  This article shows that every non-isotropic harmonic 2-torus in complex
projective space factors through a generalised Jacobi variety related to the
spectral curve. Each map is composed of a homomorphism into the variety and a
rational map off it. The same ideas allow one to construct (pluri)-harmonic
maps of finite type from Euclidean space into Grassmannians and the projective
unitary groups. Further, some of these maps will be purely algebraic. For maps
into complex projective space the algebraic maps of the plane are always doubly
periodic i.e. they yield 2-tori. The classification of all these algebraic maps
remains open.
