ON RELATIVE KODAIRA ENERGY OF QUASI-POLARIZED FIBER SPACES AND ITS APPLICATION

Let X be a smooth projective manifold over the complex number field C and L a Cartier divisor on X. Then (X, L) is called a polarized (resp. quasi-polarized) manifold if L is ample (resp. nef and big). For a po larized manifold (X, L), Takao Fujita introduced the notion of the Kod aira energy kappa epsilon(X, L). The Kodaira energy of (X, L) is thoug ht to have some interesting phenomena if kappa(X) = -infinity (for exa mple, Spectrum Conjecture which was proposed by T. Fujita). In order t o study the Kodaira energy of (X, L), we consider the case in which X has a fibre space, that is, there exist a smooth projective manifold Y with dim X > dim Y greater than or equal to 1 and a surjective morphi sm f: X --> Y with connected fibres. In this case we introduce the not ion of relative Kodaira energy and study some property of it. By using some results of relative Kodaira energy, we study some properties of the Kodaira energy.