On the subadditivity of generalized Kodaira dimensions

TL;DR

This paper introduces a new notion of generalized numerical Kodaira dimension with multiplier ideal sheaves, proves subadditivity inequalities, and uses Okounkov bodies to relate different definitions, advancing the understanding of Kodaira dimensions.

Contribution

It defines a new generalized Kodaira dimension with multiplier ideal sheaves, proves subadditivity inequalities, and compares different definitions using Okounkov bodies, providing alternative proofs and applications.

Findings

Established subadditivity inequalities for the generalized numerical Kodaira dimension.

Proved the equivalence of different definitions of generalized Kodaira dimension using Okounkov bodies.

Provided an alternative proof of Zhou-Zhu's subadditivity formula for singular metrics with analytic singularities.

Abstract

The goals of this paper are of two aspects. Firstly, we introduce the notion of generalized numerical Kodaira dimension with multiplier ideal sheaf and establish the subadditivity inequalities in terms of this notion, which can be used to give an analytic proof of O. Fujino's result on the subadditivity of the log Kodaira dimensions. Secondly, motivated by Zhou-Zhu's subadditivity of generalized Kodaira dimensions, we adopt another definition of generalized Kodaira dimension with multiplier ideal sheaf and show they are equal by using Okounkov bodies. As one application, we show that the superadditivity part in Zhou-Zhu's setting also holds true. As another application, we give an alternative proof of Zhou-Zhu's subadditivity formula, in the case when the singular metric $h_L$ has analytic singularities, by using generalized Iitaka fibrations.