  We give a new, geometric proof of the section conjecture for fixed points of
finite group actions on projective curves of positive genus defined over the
field of complex numbers, as well as its natural nilpotent analogue. As a part
of our investigations we give an explicit description of the abelianised
section map for groups of prime order in this setting. We also show a version
of the 2-nilpotent section conjecture.
