Positivity of anticanonical divisors in algebraic fibre spaces

TL;DR

This paper establishes an Iitaka-type inequality relating the Kodaira dimensions of the anticanonical divisors in algebraic fibre spaces, advancing understanding of their positivity properties in algebraic geometry.

Contribution

It proves a new inequality connecting the Kodaira dimensions of anticanonical divisors in fibre spaces, under mild conditions, and explores related positivity results.

Findings

Proves an Iitaka-type inequality for anticanonical divisors.

Establishes conditions under which positivity of -K_X and -K_Y are related.

Provides new insights into the structure of algebraic fibre spaces.

Abstract

Let $f:X\rightarrow Y$ be an algebraic fibre space between normal projective varieties and $F$ be a general fibre of $f$. We prove an Iitaka-type inequality $\kappa(X,-K_X)\leq \kappa(F,-K_F)+\kappa(Y,-K_Y)$ under some mild conditions. We also obtain some more results relates the positivity of $-K_X$ and $-K_Y$.