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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the topology of simply connected algebraic surfaces

Authors: Richard Mandelbaum and Boris Moishezon
Journal: Trans. Amer. Math. Soc. 260 (1980), 195-222
MSC: Primary 14J99; Secondary 57R99
MathSciNet review: 570786
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Abstract: Suppose x is a smooth simply-connected compact 4-manifold. Let $ p\, = \,{\textbf{C}}{P^2}$ and $ Q\, = \, - {\textbf{C}}{P^2}$ be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to $ aP\,{\text{\char93 }}\,bQ$.

By a result of Wall [W1] there always exists an integer k such that $ X\,\char93 \,(k\, + \,1)P\,\char93 kQ$ is completely decomposable. If $ X\,\char93 \,P$ is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of $ {\textbf{C}}{P^3}$ is almost completely decomposable. In this paper we generalize this result in two directions as follows:

Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer $ {m_0}\, \geqslant \,1$ such that any hypersurface section $ {V_m}$ of W of degree $ m\, \geqslant \,{m_0}$ which is nonsingular will be almost completely decomposable.

Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable.

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