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

DOI:
https://doi.org/10.1090/S0002-9947-1980-0570786-6

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 $aP {\text {\# }} bQ$. By a result of Wall [**W1**] there always exists an integer *k* such that $X \# (k + 1)P \# kQ$ is completely decomposable. If $X \# P$ 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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American Mathematical Society