In the last three decades Mori’s program, the moduli theory of varieties and complex differential geometry have identified five large and important classes of singularities. These are the basic objects of this book.
Terminal. This is the smallest class needed for running Mori’s program starting with smooth varieties. For surfaces, terminal equals smooth. These singularities have been fully classified in dimension 3 but they are less understood in dimensions ≥ 4.
Canonical. These are the singularities that appear on the canonical models of varieties of general type. The classification of canonical surface singularities by Du Val in 1934 is the first appearance of any of these classes in the literature. These singularities are reasonably ...