On the topology of simply connected algebraic surfaces

Abstract

Suppose x is a smooth simply-connected compact 4-manifold. Let p = C P 2 p\, = \,{\textbf {C}}{P^2} and Q = − C P 2 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 a P \# b Q aP\,{\text {\# }}\,bQ . By a result of Wall [W1] there always exists an integer k such that X # ( k + 1 ) P # k Q X\,\# \,(k\, + \,1)P\,\# kQ is completely decomposable. If X # P X\,\# \,P is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of C P 3 {\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 ⩾ 1 {m_0}\, \geqslant \,1 such that any hypersurface section V m {V_m} of W of degree m ⩾ m 0 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.