P = W for fixed-determinant SL_n moduli spaces

The explanation

Pick four 4-by-4 complex matrices A₁, B₁, A₂, B₂, each with determinant 1. Form the commutator [A, B] = ABA⁻¹B⁻¹ of each pair. Ask that the product of the two commutators equal i times the identity matrix. The determinant check works, since i⁴ = 1. All solutions, up to changing basis in the same way for every matrix, form a space called a character variety. It records the ways to attach matrices to the loops of a two-holed surface, which has genus 2. Here the rank is n = 4 and the genus is g = 2.

A second space looks very different. It is built from Higgs bundles on a fixed algebraic curve of the same genus. A Higgs bundle is a family of vector spaces over the curve, with an extra matrix-valued form attached. The two spaces have the same shape as smooth manifolds, by a theorem called nonabelian Hodge theory. Each space has complex dimension 2(n² − 1)(g − 1). For n = 4 and g = 2, that is 2 × 15 × 1 = 30.

Each space sorts its cohomology into layers in its own way. Cohomology is an algebraic count of holes of each dimension. On the Higgs side, a map called the Hitchin map gives the perverse filtrationSource 1. On the matrix side, the algebraic equations give the weight filtration, from Hodge theory. The two spaces are linked, but the two filtrations come from unrelated structuresSource 2. The P = W conjecture says they agree once the perverse indices are doubledSource 3. That is the meaning of P = W.

De Cataldo, Hausel and Migliorini stated the conjecture in 2012 and proved it in rank 2Source 4. For the general linear group GL_n, Maulik and Shen proved it in every rank and genus in 2024Source 5. The version here fixes the determinant and the trace, which gives SL_n. That version has an extra symmetry group Γ. It comes from line bundles on the curve whose n-th power is trivial. The group Γ has n to the power 2g elements, so 4⁴ = 256 when n = 4 and g = 2. The GL_n methods see only the part of cohomology fixed by Γ. They miss the rest, called the variant partSource 6. In 2022, de Cataldo, Maulik and Shen settled the variant part for prime ranks such as 2, 3 and 5Source 7. Composite ranks such as 4 and 6 stayed open.

This paper claims P = W for every composite rank and coprime degree, on the full cohomologySource 8. Coprime means the degree shares no factor with the rank, like 1 and 4. That makes both spaces smoothSource 9. The claim covers every character of Γ, not just the invariant partSource 10. With the prime cases, it would settle every coprime rankSource 11.

The obstacle was the variant part. No known set of generating classes reaches it. The paper builds one operator on the whole cohomology before taking invariant partsSource 12. It moves a pole of a connection around the curve and records the effect. The operator lowers the perverse index by 2 and the weight index by 4. Together with a known class, it forms a copy of a small symmetry algebra called sl₂. A standard argument then shows that both filtrations are built from the same eigenspaces of one operatorSource 13. So they must agree.

How sure is this? No Lean formalization is listed for this result. The proof combines several hard comparison theorems, each with its own hypotheses. No specialist review has been published.

What it doesn't show. This does not treat ranks and degrees with a common factor, where the spaces are singular.