Positivity of anticanonical divisors in 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 $-K_Y$.