On the superadditivity of anticanonical Iitaka dimension

TL;DR

This paper proves an inequality relating the Iitaka dimension of the anticanonical bundle of a total space to that of its fiber and base, under certain singularity conditions, extending understanding of the superadditivity property.

Contribution

It establishes a new Iitaka-type inequality for the anticanonical bundle in fibrations with good singularities, generalizing previous results to arbitrary characteristic fields.

Findings

Proves the inequality $(X,-K_X)  (X_y,-K_{X_y})+(Y,-K_Y)$ under specified conditions.

Extends superadditivity of Iitaka dimension to cases with singular fibers.

Applicable over fields of any characteristic.

Abstract

Given a fibration $f: X \to Y$ with normal general fibre $X_y$, over a field of any characteristic, we establish the Iitaka-type inequality $\kappa(X,-K_X) \leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series $|-K_X|_{\mathbb{Q}}$ has good singularities on $X_y$.