BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

The book contains a collection of articles by participants of the Working Week on Resolution of Singularities held at Obergurgl in Tirol, September 7-14, 1997. It is dedicated to Oscar Zariski, the founder of the school of algebraic geometry in the United States. During his long career as a mathematician he obtained groundbreaking results in algebra and algebraic geometry. Many years of his career were dedicated to the desingularization problem. His major achievements were the modernization of the classical theory of blow-ups and the proof of the existence of resolutions of singularities in dimension three. The main focus of the book is on the substantial recent progress in the desingularization problem resulting from a rather powerful shift of the approach to the whole subject pioneered by J. de Jong. In order to appreciate this change we need to consider the history of the subject. In geometry we are often dealing with objects which are locally similar at most points but exhibit exceptional behavior at a subset of points of smaller dimension. This subset is called the singular locus. More concretely, let us consider a simple topological version of the desingularization problem. Let X be a finite connected polyhedron of dimension d. Its smooth (or nonsingular) points are those which have a small neighborhood isomorphic to a ball. If we assume that X is a manifold without boundary and that every point ofX lies on a simplex of dimension d, then a desingularization of X is a smooth manifold X ′ (without boundary) together with a surjective map f : X ′ → X such that f is an isomorphism on an open, everywhere dense subset U ⊂ X ′. Using the desingularization (X ′, f) we can distribute the complexity concentrated at a singular point x in X over the subcomplex in X ′ containing the preimage f−1(x). Unfortunately, in such a natural geometric setting a resolution does not exist in general! The simplest counterexample is given by a union of two copies of a cone over a real manifold which is not cobordant to zero. For example, an isolated unresolvable singularity would be a real cone over the real(!) fourfold CP. Thus already in dimension 5 we can build a polyhedron which is a smooth manifold away from two singular points and which does not admit a resolution of singularities. Amazingly enough, resolutions do exist for polyhedra associated with solutions of polynomial equations with coefficients in complex numbers. These very special subsets of complex affine (resp. projective) spaces are called algebraic varieties over the complex numbers. Easy examples of highly singular varieties are subvarieties of the affine space A given by generic homogeneous polynomials of degree ≥ 2 in n-variables. In spite of the complexity of the singularities which can appear on algebraic varieties, H. Hironaka, a student of Zariski, proved in the early 1960’s the existence of a resolution in the strongest possible form. Precisely, for any