Irrational connected sums and the topology of algebraic surfaces
Author:
Richard Mandelbaum
Journal:
Trans. Amer. Math. Soc. 247 (1979), 137156
MSC:
Primary 57R15; Secondary 14J99
MathSciNet review:
517689
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Abstract: Suppose W is an irreducible nonsingular projective algebraic 3fold and V a nonsingular hypersurface section of W. Denote by a nonsingular element of . Let , , be generic elements of , , respectively such that they have normal crossing in W. Let and . Then is a nonsingular curve of genus and C is a collection of points on . By [MM2] we find that is diffeomorphic to , where is a tubular neighborhood of in , is blown up along C, is the strict image of in , is a tubular neighborhood of in and is a bundle diffeomorphism. Now is well known to be diffeomorphic to (the connected sum of and N copies of with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of to ones about we must simplify the ``irrational sum'' above. The general question we can ask is then the following: Suppose and are compact smooth 4manifolds and K is a connected qcomplex embedded in . Let be a regular neighborhood of K in and let be a diffeomorphism: Set . How can the topology of V be described more simply in terms of those of and . In this paper we show how surgery can be used to simplify the structure of V in the case and indicate some applications to the topology of algebraic surfaces.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905176892
PII:
S 00029947(1979)05176892
Article copyright:
© Copyright 1979
American Mathematical Society
