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.
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In P=W in composite rank for fixed determinant, 24 September 2026
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For every smooth projective complex curve C of genus g\ge2, every composite integer n\ge4, every integer d with \gcd(n,d)=1, and every L\in\Pic^d(C), one has
for all m,k\ge0.
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Theorem 1.1
In P=W in composite rank for fixed determinant, 24 September 2026