Configuration spaces of tori

The n-point configuration spaces E-n(T-2) = {(q(1), ..., q(n)) epsilon (T-2)(n) vertical bar q(i) not equal q(j) for all i not equal = j} and C-n(T-2) = {Q subset of T-2 | #Q = n} of a complex torus T-2 are complex manifolds. We prove that for n > 4 any holomorphic self-map F of C-n(T-2) either carries the whole of C-n(T-2) into an orbit of the diagonal (AutT(2))-action in C-n(T-2) or is of the form F(Q) = T (Q) Q, where T : C-n(T-2). AutT(2) is a holomorphic map. We also prove that for n > 4 any endomorphism of the torus braid group B-n(T-2) = pi(1)(C-n(T-2)) with a non-abelian image preserves the pure torus braid group P-n(T-2) = pi(1)(E-n(T-2)).