Knotting of Curves in Complex Algebraic Surfaces
Sergey Finashin
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{\bf Theorem.} For any $d>4$ there exists an infinite family of smooth
surfaces, $F_i$, in $CP^2$ which are homeomorphic to an algebraic
curve, $A$, of degree $d$, realize the same homology class as $A$ and
have the same as $A$ (i.e., abelian) fundamental group of the
complement, $\pi_1(CP^2-F_i)$, but with the pairs $(CP^2,F_i)$
pairwise smoothly non-equivalent.
Surfaces $F_i$ are obtained from $A$ by an annulus rim surgery.
This surgery construction is a
modification of the rim surgery used by Fintushel and Stern for
similar knotting in the case of trivial fundamental group of the
complement.
The annulus rim surgery can be also used for knottings of curves in
any complex algebraic surface provided the curves admit certain
degenerations.
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