Iitaka-Viehweg conjectures C and C++

TL;DR

This paper constructs a ramified variety over a given fiber space to relate Viehweg conjectures C and C++, demonstrating that the Viehweg dimension of the original space does not exceed that of the constructed one, using Mochizuki's Galois theory.

Contribution

It introduces a method to relate Viehweg conjectures C and C++ by constructing a ramified variety and applying Mochizuki's Galois theory to compare Viehweg dimensions.

Findings

Constructed a variety Y ramified over X/S with specific properties.

Proved Viehweg dimension of X/S is not greater than that of Y/S.

Applied Mochizuki's Galois theory to establish the inequality.

Abstract

Given a fibre space $X/S$ with the generic geometric fibre of Kodaira dimension $\geq 0$, we shall construct a variety $Y$ ramified over $X$ along such a horizontal hyperplane with respect to $X/S$ that Koll\'ar and Kawamata had proved Viehweg conjecture for $Y/S$ with the generic geometric fibre of general type or of the abundant canonical invertible sheaf where Viehweg dimensions of $X/S$ and $Y/S$ are equal, respectively. We shall show that Viehweg dimension of $X/S$ is not greater than that of $Y/S$ by Mochizuki's Galois theory.