Three-manifolds and Kähler groups

Kotschick, D.

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1081-1090.

On donne une preuve simple d’un résultat dû à Dimca et Suciu : un groupe de Kähler qui est aussi le groupe fondamental d’une variété trois-dimensionelle est fini. On montre également qu’un groupe qui est le groupe fondamental d’une variété trois-dimensionelle et en même temps le groupe fondamental d’une surface complexe compacte non-kählerienne est soit $\mathbb{Z}$ soit $\mathbb{Z}\oplus {\mathbb{Z}}_2$.

We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kähler and the fundamental group of a closed three-manifold is finite. We also prove that a group that is both the fundamental group of a closed three-manifold and of a non-Kähler compact complex surface is $\mathbb{Z}$ or $\mathbb{Z}\oplus {\mathbb{Z}}_2$.

[1] Amorós, J. On the Malcev completion of Kähler groups, Comment. Math. Helv., Volume 71 (1996), pp. 192-212 | DOI | EuDML | MR | Zbl

[2] Amorós, J.; Burger, M.; Corlette, K.; Kotschick, D.; Toledo, D. Fundamental Groups of Compact Kähler Manifolds, Mathematical Surveys and Monographs, 44, Amer. Math. Soc., Providence, R.I., 1996 | MR | Zbl

[3] Barth, W.; Peters, C.; Van de Ven, A. Compact Complex Surfaces, Mathematical Surveys and Monographs, Springer-Verlag, Berlin, 1984 | MR | Zbl

[4] Buchdahl, N. On compact Kähler surfaces, Ann. Inst. Fourier, Volume 49 (1999), pp. 287-302 | DOI | EuDML | Numdam | MR | Zbl

[5] Carlson, J. A.; Toledo, D. Harmonic mapping of Kähler manifolds to locally symmetric spaces, Publ. Math. I.H.E.S., Volume 69 (1989), pp. 173-201 | EuDML | Numdam | MR | Zbl

[6] Dimca, A.; Suciu, A. I. Which $3$-manifold groups are Kähler groups?, J. Eur. Math. Soc., Volume 11 (2009), pp. 521-528 | DOI | EuDML | MR | Zbl

[7] Gromov, M. Sur le groupe fondamental d’une variété kählérienne, C. R. Acad. Sci. Paris Sér. I Math., Volume 308 (1989), pp. 67-70 | MR | Zbl

[8] Hempel, J. Virtually Haken manifolds, Combinatorial methods in topology and algebraic geometry (Contemp. Math.), Volume 44, Providence, RI, Rochester, N.Y., 1982, Amer. Math. Soc. (1985), pp. 149-155 | MR | Zbl

[9] Hempel, J. Residual finiteness for 3-manifolds, Combinatorial group theory and topology, ed. S. M. Gersten and J. R. Stallings, Ann. Math. Stud. vol. 111, Princeton Univ. Press, 1987 | MR | Zbl

[10] Hernández-Lamoneda, L. Non-positively curved $3$-manifolds with non-Kähler ${\pi }_1$, C. R. Acad. Sci. Paris Sér. I, Volume 332 (2001), pp. 249-252 | DOI | MR

[11] Johnson, F. E. A.; Rees, E. G. On the fundamental group of a complex algebraic manifold, Bull. London Math. Soc., Volume 19 (1987), pp. 463-466 | DOI | MR | Zbl

[12] Kirby, R.; W. H. Kazez Problems in Low-Dimensional Topology, Geometric Topology (AMS/IP Studies in Advanced Mathematics), Volume 2 part 2, American Mathematical Society and International Press (1997) | MR | Zbl

[13] Kleiner, B.; Lott, J. Notes on Perelman’s papers, Geom. Topol., Volume 12 (2008), pp. 2587-2855 | DOI | MR

[14] Klingler, B. Kähler groups and duality, Preprint arXiv:1005.2836v1 [math.GR], 17 May 2010

[15] Kojima, S. Finite covers of $3$-manifolds containing essential surfaces of Euler characteristic $=0$, Proc. Amer. Math. Soc., Volume 101 (1987), pp. 743-747 | MR | Zbl

[16] Kollár, J. Shafarevich maps and automorphic forms, Mathematical Surveys and Monographs, Princeton Univ. Press, Princeton, NJ, 1995 | MR | Zbl

[17] Luecke, L. Finite covers of $3$-manifolds containing essential tori, Trans. Amer. Math. Soc., Volume 310 (1988), pp. 381-391 | MR | Zbl

[18] Milnor, J. W. A unique decomposition theorem for $3$-manifolds, Amer. J. Math., Volume 84 (1962), pp. 1-7 | DOI | MR | Zbl

[19] Morgan, J. W.; Tian, G. Ricci flow and the Poincaré conjecture, Amer. Math. Soc. and Clay Math. Institute, 2007 | MR

[20] Nakamura, I. Towards classification of non-Kählerian complex surfaces, Sugaku Exp., Volume 2 (1989), pp. 209-229 | MR | Zbl

[21] Perelman, G. Ricci flow with surgery on three-manifolds, Preprint arXiv:math/0303109v1 [math.DG], 10 Mar 2003

[22] Perelman, G. The entropy formula for the Ricci flow and its geometric applications, Preprint arXiv:math/0211159v1 [math.DG], 11 Nov 2002

[23] Scott, P. The geometries of $3$-manifolds, Bull. London Math. Soc., Volume 15 (1983), pp. 401-487 | DOI | MR | Zbl

[24] Taubes, C. H. The existence of anti-self-dual conformal structures, J. Differential Geometry, Volume 36 (1992), pp. 163-253 | MR | Zbl

[25] Toledo, D. Examples of fundamental groups of compact Kähler manifolds, Bull. London Math. Soc., Volume 22 (1990), pp. 339-343 | DOI | MR | Zbl