  The configuration space C^n of unordered n-tuples of distinct points on a
torus T^2 is a non-singular complex algebraic variety. We study holomorphic
self-maps of C^n and prove that for n>4 any such map F either carries the whole
of C^n into an orbit of the diagonal Aut(T^2) action in C^n or is of the form
F(x)=T(x)x for some holomorphic map T:C^n-->Aut(T^2). We also prove that for
n>4 any endomorphism of the torus braid group B_n(T^2) with a non-abelian image
preserves the pure torus braid group PB_n(T^2).
