×

zbMATH — the first resource for mathematics

Projective varieties with non-residually finite fundamental group. (English) Zbl 0818.14009
This paper provides an answer to a folklore problem of J.-P. Serre by showing that there are smooth complex projective varieties whose fundamental groups are not residually finite. Examples are constructed using interesting techniques: blowing down a smooth divisor, showing the result is projective, and an example is obtained by taking a hyperplane section, using Goresky-MacPherson’s Lefschetz theorem to see the fundamental group. – This technique is also used to answer a question of Gromov, showing that a smooth projective surface with fundamental group a cocompact lattice in \(SU(1,3)\) need not be birationally (or even homotopy) equivalent to a hyperplane section of the corresponding quotient of the three-ball.

MSC:
14F35 Homotopy theory and fundamental groups in algebraic geometry
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
PDF BibTeX Cite
Full Text: DOI Numdam EuDML
References:
[1] E. Ballico, F. Catanese, C. Ciliberto (Eds.), Classification of Irregular Varieties,Lect. Notes in Math.,1515, Springer, 1970.
[2] A. Borel, Compact Clifford-Klein forms of symmetric spaces,Topology,2 (1968), 111–122. · Zbl 0116.38603
[3] A. Borel,Introduction aux groupes arithmétiques, Paris, Hermann, 1969. · Zbl 0186.33202
[4] J. Carlson andD. Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces,Publ. Math. IHES,69 (1989), 173–201. · Zbl 0695.58010
[5] P. Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques,C. R. Acad. Sci. Paris, série A–B,287 (1978), 203–208. · Zbl 0416.20042
[6] M. Goresky andR. MacPherson,Stratified Morse Theory, Springer Verlag, 1988.
[7] H. Grauert, Über Modifikationen und exzeptionnelle analytische Mengen,Math. Annalen,146 (1962), 331–368. · Zbl 0173.33004
[8] M. Gromov, Sur le groupe fondamental d’une variété kählérienne,C. R. Acad. Sc. Paris, série 1,308 (1980), 67–70.
[9] R. Hartshorne, Ample Subvarieties of Algebraic Varieties,Lect. Notes in Math. 156, Springer, 1970. · Zbl 0208.48901
[10] F. Hirzebruch, Automorphe Formen und der Satz von Riemann-Roch,Symposium Internacional de Topología Algebraica, México, 1956, 129–144.
[11] S. L. Kleiman, Towards a numerical theory of ampleness,Ann. of Math.,84 (1966), 293–344. · Zbl 0146.17001
[12] K. Kodaira, On Kähler varieties of restricted type,Ann. of Math.,60 (1954), 28–48. · Zbl 0057.14102
[13] A. I. Mal’cev, On the faithful representation of infinite groups by matrices,Math. Sbornik,9 (1940), 405–422; English translation,AMS Transl.,45 (1965), 1–18.
[14] G. A. Margulis, Finiteness of quotient groups of discrete subgroups,Func. Anal. Appl.,13 (1979), 178–187.
[15] G. Mess, The Torelli group for genus 2 and 3 surfaces,Topology,31 (1992), 775–790. · Zbl 0772.57025
[16] J. Millson, Real vector bundles with discrete structure group,Topology,18 (1979), 83–89. · Zbl 0452.55014
[17] G. D. Mostow,Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies 78, Princeton Univ. Press, 1973. · Zbl 0265.53039
[18] M. S. Raghunathan, Torsion in cocompact lattices in coverings of Spin (2,n),Math. Annalen,266 (1984), 403–419. · Zbl 0529.22008
[19] J.-P. Serre, Arbres, amalgames, SL2,Astérisque, no 46, 1977; english translation :Trees, Springer, 1980.
[20] D. Toledo, Examples of fundamental groups of compact Kähler manifolds,Bull. London Math. Soc.,22 (1990), 339–343. · Zbl 0711.57024
[21] O. Zariski,Algebraic Surfaces (Second Supplemented Edition), Springer, 1971. · Zbl 0219.14020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.