Algebraic Fiber Space Whose Generic Fiber and Base Space Are of Almost General Type

. We assume that the existence and termination conjecture for °ips holds. Acomplex projective manifold is said to be of almost general type if the intersection numberof the canonical divisor with every very general curve is strictly positive. Let f be analgebraic ber space from X to Y . Then the manifold X is of almost general type if everyvery general ber F and the base space Y of f are of almost general type. 1. IntroductionIn this paper, every algebraic variety is dened over the eld C of complexnumbers. The two main conjectures of the classication theory of algebraic vari-eties are the minimal model conjecture and the abundance conjecture. The rstconsists of the existence and the termination of °ips. This conjecture appeared tobe very natural, owing to the recent progress by Shokurov [9] and Hacon-McKernan[5]. Furthermore the existence is now a theorem due to Birkar, Cascini, Hacon andMcKernan [2]. The second is the conjecture that every nef canonical divisor shouldbe semi-ample. But this is never trivial even for surfaces, as was seen in knownproofs for them. Recently Ambro ([1]) reduced this abundance conjecture to thelog minimal model conjecture and to the conjecture that every quasi-nup log canon-ical divisor should be semi-ample. Here we explain concepts related to \quasi-nup".Denition 1.1. A Q-Cartier divisor