Geometric P=W conjecture in rank three case, on the three-punctured sphere

Abstract: We describe the moduli space of gauge-equivalence classes of rank three Higgs bundles on the complex projective line with three logarithmic points, called the Dolbeault space, and also the corresponding character variety with structure group SL(3,C), called the Betti space. Our result is that the Geometric P=W conjecture holds in this setup, which asserts the existence of a homotopy commutative diagram between these spaces. Therefore we determine the homotopy type of the compactifying divisor of the Betti space, and describe the asymptotic behaviour of the non-abelian Hodge correspondence and Riemann-Hilbert correspondence, connecting the Dolbeault and Betti spaces.

Joint work with my supervisor, Szilárd Szabó.