P = W for fixed-determinant \mathrm{SL}_{n} moduli spaces

Proves P_k=W_{2k}=W_{2k+1} on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.

Manuscript

P=W in composite rank for fixed determinant

September 24, 2026 · source · cite as \cite{OAI:P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026}

We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.

Result 043 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.