Dynamic Multiplier Ideal Sheaves and the Construction of Rational Curves in Fano Manifolds

Introduction. Multiplier ideal sheaves were introduced by Kohn [Kohn1979] and Nadel [Nadel1990] to identify the location and the extent of the failure of crucial estimates. Such multiplier ideal sheaves are defined by a family or a sequence of inequalities instead of a single inequality. In Kohn’s definition there is one inequality for every test function (or test form) and the multiplier has to make all the inequalities hold for all the test functions (or test forms) at the same time. Nadel’s definition is designed for the continuity method for the problem of the existence of Kähler-Einstein metrics on Fano manifolds. Nadel’s multiplier has to make the uniform finiteness of the integral from the crucial zero-order estimate hold for the entire sequence of perturbations of Kähler potentials occurring in the closed part of the continuity method.