JournalsjemsVol. 10 , No. 3DOI 10.4171/jems/124

Spaces of geometrically generic configurations

  • Yoel Feler

    Weizmann Institute of Science, Rehovot, Israel
Spaces of geometrically generic configurations cover

Abstract

Let XX denote either CPm\Bbb{CP}^m or Cm\Bbb{C}^m. We study certain analytic properties of the space \CalEn(X,gp)\Cal{E}^n(X,gp) of ordered geometrically generic nn-point configurations in XX. This space consists of all q=(q1,...,qn)Xnq=(q_1,...,q_n)\in X^n such that no m+1m+1 of the points q1,...,qnq_1,...,q_n belong to a hyperplane in XX. In particular, we show that for a big enough nn any holomorphic map f ⁣:\CalEn(CPm,gp)\CalEn(CPm,gp)f\colon\Cal{E}^n(\Bbb{CP}^m,gp)\to\Cal{E}^n(\Bbb{CP}^m,gp) commuting with the natural action of the symmetric group S(n)\mathbf{S}(n) in \CalEn(CPm,gp)\Cal{E}^n(\Bbb{CP}^m,gp) is of the form f(q)=τ(q)q=(τ(q)q1,...,τ(q)qn)f(q)=\tau(q)q=(\tau(q)q_1,...,\tau(q)q_n), q\CalEn(CPm,gp)q\in \Cal{E}^n(\Bbb{CP}^m,gp), where τ ⁣:\CalEn(CPm,gp)PSL(m+1,C)\tau\colon\Cal{E}^n(\Bbb{CP}^m,gp) \to{\mathbf{PSL}(m+1,\mathbb C)} is an S(n)\mathbf{S}(n)-invariant holomorphic map. A similar result holds true for mappings of the configuration space \CalEn(Cm,gp)\Cal{E}^n(\Bbb{C}^m,gp).