, Bull. Am. Math. Soc., Tome 84 (1978) no. 4, pp. 513-530 | Zbl 0399.55009