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

This is the second one quantity of the hot subseries "Invariant concept and Algebraic Transformation Groups". the purpose of the survey by way of A. Bialynicki-Birula is to give the most tendencies and achievements of study within the concept of quotients by way of activities of algebraic teams. This idea includes geometric invariant concept with a number of purposes to difficulties of moduli concept. The contribution via J. Carrell treats the topic of torus activities on algebraic kinds, giving a close exposition of some of the cohomological effects one obtains from having a torus motion with fastened issues. Many examples, resembling toric types and flag forms, are mentioned intimately. W.M. McGovern experiences the activities of a semisimple Lie or algebraic crew on its Lie algebra through the adjoint motion and on itself through conjugation. His contribution focuses totally on nilpotent orbits that experience came upon the widest software to illustration thought within the final thirty-five years.

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**Extra info for Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action**

**Sample text**

Then there exists a one-parameter subgroup ¢ : k* --+ G such that lim ¢(t)x = y. 2. G-Maximal Subsets The main problem of the theory of good quotient can be formulated as follows: For a given action of a reductive group G on an algebraic variety X (or more generally: on an algebraic space) find all open G-invariant subsets which admit a good quotient where the quotient space is of the given type (quasi-projective, algebraic variety or more generally: an algebraic space). e. if V is open, admits a good quotient by G and there exists no properly bigger open subset U of X such that the quotient U ~ UIIG exists and V is an inverse image under rru of some open subset of U II G.

Thefield k(XI) of rational functions on Xl is a Galois extension of the field k(X) with Galois group g. and the group g. acts on X I 3. there exists a free action of G on X I so that the action commutes with the action ofg. and K is G-equivariant, 4. there exists a good geometric quotient rr : XI ~ XI/G, where XI/G is a prevariety. 5. the morphism rr defines a structure of a locally (in Zariski topology) trivial principal fibration with structure group G, 6. if the action of G on X is proper, then the action of G on X I is proper and X I I G is a variety, 7.

3. Quotients of Open Subsets of Projective and Affine Spaces The described above methods of finding some open subsets of projective varieties can be used to describe all T -maximal subsets of projective spaces (see [BB,Sw 9 D. We shall sketch below the main results obtained in the case. Let T = (k*Y act on a projective space pn = Proj(vn+I). Suppose that we have fixed a lifting of the action to Vn+l . Then to every point x E pn, we may associate a cell c(x) and convex set JL(c(x» C Zs ® R (see page 47).