The Campana–Peternell conjecture in dimension six

The explanation

Start with a small equation that sits at the heart of the proof: 32 − 20u + 3u² = 0. Its discriminant is 20² − 4 × 3 × 32 = 400 − 384 = 16. So u = (20 ± 4)/6, which gives u = 4 or u = 8/3. Now take the product P¹ × P² of a projective line and a projective plane. The Segre map places it inside five-dimensional projective space P⁵ by multiplying coordinates in pairs. On this threefold, write c₁ for its first Chern class, a basic curvature class, and L for the hyperplane class. Multiplying out gives L³ = 3 and c₁ · L² = 8. Their ratio is 8/3, one of the two roots.

The setting is a Fano manifold. That is a smooth, closed algebraic space whose anticanonical class is positive on every curve. Its tangent bundle collects the tangent directions at all its points. The bundle is nef when it is never negative along any curve, in a precise sense. A variety is rational homogeneous when a group of symmetries can carry any point to any other. Projective spaces, smooth quadrics and Grassmannians are examples. The Grassmannian Gr(2,5) is the space of two-dimensional planes through the origin in five-dimensional space. It has dimension 6.

Tangent bundles of rational homogeneous varieties are always nefSource 1. In 1991 Campana and Peternell asked whether the converse holds for every Fano manifoldSource 2. They proved it in dimension at most three and for fourfolds of Picard number above oneSource 3. The Picard number counts independent ways to measure curves. Work of Mok and Hwang settled the remaining fourfoldsSource 4. Watanabe proved the fivefold case of Picard number above oneSource 5. Kanemitsu completed dimension fiveSource 6. Kanemitsu also proved the conjecture whenever the Picard number exceeds the dimension minus five. For sixfolds, that leaves only Picard number oneSource 7.

This paper claims the conjecture in dimension sixSource 8. With Picard number one, standard results reduce the problem to pseudoindex four or fiveSource 9. The pseudoindex is the smallest anticanonical degree of a rational curve, a curve shaped like a sphere. Watanabe had treated pseudoindex four, and the paper gives its own argument for itSource 10. The main new step is pseudoindex fiveSource 11.

The obstacle is to identify the tangent directions of the shortest rational curves through a pointSource 12. At a general point, those directions form a threefold inside P⁵. It is called the variety of minimal rational tangents. Positivity alone does not say which threefold it isSource 13. The paper uses the family of these curves in two ways at once. That gives the identity 32 − 20u + 3u² = 0 for a ratio u of curvature numbersSource 14. A sectional-genus formula rules out u = 4 and forces L³ = 3 when u = 8/3Source 15. So the threefold of directions is the Segre threefold P¹ × P²Source 16. Mok's recognition theorem then identifies the sixfold with Gr(2,5)Source 17. The paper prints a complete program that checks the finite calculation behind the identitySource 18.

The paper also claims a consequence for compact Kähler manifolds, a class of complex spaces that need not be algebraic. It covers such manifolds with nef tangent bundle, in particular every one of dimension at most sixSource 19. For them, the paper describes the universal cover as a rational homogeneous space times a complex coordinate space.

How sure is this? There is no Lean scope note for this result, so no computer proof checker has tested the proof. The printed program checks only the finite calculation, not the geometric argument. No specialist review has been published.

What it doesn't show. This does not settle the conjecture in dimension seven or higher.