On a Non-vanishing Conjecture of Kawamata and the Core of an Ideal

We show, under suitable hypothesis which are sharp in a certain sense, that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power. This generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to Huneke and Swanson. We also find a generalization of this result to singular (non-regular) settings, which we show to be intimately related to the problem of finding non-zero sections of ample line bundles on projective varieties. In particular, we show that a graded analog of our formula for core would imply a remarkable conjecture of Kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non-zero section.