Quotients of group completions by spherical subgroups

Let <f>G</f> be a semi-simple algebraic group and let <f>H</f> be a spherical subgroup. The ground field <f>k</f> is algebraically closed and of characteristic zero. This article is concerned with projective embeddings <f>Y</f> of spherical homogeneous spaces <f>G/H</f>. Our approach in the study of such a variety <f>Y</f> is to realize them as quotients under the action of <f>H</f> of projective embeddings of <f>G</f>. First, we give a more precise sense to this project by defining the quotient of a <f>G</f>-variety by a spherical subgroup <f>H</f>. Then, we give a condition, in terms of <f>G</f>-invariant valuations, under which <f>Y</f> can be obtained by quotient of an embedding of <f>G</f>. Finally, if the index of <f>H</f> in its normalizer is finite, we show that an important class of embeddings of <f>G/H</f> (toroidal and liftable) geometric quotients of embeddings of <f>G</f>. [Copyright &y& Elsevier]