Hyperkähler metrics on the regular nilpotent adjoint orbit

This thesis studies the Kronheimer hyperkähler metric on the adjoint orbit of the classical Lie group SL_n (C) of a regular, nilpotent element in its Lie algebra sl_n(C). We describe a Kähler potential of this hyperkähler metric in terms of the theta function on the Jacobian, consisting of invertible sheaves of degree g - 1, of the nilpotent, spectral curve. By using an explicit description of matricial polynomials of degree two corresponding to invertible sheaves of degree g - 1 without a non-trivial, global section on the nilpotent, spectral curve we construct some explicit solutions to Nahm’s equations.