Abstract
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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References
Alper, J.D., Easton, R.W.: Recasting results in equivariant geometry: affine cosets, observable subgroups and existence of good quotients. Transform. Groups 17(1), 1–20 (2012)
Artin, M.: Algebraization of formal moduli. II. Existence of modifications. Ann. Math. 2, 91 (1970)
Bäker, H.: Good quotients of Mori dream spaces. Proc. Am. Math. Soc. 139(9), 3135–3139 (2011)
Białynicki-Birula, A.: Quotients by actions of groups. In: Algebraic Quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia of Mathematical Science, vol. 131, pp. 1–82. Springer, Berlin (2002)
Białynicki-Birula, A., Sommese, A.J.: Quotients by \(\mathbb{C}^{\ast }\) and \({\rm SL}(2, \mathbb{C}) \) actions. Trans. Am. Math. Soc. 279(2), 773–800 (1983)
Białynicki-Birula, A., Sommese, A.J.: Quotients by \(\mathbb{C}^* \times \mathbb{C}^*\) actions. Trans. Am. Math. Soc. 289(2), 519–543 (1985)
Białynicki-Birula, A., Świecicka, J.: On complete orbit spaces of \({\rm SL}(2)\) actions. II. Colloq. Math. 63(1), 9–20 (1992)
Białynicki-Birula, A., Świecicka, J.: Open subsets of projective spaces with a good quotient by an action of a reductive group. Transform. Groups 1(3), 153–185 (1996)
Białynicki-Birula, A., Świecicka, J.: Three theorems on existence of good quotients. Math. Ann. 307(1), 143–149 (1997)
Białynicki-Birula, A., Świecicka, J.: A recipe for finding open subsets of vector spaces with a good quotient. Colloq. Math. 77(1), 97–114 (1998)
Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995)
Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties. Graduate studies in mathematics, vol. 124. American Mathematical Society, Providence (2011)
Grauert, H.: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81, 377–391 (1963)
Grauert, H., Remmert, R.: Theory of Stein Spaces. Classics in Mathematic. Springer, Berlin (2004)
Greb, D.: Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory. Adv. Math. 224(2), 401–431 (2010)
Greb, D.: Projectivity of analytic Hilbert and Kähler quotients. Trans. Am. Math. Soc. 362, 3243–3271 (2010)
Greb, D.: Rational singularities and quotients by holomorphic group actions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) X(2), 413–426 (2011)
Greb, D., Heinzner, P.: Kählerian reduction in steps. In: Campbell, E., Helminck, A.G., Kraft, H., Wehlau, D. (eds.) Symmetry and Spaces—Proceedings of a workshop in honour of Gerry Schwarz, Progress in Mathematics, vol. 278, pp. 63–82. Birkhäuser, Boston (2010)
Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. 24 (1965)
Hacon, C.D., Kovács, S.J.: Classification of higher dimensional algebraic varieties. Oberwolfach seminars, vol. 41. Birkhäuser, Basel (2010)
Hartshorne, R.: Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics, vol. 156. Springer, Berlin (1970)
Hartshorne, R.: Algebraic Geometry. Graduate texts in mathematics, vol. 52. Springer, New York (1977)
Hausen, J.: Complete orbit spaces of affine torus actions. Int. J. Math. 20(1), 123–137 (2009)
Hausen, J.: Three Lectures on Cox Rings. In: Torsors, Étale Homotopy and Applications to Rational Points. LMS Lecture Note Series, vol. 405, pp. 3–60. Cambridge University Press (2013)
Heinzner, P.: Fixpunktmengen kompakter Gruppen in Teilgebieten Steinscher Mannigfaltigkeiten. J. Reine Angew. Math. 402, 128–137 (1989)
Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann. 289(4), 631–662 (1991)
Heinzner, P., Loose, F.: Reduction of complex Hamiltonian \(G\)-spaces. Geom. Funct. Anal. 4(3), 288–297 (1994)
Heinzner, P., Huckleberry, A.T., Loose, F.: Kählerian extensions of the symplectic reduction. J. Reine Angew. Math. 455, 123–140 (1994)
Heinzner, P., Migliorini, L., Polito, M.: Semistable quotients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(2), 233–248 (1998)
Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000)
Ivashkovich, S.: Limiting behavior of trajectories of complex polynomial vector fields (2010). arXiv:1004.2618
King, A.D.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxf. Ser. (2) 45(180), 515–530 (1994)
Knutson, D.: Algebraic Spaces. Lecture Notes in Mathematics, vol. 203. Springer, Berlin (1971)
Luna, D.: Slices étales. Bull. Soc. Math. France 33, 81–105 (1973)
Luna, D.: Fonctions différentiables invariantes sous l’opération d’un groupe réductif. Ann. Inst. Fourier (Grenoble) 26(1), ix, 33–49 (1976)
Lopez, A.F.: Noether-Lefschetz theory and the Picard group of projective surfaces. Mem. Am. Math. Soc. 89(438) (1991)
Matsushima, Y.: Espaces homogènes de Stein des groupes de Lie complexes. Nagoya Math. J. 16, 205–218 (1960)
Mumford, D.: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol. 1358. Springer, Berlin (1999)
Mumford, D., Fogarty, J., Kirwan, F.C.: Geometric Invariant Theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, vol. 34, 3rd edn. Springer, Berlin (1994)
Nemirovski, S.: The Levi problem and semistable quotients. Complex Var. Elliptic Equ. 58(11), 1517–1525 (2013)
Onishchik, A.L., Vinberg, E.B.: Lie Groups and Algebraic Groups. Springer series in Soviet mathematics. Springer, Berlin (1990)
Popov, V., Vinberg, E.B.: Invariant Theory. Algebraic geometry IV. In: Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123–284. Springer, Berlin (1994)
Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443 (1956)
Serre, J.P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier (Grenoble) 6, 1–42 (1955–1956)
Shafarevich, I.R.: Basic Algebraic Geometry, 2nd edn. Springer, Berlin (1994)
Snow, D.M.: Reductive group actions on Stein spaces. Math. Ann. 259(1), 79–97 (1982)
Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ. 14, 1–28 (1974)
Acknowledgments
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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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”.
