Do the forms add up?

0:00Counting forms

Here’s one way to measure how complicated a shape is. On a smooth projective variety, look at its pluricanonical forms: locally, a function times the m-th power of a top-degree differential form. Count how many independent ones there are for each m, and watch how fast that count grows. Take curves. On the projective line, there are none at all. On a plane cubic, an elliptic curve, there’s exactly one for every m. On a smooth plane quartic, of genus three, the count is 4m − 2, from m = 2 on: it grows like m. The growth exponent is the Kodaira dimension: here minus infinity, zero, and one. In general it’s minus infinity, or a whole number from zero up to the dimension. For a product of two shapes, the counts multiply, so Kodaira dimensions add. Two quartic curves give a surface whose counts grow like m²: Kodaira dimension two.

0:57Fibered shapes

Most varieties aren’t products. But many are fibered: a map f from X onto a base Z, whose fibers can vary from point to point, like a family of curves over a curve. Write F for a typical fiber. In the 1970s, Shigeru Iitaka asked whether Kodaira dimension is at least additive in a fibration: is κ(X) always at least κ(F) + κ(Z)? This is Iitaka’s subadditivity conjecture, often called C(n,m), after the dimensions n and m of X and Z. Many cases were proved. Kawamata handled curve bases, and fibers with good minimal models. Viehweg did bases of general type, and Kollár fibers of general type. Birkar proved it in total dimension up to six. Cao and Păun did abelian varieties as bases, and Hacon, Popa and Schnell bases of maximal Albanese dimension. Earlier preprints by Tsuji and Maehara state general versions, and the new papers don’t rely on them. There are versions for open varieties too. Allow the forms simple poles at punctures: the projective line minus two points then has log Kodaira dimension zero, and minus three points it has one. And Campana’s orbifold version also weighs multiple fibers: a fiber whose components all have multiplicity at least m adds at least 1 − 1/m to the base, at its point.

2:22The claim

Result 033 claims the whole family. Across five OpenAI preprints dated September 2026, the main one claims a proof of Campana’s orbifold conjecture for compact complex manifolds in Fujiki’s class C, those bimeromorphic to Kähler manifolds, with rational simple normal-crossing boundaries. The logarithmic and ordinary conjectures follow: κ(X) ≥ κ(F) + κ(Z), in every dimension. Another paper gives a separate projective proof of the ordinary case. Two of the others claim refinements proposed by Mihnea Popa. When the base has nonnegative log Kodaira dimension, the bound improves to the fiber plus the larger of the base and the variation, which counts how many parameters the fibers depend on, up to birational equivalence. And when the family and all its boundary strata are smooth over the open base, the inequality also reverses, giving exact additivity.

3:23How the proof goes

Here is the route in the orbifold paper. Suppose the fiber has Kodaira dimension k. First, a relative Iitaka reduction factors the map through an intermediate space W, with k-dimensional fibers over the base, so that the remaining fibers G, over W, have Kodaira dimension zero. Kodaira dimension zero means that, in suitable degrees, each G carries one pluricanonical form, up to scale. Taking a root of it on a cyclic cover turns it into an ordinary top-degree form, with at most logarithmic poles, and the line it spans in cohomology varies over W: a Hodge line. The orders of that form give an exact comparison. The forms on X match the forms of K + T + M on W, where T is a boundary and M is the line bundle of the Hodge line. Second, positivity of the Hodge line. The period map records how the Hodge structures vary. The paper uses it to show that M is pulled back from a nef line on a new base S, on which the canonical class plus a multiple of that line is big. The delicate part is the boundary, where the paper shows the map remembering the line loses rank. Third, addition. On the fibers of W over the base, K + T + M is big, so an addition theorem for log general type fibers applies. Weak positivity supplies one more fixed section, and multiplying sections cancels the ample correction terms. The forms on X then grow at least as fast as the fiber and the base require.

5:00Where it stands

These are preprints, not yet peer reviewed. OpenAI’s Lean formalization covers only one narrow branch of the additivity paper: the case where the fiber’s log Kodaira dimension is minus infinity. None of the five papers has been withdrawn or revised, and we found no public assessment by specialists yet. Other results in the release build on this one, including the abundance claims in our next episode. If it holds, the forms on a fibered variety always account for at least those of its fiber and its base: a question at the heart of the classification of algebraic varieties for half a century.