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.


14F35 Homotopy theory and fundamental groups in algebraic geometry
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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