Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups
Annales de l'Institut Fourier, Volume 50 (2000) no. 2, p. 633-675

We prove fibration theorems on compact Kähler manifolds with conditions on first cohomology groups of fundamental groups with respect to unitary representations into Hilbert spaces. If the fundamental group T of compact Kähler manifold X violates Property (T) of Kazhdan’s, then H 1 (Gamma,Φ)0 for some unitary representation Φ. By our earlier work there exists a d-closed holomorphic 1-form with coefficients twisted by some unitary representation Φ , possibly non-isomorphic to Φ. Taking norms we obtains a positive semi-definite d-closed (1,1)-form ν sur x, which underlies a semi-Khäler structure. We study meromorphic foliations related to this semi-Khäler structure and another semi-Khäler structure related to the Ricci form to prove fibration theorems on some modification of an unramified finite cover of x. The base manifold is shown to be either a compact complex torus or a variety of logarithmic general type with respect to the multiplicity locus of the holomorphic fibration.

Nous démontrons des théorèmes de fibrations pour des variétés kählériennes compactes, sous des hypothèses sur les premiers groupes de cohomologie des groupes fondamentaux par rapport aux représentations unitaires dans des espaces de Hilbert. Si le groupe fondamental d’une variété kälérienne compacte ne satisfait pas la propriété (T) de Kazhdan, on a H 1 (Γ,Φ)0 pour une certaine représentation unitaire Φ. Dans nos travaux antérieurs nous avons montré l’existence d’une forme 1-forme holomorphe d-fermée non triviale à coefficients tordus selon une certaine représentation unitaireΦ éventuellement non isomorphe à Φ. En prenant des normes nous obtenons une (1,1)-forme fermée semi-positive ν sur X, qui est sous-jacente à une structure semi-kählérienne. Nous étudions les feuilletages méromorphes provenant de cette structure semi-kählérienne et d’une autre structure semi-kählérienne liée à la courbure de Ricci pour démontrer des théorèmes de fibration, en passant à une modification d’un revêtement fini non ramifié de X. La base de cette fibration est soit un tore compact complexe, soit une variété de type général logarithmique par rapport au lieu de multiplicités de la fibration holomorphe construite.

@article{AIF_2000__50_2_633_0,
     author = {Mok, Ngaiming},
     title = {Fibrations of compact K\"ahler manifolds in terms of cohomological properties of their fundamental groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     pages = {633-675},
     doi = {10.5802/aif.1767},
     zbl = {0986.53023},
     mrnumber = {2001k:32035},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_2_633_0}
}
Mok, Ngaiming. Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups. Annales de l'Institut Fourier, Volume 50 (2000) no. 2, pp. 633-675. doi : 10.5802/aif.1767. http://www.numdam.org/item/AIF_2000__50_2_633_0/

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