The first edition of this book was published in 1984
[W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces,
Springer, Berlin, 1984; MR0749574 (86c:32026)]. This second enlarged
edition reflects several important developments in the theory of
compact complex surfaces. A whole new chapter has been added. It
concerns the differentiable structures on surfaces, including the
theories of Donaldson and Seiberg-Witten invariants. The new approach
to the theory of pluricanonical maps of surfaces is based on Reider's
method. Some new results about the geography of surfaces of general
type have been added. The chapter on K3 surfaces now has a new part
discussing mirror symmetry for lattice polarized K3 surfaces,
Mumford's result on the existence of rational curves on K3 surfaces
and applications to hyperbolic structures on algebraic surfaces. Many
historical remarks and an added bibliography play important roles in
updating the material of the book. Still lacking is the theory of
rational surfaces, in particular del Pezzo surfaces and Cremona
transformations. This is surprising, taking into account the recent
active study of differential geometry of such surfaces as well as
their important role in multi-dimensional complex dynamics. In spite
of this gap the book has truly become an encyclopedia of the theory of
compact complex surfaces.
Reviewed by I. Dolgachev