Generalized Nonvanishing Conjecture and Iitaka Conjecture

Abstract

In this article, we will prove the Generalized Nonvanishing Conjecture holds for threefolds with either $κ>0$ or $q>0$. As a result, we can prove the Iitaka conjecture $C_{n,m}$ holds for $n=7$ if the source space has non-negative Kodaira dimension, if the general fibre has positive Kodaira dimension, or if the base space is not threefold with $κ=q=0$. In particular, $C^-_{n,m}$ holds if $n\leq 7$.

Theorems & Definitions (42)

• Conjecture 1.1: Iitaka Conjecture
• Conjecture 1.2: Ueno Conjecture K
• Conjecture 1.3: Generalized Nonvanishing Conjecture
• Conjecture 1.4: Generalized Abundance Conjecture
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Definition 2.2