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A general Hilbert-Mumford criterion

Jürgen Hausen (2003)

Annales de l’institut Fourier

Let a reductive group G act on an algebraic variety X . We give a Hilbert-Mumford type criterion for the construction of open G -invariant subsets V X admitting a good quotient by G .

Adjoint representation of E 8 and del Pezzo surfaces of degree 1

Vera V. Serganova, Alexei N. Skorobogatov (2011)

Annales de l’institut Fourier

Let X be a del Pezzo surface of degree 1 , and let G be the simple Lie group of type E 8 . We construct a locally closed embedding of a universal torsor over X into the G -orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Néron-Severi torus T of X identified with a maximal torus of G extended by the group of scalars. Moreover, the T -invariant hyperplane sections of the torsor defined by the roots of G are the inverse images...

Affine compact almost-homogeneous manifolds of cohomogeneity one

Daniel Guan (2009)

Open Mathematics

This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem...

Algebras with finitely generated invariant subalgebras

Ivan V. Arzhantsev (2003)

Annales de l’institut Fourier

We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

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