Remark: 033-05. B-semiampleness for compact log-smooth Kähler fibrations

Authors: OpenAI. Family 033.

Title: B-semiampleness for compact log-smooth Kähler fibrations.

Abstract: 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 MS 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.

This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.