By Andrzej Białynicki-Birula, James B. Carrell, William M. McGovern (auth.)

ISBN-10: 3642077455

ISBN-13: 9783642077456

ISBN-10: 3662050714

ISBN-13: 9783662050712

This is the second one quantity of the recent subseries "Invariant thought and Algebraic Transformation Groups". the purpose of the survey via A. Bialynicki-Birula is to provide the most developments and achievements of analysis within the conception of quotients by way of activities of algebraic teams. This thought comprises geometric invariant concept with a variety of purposes to difficulties of moduli concept. The contribution via J. Carrell treats the topic of torus activities on algebraic kinds, giving an in depth exposition of a few of the cohomological effects one obtains from having a torus motion with fastened issues. Many examples, akin to toric forms and flag types, are mentioned intimately. W.M. McGovern experiences the activities of a semisimple Lie or algebraic team on its Lie algebra through the adjoint motion and on itself through conjugation. His contribution focuses totally on nilpotent orbits that experience came across the widest program to illustration conception within the final thirty-five years.

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**Example text**

Let X = Gr(2, 4) be the Grassmanian of 2-dimensional vector subspaces (planes) in V 4 (k). Let the plane span by vectors ai, a2 be denoted by al 1\ a2. Then consider the action of one-dimensional torus T ~ k* on X induced by the action on V4(k) defined by where i = 1,2,3,4 and nj are integers such that nl < n2 < n3 < n4. There are exactly six fixed points of the action on X. These are ej 1\ ej. where i < j and i, j = 1,2,3,4. Then there are three cases: 1. nl +n4 < n2 +n3, 2. nl + n4 = n2 + n3, 3.

Xil = Xi, Xis = Xj, 2. Xi p -

Says that there exists exactly one such a maximal subset. It is not known, if the number of maximal (with respect to G-saturated inclusion) subsets in the family of all open G-invariant subsets of X admitting good quotient with quasi-projective variety as the quotient space, is finite. No general method of finding the family of G-maximal subsets is known. However, in some special cases such methods can be described. 2 ([Sw 1]). Let U C X be G-maximal. Let H be a connected algebraic group acting on X so that the image of H in the group Aut(X) normalizes the image ofG.

### Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action by Andrzej Białynicki-Birula, James B. Carrell, William M. McGovern (auth.)

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