Termination of projective and Kähler fourfold minimal model programs

The explanation

Start with a smooth surface and blow up one point. Blowing up replaces the point by a whole curve E, one point for each direction through it. This curve has self-intersection E·E = −1. Let K be the canonical divisor and g the genus of E. The adjunction formula says K·E + E·E = 2g − 2. Here E is a sphere, so g = 0 and the right side is −2. That gives K·E = −2 − (−1) = −1. The canonical divisor is negative on E. The classical first step of simplifying the surface is to contract E back to a point.

The minimal model program, or MMP, does this in every dimension. It repeatedly finds curves where the canonical divisor, or an adjusted version K + B, is negative. It then performs an operation that removes that negativity. A divisorial contraction collapses a whole hypersurfaceSource 1. A flip is subtler. It cuts out a small piece and glues in a different small piece where K + B becomes positiveSource 2. The program should end at one of two kinds of modelSource 3. In one, K + B is nef, never negative on curves. The other is a Mori fibre space, a fibration of a standard shape.

The danger is that the program might never end. Termination asks that every sequence of choices reaches an endpoint after finitely many stepsSource 4. That is stronger than finding one good sequenceSource 4. Flips make this hard. Since flips contract no divisors, their number cannot be bounded by counting divisors that disappearSource 5.

The standard tool is a difficulty, a whole-number count that should drop at each step and never go below zero. A count like that cannot drop forever. Shokurov introduced such a countSource 6. Kawamata, Matsuda and Matsuki used it for terminal fourfold flips, and Alexeev, Hacon and Kawamata refined itSource 7,Source 8. More recent work proved termination for pseudo-effective log canonical fourfoldsSource 9. Han, Liu and Zhuang proved it for klt fourfold programs with big boundarySource 10. Those results needed an extra positivity hypothesis.

The third paper in this family claims termination for projective log canonical fourfolds with rational boundary. It covers every permitted program, with no pseudo-effectivity hypothesisSource 11. Log canonical is a broad class of mild singularities. The result holds over every algebraically closed field of characteristic zeroSource 12. The proof first works over the complex numbers and then transfers to other such fieldsSource 13.

The obstacle shows up in rounding. The difficulty uses weights rounded up to whole numbersSource 14. Rounding up a sum can lose ground against rounding up its parts. For example, ⌈0.4 + 0.4⌉ = 1, while ⌈0.4⌉ + ⌈0.4⌉ = 2. The paper’s inequality says the loss with r parts is at most r − 1Source 15. Here r = 2 and the loss is exactly 1. The new geometric input pays for this loss using other positive terms of the countSource 16.

The other four papers move to compact Kähler fourfolds. These are complex spaces that need not be cut out by polynomial equations. The first paper claims that every sequence of generalized log canonical flips terminates on such spacesSource 17. It assumes the flips are projective small diagrams with the stated signsSource 17. It also assumes a global form of Q-factoriality, a technical condition on divisors. The theorem concerns existing flip sequences. It does not assert that every needed contraction or flip existsSource 18. The second paper claims the same for generalized-canonical flips, a narrower classSource 19.

The last two papers treat ordinary programs for klt pairs, a slightly narrower class of mild singularities. One claims that every maximal ordinary program terminatesSource 20. The other claims that some finite ordinary program exists, starting on the given pairSource 21. Both end at a nef model when the adjoint is pseudo-effective and at a Mori fibre space otherwiseSource 22,Source 23.

How sure is this? No Lean formalization is listed for these five papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not settle termination in dimension five or higher, or prove that the Kähler flips exist.