Iitaka subadditivity, variation, and logarithmic additivity

We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.

We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism f:(X,E)→(Y,D) of smooth projective reduced simple-normal-crossing pairs, with \mathop{Supp}\nolimits (f^*D)⊆\mathop{Supp}\nolimits E, such that X and every boundary stratum are smooth over Y\setminus\mathop{Supp}\nolimits D. Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.

We prove the compact log-smooth Kähler case of b-semiampleness. Let f:Y→ 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∈\mathop{Pic}\nolimits (X)_{\mathbb Q}. There is a smooth compact Kähler modification S→ X for which the threshold-moduli line satisfies M_{S_1}=\nu^*M_S in \mathop{Pic}\nolimits (S_1)_{\mathbb Q} for every smooth compact Kähler modification \nu:S_1→ 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.