Kodaira additivity, birational isotriviality and specialness

TL;DR

This paper proves a key additivity property of logarithmic Kodaira dimensions for certain complex fibrations, extending conjectures and using novel approaches involving birationally isotrivial fibrations and special manifolds.

Contribution

It establishes the equality of Kodaira dimensions under broader conditions, utilizing new methods related to birational isotriviality and the core map.

Findings

Proves Kodaira dimension additivity for fibrations with fibers admitting good minimal models.

Extends conjectures to cases without the assumption of good minimal models.

Introduces new techniques involving special manifolds and the core map.

Abstract

We show, using [14], that a smooth projective fibration f : X $\rightarrow$ Y between connected complex quasi-projective manifolds satisfies the equality $\kappa$(X) = $\kappa$(X y) + $\kappa$(Y) of Logarithmic Kodaira dimensions if its fibres X y admit a good minimal model. Without the last assumption, this was conjectured in [11]. Several cases are established in [13], which inspired the present text. Although the present results overlap with those of [13] in the projective case, the approach here is different, based on the r{\^o}le played by birationally isotrivial fibrations, special manifolds and the core map of Y introduced and constructed in [3].