On sums of algebraic surfaces (Q1112381)

We examine diffeomorphism types of connected sums of algebraic surfaces. We consider a collection S consisting of all simply connected algebraic surfaces except for \({\mathbb{C}}P^ 2\) and possibly some surfaces of general type. All complete intersections except for \({\mathbb{C}}P^ 2\) are included in S, as are all simply connected elliptic surfaces. We prove that for \(M,N\in S\), if M or N has odd intersection form then the connected sum \(M{\#}\bar N\) decomposes as a connected sum of \(\pm {\mathbb{C}}P^ 2\)'s. Here, \(\bar N\) denotes N with orientation opposite to the canonical complex orientation. This choice of orientation seems to be crucial, since we exploit embedded spheres of positive square in \(\bar N,\) and these rarely exist in algebraic surfaces with their usual orientations. In more recent work (to appear), the case where both M and N have even forms has been examined. In particular, if M and N are even and elliptic, \(M{\#}\bar N\) decomposes as a connected sum of K3 surfaces and \(S^ 2\times S^ 2\)'s. Similarly, if S is a ``fiber sum'' of simply connected elliptic surfaces with incompatible orientations, and if S is simply connected, then S admits one of the above two types of decompositions (depending on whether S is even or odd).