Approximating Zariski decomposition of big line bundles

Abstract

In this note we prove the following THEOREM.Let L be a line bundle on a variety V with n = dimV.Suppose that h\V, tL)^dί n /n !+ύ(t) for infinitely many positive integers t, where φ(ΐ) is a function with \im t ->^φ(t)/t n =Q and d is a positive real number.Then, for any ε>0, there is a birational morphism π : M-+V together with an effective Q-divisor E on M such that H-π^L-E is a semiample Q-bundle with H n >d -ε.