Ph.D Thesis


Ph.D StudentFeler Yoel
SubjectSome Analytic Properties of Configuration Spaces
DepartmentDepartment of Mathematics
Supervisors PROFESSOR EMERITUS Vladimir Lin
PROF. Michael Entov


Abstract

The configuration space Cn(X) of a complex space X consists of all n point subsets (``configurations") Q={q1,...,qn}X. If X carries an additional geometric structure, it may be taken into account. Say if X is either the projective space CPm or the affine space Cm 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,...,qn)Xn such that the set Q={q1,...,qn}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 Bn(T2)=π1(Cn(T2)) with a non-abelian image preserves the pure torus braid group Pn(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(CPm,gp) commuting with the natural action of the symmetric group S(n) in En(CPm,gp) is of the form f(q)=τ(q)q=(τ(q)q1,..., τ(q)qn), q=(q1,...,qn)En(CPm,gp), where

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