Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

The explanation

Take a curve of genus 2, a surface with two holes, and multiply it by a doughnut-shaped elliptic curve. The result is a complex surface S. Its canonical divisor K is pulled back from the genus-2 curve, where it has degree 2g − 2 = 2. Count sections of the multiple 3K, which act like functions with allowed poles. On S there are (2·3 − 1)(2 − 1) = 5 of them, the same as on the curve. Those 5 sections define a map from S onto a copy of the genus-2 curve. Every fiber of the map is a copy of the doughnut.

That map is what abundance predicts in general. A divisor is nef if its degree on every curve is at least zero. It is semiample if some multiple has enough sections to define a map with no base points. The log abundance conjecture asks whether every nef log canonical divisor K_X + B is semiampleSource 1. Here B is a boundary, a weighted sum of hypersurfaces, and log canonical names the mildest allowed singularities. Semiampleness turns a numerical sign into an actual fibrationSource 2. On S, the fibration is the projection to the genus-2 curve.

For threefolds, the conjecture was settled in the 1980s and 1990s. Miyaoka proved one key case in 1988Source 3. Kawamata finished the minimal case, and Keel, Matsuki and McKernan proved the log version in 1994Source 4. In higher dimensions, Birkar, Cascini, Hacon and McKernan built minimal models in the log-general-type range in 2010Source 5. The general case stayed open. Nonvanishing, having at least one section, was a key missing step. Even sections alone do not make a nef divisor semiampleSource 6.

This family of 14 papers makes several linked claims. The central paper claims log abundance for rational boundaries in every dimension, in characteristic zeroSource 7. Its proof is an induction on dimension over the complex numbersSource 8. Each step first shows that smooth varieties have pluricanonical sections, then upgrades minimal models to good ones.

A second paper moves to compact Kähler spaces. These carry a compatible metric but need not be cut out by polynomials. There, nefness is a statement about a cohomology class, while semiampleness needs an actual line bundleSource 9. The paper claims log abundance there, but only assuming logarithmic Iitaka subadditivitySource 10. That assumption is a separate claimed theorem in another OpenAI family, also unreviewed.

A third paper claims the effective Iitaka fibration conjecture. For each dimension d, one number m(d) should work for every smooth projective variety of that dimension with sections. The multiple m(d) times K then defines that variety's Iitaka fibration, the map that records how its sections grow. Hacon and McKernan, Takayama and Tsuji had proved this for varieties of general typeSource 11. The paper claims the full statement, using the log abundance paper as inputSource 12. It gives no practical size for m(d)Source 13. A fourth paper claims a uniform index for semi-log-canonical log Calabi–Yau pairs. That index is a single multiple that makes the divisor trivial, independent of the number of componentsSource 14. The other ten papers supply inputs: nonvanishing for fourfolds, metric tools, lifting of sections, and bounds for related fibrations.

The papers lean heavily on each other. A gap in the central log abundance paper would affect most of the rest.

How sure is this? No Lean formalization is listed for any paper in this family. The arguments are long and interlocking. No specialist review has been published.

What it doesn't show. This does not cover positive characteristic, and its Kähler result assumes logarithmic Iitaka subadditivity.