Rational Singularities Associated to Pairs

Rational singularities are a class of singularities which have been heavily studied since their introduction in the 1960s. Roughly speaking, an algebraic variety has rational singularities if its structure sheaf has the same cohomology as the structure sheaf of a resolution of singularities. Rational singularities enjoy many useful properties, in particular they are both normal and Cohen-Macaulay. Furthermore, many common varieties have rational singularities, including toric varieties and quotient varieties. Rational singularities are also known to be closely related to the singularities of the minimal model program. In particular, it is known that log terminal singularities are rational and that Gorenstein rational singularities are canonical. There is, however, an important distinction between rational singularities and singularities of the minimal model program. In the minimal model program, it is very natural to consider pairs (X,D) where X is a variety and D is a Q-divisor. In recent years, the study of pairs (X, a) where a is an ideal sheaf and c is a positive real number, has also become quite common. Thus it is very natural to try to extend the notion of rational singularities to pairs. We define two notions of rational pairs. First we define a rational pair which is analogous to a Kawamata log terminal (klt) pair, and then we define a purely rational triple which is analogous to purely log terminal (plt) triple (we will discuss the characteristic p analogues later). It is hoped that these definitions and their study will help further the understanding both of rational singularities and log terminal pairs. In characteristic zero, defining rational singularities for pairs has one distinct advantage over the corresponding variants of log terminal singularities. In order for (X,D) to be log terminal, one necessarily must have KX + D a Q-Cartier divisor. Likewise, for the pair (X, a) to be log terminal, X must necessarily be Q-Gorenstein. One can define rational singularities for a pair (X, a) without any such conditions on X. Virtually all standard properties of rational singularities transfer to pairs, as we show. In particular, summands and deformations behave well, see Corollary 4.11 and Theorem 4.13, as do various implications between log terminal and rational pairs, see Proposition 4.1 and Proposition 4.2. For the most part, the proofs are generalizations of proofs of the analogous properties of rational singularities. Since singularities of pairs come up very naturally in theorems related to adjunction and inversion of adjunction, we prove that several of these results extend to rational pairs as well. In particular, we are able to prove a