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Quotients by $${\mathbb{C}}^*$$ and SL(2,$${\mathbb{C}})$$ actions. (English) Zbl 0566.32026
Let the notions be as in the preceding review. The main theorem states the following: There is a bijective correspondence between the set of cross sections $$(A^+,A^-)$$ and the open sets $$U\subset X\setminus X^ T$$ with the properties: (i) U is T-invariant, (ii) the geometric quotient $$U\to U/T$$ exists and U/T is a compact complex space. The correspondence is given by $$U=\cup_{i\in A^-,j\in A^+}C_{ij}$$. Furthermore all such sets U are Zariski open in X. They are called sectional open sets. If X is a projective algebraic variety or a Kähler manifold then $$X\setminus X^ T$$ is the union of all sectional open sets. In the special case that $$\rho$$ is an algebraic T-action on an algebraic manifold X the following cohomology formula for the quotient of a sectional set U is proved: $$P(U/T)=\sum_{i\in A^-}P(F_ i)(t^{2d_ i^+}-t^{2d^-_ i})/t^ 2-1)=\sum_{j\in A^+}(t^{2d_ j^-}-t^{2d_ j^+})/(t^ 2-1)$$, where P denotes the Poincaré polynomial and $$d_ i^{\pm}=\dim X_ i^{\pm}-\dim F_ i.$$ Finally the following conjecture of D. Mumford is settled: Consider the diagonal $$SL_ 2({\mathbb{C}})$$-action on $$({\mathbb{P}}_ 1({\mathbb{C}}))^ n$$ (n$$\geq 3)$$. Let U be a $$SL_ 2({\mathbb{C}})$$-invariant Zariski open set, which is also invariant under coordinate interchanging. Assume that the geometric quotient $$U/SL_ 2({\mathbb{C}})$$ exists and is an algebraic variety in the sense of Artin. Then n is odd and U is the set of points with at most (n-1)/2 coordinates the same.
Reviewer: K.Oeljeklaus

##### MSC:
 32M05 Complex Lie groups, group actions on complex spaces 14L30 Group actions on varieties or schemes (quotients) 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32C20 Normal analytic spaces
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