Configuration spaces of tori (Q931298)

Summary: The \(n\)-point configuration spaces \(\mathcal{E}^{n} (\mathbb{T}^{2}) = \{(q_{1},\dots, q_{n}) \in (\mathbb{T}^{2})^{n}| q_{i} \neq q_{j} \, \forall i \neq j\}\) 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 \(({\text{Aut}} \mathbb{T}^{2})\)-action in \(\mathcal{C}^{n} (\mathbb{T}^{2})\) or is of the form \(F(Q)=T(Q)Q\), where \(T: \mathcal{C}^{n} (\mathbb{T}^{2}) \to {\text{Aut}} \mathbb{T}^2\) 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 \(P_{n} (\mathbb{T}^{2}) = \pi_{1} (\mathcal{E}^{n} (\mathbb{T}^{2}))\).