Kodaira energies of polarized log varieties

Abstract

Let L be an ample line bundle on a log variety (V, D) having only log terminal singularities.The Kodaira energy of such a triple (V, D, L) is defined as follows: κε=-Inf{t\in Q | K(V,D)+tL is big}. Here K(V,D)=K_V+D is the log canonical bundle of (V,D) and "big" means κ=dim V. We first show that the rationality of κε, when it is negative, follows from log Flip theorems (thus it is proved for dim V\le 3 at present). Second we consider the case in which D=0, dim V=3 and V has only terminal singularities. Let S be the set of all the possible values of Kodaira energies of such polarized log threefolds. We prove that S has no negative limit point. This result is closely related to the boundedness of Fano 3-folds.