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Complex-analytic quotients of algebraic \(G\)-varieties


It is shown that any compact semistable quotient of a normal variety by a complex reductive Lie group \(G\) (in the sense of Heinzner and Snow) is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski’s class \(\fancyscript{Q}_G\) has a realisation as a good quotient, and that every complete algebraic variety in \(\fancyscript{Q}_G\) is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class \(\fancyscript{Q}_T\), where \(T\) is an algebraic torus, is a toric variety.

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The author wants to thank Peter Heinzner, Christian Miebach, Stefan Nemirovski, and Karl Oeljeklaus for interesting and stimulating discussions. Furthermore, he is grateful to the organisers of the “Russian–German conference on Several Complex Variables” at Steklov Institute, during which some of these discussions took place, for the invitation and for their hospitality.

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Correspondence to Daniel Greb.

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During the preparation of this paper, the author was partially supported by the DFG-Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds”, the DFG-Graduiertenkolleg 1821 “Cohomological Methods in Geometry”, as well as by the Baden-Württemberg-Stiftung through the “Eliteprogramm für Postdoktorandinnen und Postdoktoranden”.

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Greb, D. Complex-analytic quotients of algebraic \(G\)-varieties. Math. Ann. 363, 77–100 (2015).

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Mathematics Subject Classification

  • 32M05
  • 14L24
  • 14L30