Iitaka subadditivity, variation, and logarithmic additivity

What OpenAI claims

Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-𝒞 manifolds with rational simple-normal-crossing boundaries. For projective fibrations f:U → V of smooth complex quasi-projective varieties with connected fibers, general fiber F, and κ̄(V) ≥ 0, proves Popa's inequality κ̄(U) ≥ κ(F)+max {κ̄(V),Var (f)}, where variation measures the whole geometric generic fiber. (Lean)

Papers (5)

1. Orbifold and logarithmic Iitaka subadditivity

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class 𝒞, including coefficient one. Ordinary and logarithmic subadditivity follow.

2. Logarithmic Kodaira dimension and whole-fiber variation

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

3. The reverse logarithmic Kodaira inequality and additivity

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 Supp (f^*D) ⊆ Supp E, such that X and every boundary stratum are smooth over Y∖Supp 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.

4. Projective Hodge lines and ordinary Iitaka subadditivity

   27 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 → Z, then κ(X) ≥ κ(F)+κ(Z).

5. B-semiampleness for compact log-smooth Kähler fibrations

   10 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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+Δ ∼ _ℚf^*L for L ∈ Pic (X)_ℚ. There is a smooth compact Kähler modification S → X for which the threshold-moduli line satisfies M_S1=ν^*M_S in Pic (S₁)_ℚ for every smooth compact Kähler modification ν:S₁ → 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.