Yoel Feler

The configuration space Cⁿ(X) of a complex space X consists of all n point subsets (``configurations") Q={q₁,...,q_n}X. If X carries an additional geometric structure, it may be taken into account. Say if X is either the projective space CP^m or the affine space C^m and n>m then the space Cⁿ(X,gp) of geometrically generic configurations consists of all n point subsets QX such that no hyperplane in X contains more than m points of Q. The corresponding ordered configuration spaces Eⁿ(X) and Eⁿ(X,gp) consist of all q=(q₁,...,q_n)Xⁿ such that the set Q={q₁,...,q_n}X belongs to Cⁿ(X)  and Cⁿ(X,gp), respectively.

In the first part of my work, it is proved that for n>4 any holomorphic self-map F of the configuration space Cⁿ(T²) of a complex torus T² either carries the whole of Cⁿ(T²) into an orbit of the diagonal Aut(T²) action in Cⁿ(T²) or is of the form F(Q)=T(Q)Q, where T: Cⁿ(T²)Aut T² is a holomorphic map. We also prove that for n>4 any endomorphism of the torus braid group B_n(T²)=π₁(Cⁿ(T²)) with a non-abelian image preserves the pure torus braid group P_n(T²)=π₁(Eⁿ(T²)).

In the second part of my work, I study certain analytic properties of the spaces of geometrically generic configurations.

In particular, it is shown that for n big enough any holomorphic self-map f of  Eⁿ(CP^m,gp) commuting with the natural action of the symmetric group S(n) in Eⁿ(CP^m,gp) is of the form f(q)=τ(q)q=(τ(q)q₁,..., τ(q)q_n), q=(q₁,...,q_n)Eⁿ(CP^m,gp), where

τ: Eⁿ(CP^m,gp)PSL(m+1,C) is an S(n)-invariant  holomorphic map. A similar result holds true for mappings of the configuration space Eⁿ(C^m,gp).