Notes on Kodaira Energies of Polarized Varieties

§1. Conjectures Let V be a variety over C and let B = ∑ biBi be an effective Q-Weil divisor on V such that bi ≤ 1 for any i. Such a pair (V,B) will be called a log variety. It is said to be log terminal if it has only weak log terminal singularities in the sense of [KMM]. In this case, the Q-bundle KV + B is called the log canonical bundle of (V,B) and will be denoted by K(V,B). A Q-bundle L on a log terminal variety (V,B) is said to be log ample if there is an effective Q-divisor E such that (V,B + E) is log terminal and L− ǫE is ample for any 0 < ǫ ≤ 1. Note that “log ample” implies “nef big”, and the converse is also true if bi < 1 for all i. For a big Q-bundle L on a log terminal variety (V,B), we define