Configuration spaces of tori
  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). 
Yoel Feler