The generalized Mukai conjecture

The explanation

Take the projective plane P², an ordinary plane with a line at infinity added. It has dimension n = 2. A rational curve is a curve shaped like a sphere, such as a line. The anticanonical degree of a curve measures how strongly the space bends positively along it. For a line in P² that degree is 3, and no rational curve has less. The smallest such degree is called the pseudoindex, written ι. The Picard number ρ counts the independent ways to measure curves. For P², ρ = 1. So ρ(ι − 1) = 1 × 2 = 2, which equals n.

Now take P¹ × P¹, a product of two lines. Here n = 2 and ρ = 2. A line in either direction has anticanonical degree 2, so ι = 2. Then ρ(ι − 1) = 2 × 1 = 2, again equal to n. The product P¹ × P² has n = 3, ρ = 2 and ι = 2. There ρ(ι − 1) = 2, which is below 3. Blow up one point of the plane, replacing it by a small sphere. That sphere has anticanonical degree 1, so ι = 1 and ρ(ι − 1) = 0.

All of these are Fano manifolds. A Fano manifold is a smooth, closed algebraic space whose anticanonical class is ample, meaning positive on every curve. That positivity forces rational curves to exist, so ι makes sense. The generalized Mukai conjecture says ρ(ι − 1) ≤ n for every such space of positive dimension n. It adds that equality happens only for a product of ρ copies of the projective space P^(ι−1). The examples fit: P² is one copy of P², and P¹ × P¹ is two copies of P¹.

Shigeru Mukai posed the first version in 1988, using the Fano indexSource 1. The Fano index is the largest integer dividing the anticanonical class, and it can differ from ιSource 2. Jarosław Wiśniewski introduced the pseudoindex and proved that 2ι > n + 2 forces ρ = 1Source 3. Bonavero, Casagrande, Debarre and Druel stated the pseudoindex form and proved it up to dimension fourSource 4. Andreatta, Chierici and Occhetta extended this to dimension fiveSource 5. Casagrande settled a broad class of toric Fano varieties, built from combinatorial data, in every dimensionSource 6. Novelli proved both parts whenever ι > n/3Source 7. Each of these results needs a limit on dimension, symmetry or ι. The paper claims a proof with no such limitSource 8,Source 9.

The approach uses quantum cohomology. Ordinary cohomology records how subspaces meet. Quantum cohomology also counts rational curves that run between them. Multiplying by a divisor class, a basic way to measure curves, then gives a matrix. Its entries mix ordinary intersection with counts of curves of each degree. A curve count of anticanonical degree d lowers codimension by d − 1Source 10. Codimension is the number of conditions that cut out a subspace. Each step lowers it by at least ι − 1, and it can only fall from n to 0. So a nonzero product built from ρ independent curve degrees gives ρ(ι − 1) ≤ n.

The main obstacle is producing that nonzero product without assuming semisimplicitySource 11. Semisimple roughly means the matrices split into independent one-by-one blocks. The paper first claims a lower bound for certain curve counts called point descendantsSource 12. A recurrence then shows that one joint eigenvalue list has ρ algebraically independent entriesSource 13. A trace argument turns that independence into the needed product of curve countsSource 14. At equality, the paper builds chains of actual rational curves. It then applies Occhetta's description of products of projective spacesSource 15.

How sure is this? There is no Lean scope note for this result, so no computer proof checker has tested it. The paper claims to prove both the bound and the equality case. No specialist review has been published.

What it doesn't show. This does not come with a computer-checked proof, and no Lean scope note exists for it.