Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-\mathcal C manifolds with rational simple-normal-crossing boundaries. For projective fibrations f:U\to V of smooth complex quasi-projective varieties with connected fibers, general fiber F, and \bar\kappa(V)\ge0, proves Popa's inequality \bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathop{\mathrm{Var}}\nolimits (f)\}, where variation measures the whole geometric generic fiber. (Lean)
Manuscripts (5)
Orbifold and logarithmic Iitaka subadditivity
September 26, 2026 · source · cite as \cite{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}
We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class \mathcal C, including coefficient one. Ordinary and logarithmic subadditivity follow.
Logarithmic Kodaira dimension and whole-fiber variation
September 26, 2026 · source · cite as \cite{OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026}
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.
The reverse logarithmic Kodaira inequality and additivity
September 26, 2026 · source · cite as \cite{OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026}
We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism f:(X,E)\to(Y,D) of smooth projective reduced simple-normal-crossing pairs, with \mathop{\mathrm{Supp}}\nolimits (f^*D)\subseteq\mathop{\mathrm{Supp}}\nolimits E, such that X and every boundary stratum are smooth over Y\setminus\mathop{\mathrm{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.
Projective Hodge lines and ordinary Iitaka subadditivity
September 27, 2026 · source · cite as \cite{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026}
We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of f:X\to Z, then \kappa(X)\geq\kappa(F)+\kappa(Z).
B-semiampleness for compact log-smooth Kähler fibrations
September 10, 2026 · source · cite as \cite{OAI:B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026}
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.
Result 033 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md