TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 260, Number 1, July 1980 ON THE TOPOLOGY OF SIMPLY-CONNECTED ALGEBRAIC SURFACES BY RICHARD MANDELBAUM AND BORIS MOISHEZON Abstract. Suppose A" is a smooth simply-connected compact 4-manifold. Let P = CP2 and Q = -CP2 be the complex projective plane with orientation oppo- site to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to aP %bQ. By a result of Wall [Wl] there always exists an integer k such that X Jf (A: + \)P 8 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 CP3 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„ > 1 such that any hypersurface section Vm of W of degree m > mg 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. Introduction. Suppose A" is a simply-connected compact 4-manifold. Let P = CP2 and Q — — CP2 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 s» aP # bQ. (Read '» ' as 'is diffeomorphic to'.) By a result of Wall [Wl], [W2] there always exists an integer k such that X # (k + l)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 nonsingu- lar hypersurface of CP3 is almost completely decomposable. There are several possible ways of generalizing these results. One can consider hypersurface sections of a simply-connected algebraic 3-fold W instead of those of CP3, or one can consider nonsingular algebraic surfaces defined by the intersection of k hyper- surfaces of CPk+2 (so-called complete intersections). For these two possible generalizations we obtain the following results. Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer m0 > 1 such that any hypersurface section Vm of degree m > m0 which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular compact complex algebraic surface which is a complete intersection. Then V is almost completely decomposable. To introduce our other results we must first establish some more terminology. We recall that the field F is called an algebraic function field of two variables over Received by the editors February 20, 1978 and, in revised form, May 8, 1979 and September 6, 1979. AMS (MOS) subject classifications (1970). Primary 57D55, 57A15, 14J99. © 1980 American Mathematical Society O0O2-9947/8O/OOO0-03O9/SO8.00 195 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use