unique3 - UNIQUENESS QUESTIONS IN REAL ALGEBRAIC...

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UNIQUENESS QUESTIONS IN REAL ALGEBRAIC TRANSFORMATION GROUPS KARL HEINZ DOVERMANN AND MIKIYA MASUDA Abstract. Let G be a compact Lie group and H a closed subgroup. We show that the homogeneous space G/H has a unique structure as a real algebraic G -variety. For a real algebraic H -variety we show that the balanced product G × H X has the structure of a real algebraic G - variety, and this structure is uniquely determined by the structure on X . On the other hand, suppose that M is a closed smooth G -manifold with positive dimensional orbit space M/G .I f M has a G -equivariant real algebraic model, then it has an uncountable family of birationally inequivalent such models. 1. Introduction Suppose G is a compact Lie group. We like to address the question to which extent an equivariant real algebraic structure 1 on a compact smooth G -manifold is unique. There are answers at both extremes. First we consider a situation in which the structure is unique. The following classical result serves as a starting point. Theorem 1.1. A compact Lie group has the structure of a real linear alge- braic group, and this structure is unique. The first part of the theorem appears to have been invisioned as a final conclusion in Chevalley’s book, see [9, p. viii and Chapter VI]. As stated, the result can be found in [21, p. 246]. Using classical techniques from [9] and [17] combined with the idea of algebraic quotients from [25], we show (see Corollary 4.4): Theorem 1.2. Let G be a compact Lie group and H a closed subgroup. The homogeneous space G/H has the structure of a non-singular real algebraic G -variety, and this structure is unique. If K is another closed subgroup of G and η : G/H G/K is an equivariant map, then η is a regular map. Date : March 28, 2006. 1991 Mathematics Subject Classification. Primary 14P25, 57S15; Secondary 57S25. Key words and phrases. Real Algebraic Geometry, Quotients, Induction. 1 Precise definitions of some of the terms used in this introduction are given in Section 2. In particular, real algebraic structures and varieties in this introduction are assumed to be affine. 1
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2 KARL HEINZ DOVERMANN AND MIKIYA MASUDA In complex algebraic geometry and transformation groups, the study of the structure of homogeneous spaces is a standard topic, see [5, Chapter II] and [18, Chapter IV]. Our proof of Theorem 1.2 also provides a simple real proof of Theorem 1.1, if one assumes the existence of a faithful representation of G . It would be interesting to have a strictly real proof of the existence of a faithful representation of a compact Lie group. Using additional ideas from algebraic geometry and another result of Schwarz [26], one can show a stronger result. Theorem 1.3. Suppose G is a compact Lie group, H ac lo sedsub g roup , and X a real algebraic H -variety. Then G × H X has the structure of a real algebraic G -variety, and this structure is uniquely determined by the structure of X as a real algebraic H -variety. If X is non-singular, then so is G × H X .
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unique3 - UNIQUENESS QUESTIONS IN REAL ALGEBRAIC...

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