The aim of this talk is to answer a question on the geometry of the moduli space M_H of \SL(2,\mathbb C) Higgs bundles on a compact Riemann surface \Sigma of genus g>1. M_H admits a \mathbb C^{\star}-action and to each stable \mathbb C^{\star}-fixed point [(\bar\partial_0,\Phi_0)] one associates a holomorphic Lagrangian submanifold W^1(\bar\partial_0,\Phi_0) inside the de Rham moduli space M_{dR} of complex flat connections. Carlos Simpson conjectured that W^1(\bar\partial_0,\Phi_0) is always closed in the topology of M_{dR}. After motivating the question, I will present the main ideas that go into the proof.