B-semiampleness for compact log-smooth Kähler fibrations

Abstract — v1

We prove the compact log-smooth Kähler case of b-semiampleness. Let $f:Y\to X$ be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in $[0,1]$, with $K_Y+\Delta\sim_{\mathbb Q}f^*L$ for $L\in\mathop{\mathrm{Pic}}\nolimits (X)_{\mathbb Q}$. There is a smooth compact Kähler modification $S\to X$ for which the threshold-moduli line satisfies $M_{S_1}=\nu^*M_S$ in $\mathop{\mathrm{Pic}}\nolimits (S_1)_{\mathbb Q}$ for every smooth compact Kähler modification $\nu:S_1\to S$, and some positive multiple of $M_S$ is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.