We use abelianization of Higgs bundles away from the ramification divisor and fiducial solutions to analyze the large scale behaviour of Fenchel--Nielsen co-ordinates on the moduli space of rank $2$ Higgs bundles on the Riemann sphere with $5$ punctures. We solve the related Hitchin WKB problem and prove the lowest degree weighted pieces of the $P=W$ conjecture in this case.
Categories: math.AG math.DG