Symplectic geometry of the moduli space of framed Higgs bundles

Let X be a compact Riemann surface and D an effective divisor on X. The moduli space of D-twisted stable Higgs bundles on X is known to have a holomorphic Poisson structure which is in fact symplectic if and only if D is the zero divisor. We prove that this moduli space admits a natural enhancement to a holomorphic symplectic manifold. This is based on joint work in collaboration with I. Biswas and A. Peón-Nieto.