From Magnetically Twisted to Hyperkähler

The tangent bundle of a Kähler manifold admits in a neighborhood of the zero section a hyperkähler structure. From a symplectic point of view, this means we have three symplectic structures all compatible with a single metric. Two of the three symplectic structures are easy to describe in terms of the canonic symplectic structure. The third one is harder to describe, but in the case of hermitian symmetric spaces, there is an explicit formula found by Biquard and Gauduchon. In this talk, I will construct a surprising diffeomorphism of the tangent bundle of a hermitian symmetric space that identifies this third symplectic structure with the magnetically twisted symplectic structure, where the twist is given by the Kähler form on the base.