Groups Which Cannot be Realized as Fundamental Groups of the Complements to Hypersurface in CN
Finding restrictions imposed on a group by the fact that it can appear as a fundamental group of a smooth algebraic variety is an important problem particularly attributed to J.P. Serre. It has rather different aspects in characteristic p and zero and here we will address exclusively the latter case. Most restrictions described in the literature seem to rely on Hodge theory or some clever use of it (cf. ). A prototype of such restrictions is evenness of rk(π1/π′1 ⊗ Q)where π′1 is the commutator subgroup of the fundamental group 1rl. In the case of open non-singular varieties one can apply mixed Hodge theory. This was done by J. Morgan who obtained restrictions on the nilpotent quotients of the fundamental groups . Here I shall describe a different (but also by no means complete) type of restriction on the fundamental groups of open varieties which are complements to hypersurfaces in C n It is implicitly contained in previous work on Alexander polynomial of plane curves . For example many knot groups cannot occur as fundamental groups of the complement to an algebraic curve. This gives automatically the same restrictions on the fundamental groups of complements to arbitrary hypersurfaces in C n as follows from the well-known argument using Zariski Lefschetz type theorem: For a generic plane H relative to given hypersurface V in C n the natural map \(\pi _1 (H - H \cap V,p_0 ) \to \pi _1 (C^n - V,p_0 )(p_0 \in H)\)is an isomorphism, i.e., possible fundamental groups of the complement to hypersurfaces in C n are precisely the fundamental groups of the complements to plane curves. Therefore from now on I shall work with plane curves only.
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