Ph.D Thesis | |

Ph.D Student | Feler Yoel |
---|---|

Subject | Some Analytic Properties of Configuration Spaces |

Department | Department of Mathematics |

Supervisors | PROFESSOR EMERITUS Vladimir Lin |

PROF. Michael Entov |

The configuration space C^{n}(X) of a complex
space X consists of all n point subsets (``configurations") Q={q_{1},...,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 C^{n}(X,gp) of geometrically generic configurations
consists of all n point subsets Q_{}X such that no hyperplane in X contains
more than m points of Q. The corresponding ordered configuration spaces E^{n}(X)
and E^{n}(X,gp) consist of all q=(q_{1},...,q_{n})_{}X^{n}
such that the set Q={q_{1},...,q_{n}}_{}X belongs to C^{n}(X)
and C^{n}(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 C^{n}(**T**^{2})
of a complex torus **T**^{2} either carries the whole of C^{n}(**T**^{2})
into an orbit of the diagonal Aut(**T**^{2}) action in C^{n}(**T**^{2})
or is of the form F(Q)=T(Q)Q, where T: C^{n}(**T**^{2})_{}Aut **T**^{2}
is a holomorphic map. We also prove that for n>4 any endomorphism of the
torus braid group B_{n}(**T**^{2})=π_{1}(C^{n}(**T**^{2}))
with a non-abelian image preserves the pure torus braid group P_{n}(**T**^{2})=π_{1}(E^{n}(**T**^{2})).

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 E^{n}(**CP**^{m},gp) commuting
with the natural action of the symmetric group **S**(n) in E^{n}(**CP**^{m},gp)
is of the form f(q)=τ(q)q=(τ(q)q_{1},..., τ(q)q_{n}),
q=(q_{1},...,q_{n})_{}E^{n}(**CP**^{m},gp),
where

τ: E^{n}(**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 E^{n}(**C**^{m},gp).