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

We show that under suitable hypothesis (which are sharp in 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, generalizing a result of Huneke and Swanson in dimension two. We also prove a version of this in the singular setting, which we show to be intimately related to the problem of finding global 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 ample adjoint bundle has a non-trivial section.