Rational Singularities Associated to Pairs

摘要

Rational singularities are a class of singularities that 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.