Algebraic density property of homogeneous spaces. (English) Zbl 1201.14041
Let \(X\) be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that \(X\) is equipped with several fixed point free nondegenerate \(\mathrm{SL}_{2}\)-actions satisfying some mild additional assumption.
The authors prove that the Lie algebra generated by completely integrable algebraic vector fields on \(X\) coincides with the space of all algebraic vector fields. In particular, it is shown that apart from a few exceptions this fact is true for any homogeneous space of form \(G/R\) where \(G\) is a linear algebraic group and \(R\) is a closed proper reductive subgroup of \(G\).
The authors prove that the Lie algebra generated by completely integrable algebraic vector fields on \(X\) coincides with the space of all algebraic vector fields. In particular, it is shown that apart from a few exceptions this fact is true for any homogeneous space of form \(G/R\) where \(G\) is a linear algebraic group and \(R\) is a closed proper reductive subgroup of \(G\).
Reviewer: A. Arvanitoyeorgos (Patras)
MSC:
| 14R20 | Group actions on affine varieties |
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
| 14M17 | Homogeneous spaces and generalizations |
Keywords:
affine algebraic variety; algebraic vector field; homogeneous space; linear algebraic groupReferences:
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