On subadditivity of the logarithmic Kodaira dimension

TL;DR

This paper proves the subadditivity of the logarithmic Kodaira dimension for affine varieties by linking it to the generalized abundance conjecture and utilizing Nakayama's theory, advancing the understanding of algebraic geometry.

Contribution

It reduces Iitaka's subadditivity conjecture to a special case of the generalized abundance conjecture using Nakayama's numerical Kodaira dimension and applies the minimal model program for affine varieties.

Findings

Established an Iitaka type inequality for Nakayama's numerical Kodaira dimension.

Proved subadditivity of the logarithmic Kodaira dimension for affine varieties.

Connected the conjecture to the generalized abundance conjecture.

Abstract

We reduce Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension to a special case of the generalized abundance conjecture by establishing an Iitaka type inequality for Nakayama's numerical Kodaira dimension. Our proof heavily depends on Nakayama's theory of $\omega$-sheaves and $\widehat{\omega}$-sheaves. As an application, we prove the subadditivity of the logarithmic Kodaira dimension for affine varieties by using the minimal model program for projective klt pairs with big boundary divisor.