The authors survey recent developments in the differential topology of complex algebraic surfaces, and propose a number of wide-ranging conjectures as goals for future research.
The signal feature of these recent developments is the close interaction between the complex geometry of surfaces and their differential topology. For general classes of 4-manifolds new information has been obtained by exploiting the moduli spaces of anti-self-dual Yang-Mills connections, defined with respect to some Riemannian metric. When the 4-manifold is a complex surface and the metric is Kähler the moduli spaces can be identified with those of stable holomorphic bundles, and this brings the complex geometry to the fore. Friedman and Morgan explain that this relation between topology and geometry is analogous to that, stemming from the uniformization theorem, for Riemann surfaces in two dimensions.
In the Enriques-Kodaira rough classification there are just 4 classes of simply connected surfaces: rational, elliptic, K3 and general type. Two complex surfaces are said to be deformation equivalent if they occur as members of a holomorphic family of surfaces over a connected base space. This is an equivalence relation which is a priori finer than that of diffeomorphism. Any K3 surface is deformation equivalent to an elliptic surface, so the rough classification leads to just three categories, in which the rational case is very well understood.
The authors’ conjectures are all in the general direction that the diffeomorphism and deformation classifications of surfaces should be very similar. This is in line with the classical case of Riemann surfaces, but quite opposite to what happens in higher dimensions. One of their main conjectures is that the forgetful map taking deformation classes of simply connected surfaces to diffeomorphism classes is finite-to-one. This conjecture has recently been proved by Friedman and Morgan. (It is possible that the map is one-to-one, so that surfaces are diffeomorphic if and only if they are deformation equivalent, but this seems to lie well beyond present techniques.)
In a similar vein the authors discuss the \(C^{\infty}\)-invariance of the canonical class \(c_ 1(S)\) of a minimal complex surface S. They conjecture that if f: \(S\to S\) is an orientation-preserving diffeomorphism then \(f^*(c_ 1(S))=\pm c_ 1(S).\) This conjecture has been proved for many surfaces. For a non-minimal surface there is a distinguished set of homology classes corresponding to the exceptional curves: Friedman and Morgan conjecture that, if the surface is irrational, the subspace spanned by this set is invariant under diffeomorphisms. They note that this would imply that the only non-zero classes in \(H_ 2(S)\) which can be realized by embedded 2-spheres are those of the exceptional curves.
Moving to the diffeomorphism classification of arbitrary smooth, closed, simply connected oriented, 4-manifolds: an extreme possibility is that any such manifold is diffeomorphic to a connected sum of algebraic surfaces, each with either the standard orientation or the reverse. A weaker conjecture is that any such manifold is homotopy equivalent to a connected sum of this kind. The authors point out that this second conjecture is equivalent to the “eleven-eights” conjecture - that if the intersection form of such a manifold M is even then the second Betti number of M is at least 11/8 of the absolute value of the signature. This reduction is made using the Bogomolov-Miyaoka-Yau inequality for complex surfaces.