Configuration spaces of tori

Abstract

The $n$-point configuration spaces ${\mathcal E}^n({\mathbb T}^2)=\{(q_1,...,q_n) \in ({\mathbb T}^2)^n\,|\ q_i\ne q_j \ \forall\,i\ne j\}$ and ${\mathcal C}^n({\mathbb T}^2) =\{Q\subset {\mathbb T}^2\,|\ \#Q=n\}$ of a complex torus ${\mathbb T}^2$ are complex manifolds. We prove that for $n>4$ any holomorphic self-map $F$ of ${\mathcal C}^n({\mathbb T}^2)$ either carries the whole of ${\mathcal C}^n({\mathbb T}^2)$ into an orbit of the diagonal $\Aut({\mathbb T}^2)$ action in ${\mathcal C}^n({\mathbb T}^2)$ or is of the form $F(Q)=T(Q)Q$, where $T\colon{\mathcal C}^n({\mathbb T}^2)\to\Aut({\mathbb T}^2)$ is a holomorphic map. We also prove that for $n>4$ any endomorphism of the torus braid group $B_n({\mathbb T}^2)=\pi_1({\mathcal C}^n({\mathbb T}^2))$ with a non-abelian image preserves the pure torus braid group $P_n({\mathbb T}^2) =\pi_1({\mathcal E}^n({\mathbb T}^2))$.