טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentYoel Feler
SubjectSome Analytic Properties of Configuration Spaces
DepartmentDepartment of Mathematics
Supervisors Professor Emeritus Lin Vladimir
Full Professor Entov Michael


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).