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Lifting mappings over invariants of finite groups. (English) Zbl 1164.14009

Let \(G\) be a finite group of automorphisms of a complex vector space \(V\). The orbit space \(Z = V/G\) is a normal affine variety and has the structure of a complex analytic space. So one can speak about regular, holomorphic and formal maps of \(\mathbb {C}^p\) into \(Z\). The authors study the problem of lifting of a morphism \( f \: \mathbb {C}^p \rightarrow V/G\) near the origin to a morphism \(\tilde f \: \mathbb {C}^p \rightarrow V\). They consider cases when \(f\) is a regular, holomorphic or formal morphism and give sufficient conditions for existence of a lift in terms of the jet spaces \(J^q(\mathbb {C}^p, Z)\). Algebraically, the problem can be considerered as a special case of the problem of extension of morphisms in the category of \(\mathbb {C}\)-algebras from a subalgebra to the whole algebra.
These conditions are checking for some complex reflection groups and for the dihedral groups.

MSC:

14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
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