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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

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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