Projectivity of analytic Hilbert and Kähler quotients

Greb, Daniel

doi:10.1090/S0002-9947-10-05000-2

Projectivity of analytic Hilbert and Kähler quotients
HTML articles powered by AMS MathViewer

by Daniel Greb PDF

Trans. Amer. Math. Soc. 362 (2010), 3243-3271

Abstract:

We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups $G$. We obtain a projectivity result for compact momentum map quotients of algebraic $G$-varieties. Furthermore, we prove equivariant versions of Kodaira’s Embedding Theorem and Chow’s Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient.

References

Daniel Barlet, Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie, Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975) Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975, pp. 1–158 (French). MR 0399503
A. Białynicki-Birula and J. Święcicka, Three theorems on existence of good quotients, Math. Ann. 307 (1997), no. 1, 143–149. MR 1427680, DOI 10.1007/s002080050027
Jean-François Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), no. 1, 65–68 (French). MR 877006, DOI 10.1007/BF01405091
Henri Cartan, Quotients of complex analytic spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 1–15. MR 0139769
Gerd Fischer, Ein relativer Satz von Chow und die Elimination der Unbestimmtheitsstellen meromorpher Funktionen, Math. Ann. 217 (1975), no. 2, 145–152. MR 397031, DOI 10.1007/BF01351292
Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR 755331, DOI 10.1007/978-3-642-69582-7
Daniel Greb, Projectivity of analytic Hilbert quotients, Dissertation, Ruhr-Universität Bochum, 2008, electronic version available at: http://www-brs. ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/GrebDaniel/diss.pdf.
—, $1$-rational singularities and quotients by reductive groups, http://arxiv. org/abs/0901.3539, 2009.
V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
Peter Heinzner, Linear äquivariante Einbettungen Steinscher Räume, Math. Ann. 280 (1988), no. 1, 147–160 (German). MR 928302, DOI 10.1007/BF01474186
Peter Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), no. 4, 631–662. MR 1103041, DOI 10.1007/BF01446594
Peter Heinzner and Alan Huckleberry, Kählerian potentials and convexity properties of the moment map, Invent. Math. 126 (1996), no. 1, 65–84. MR 1408556, DOI 10.1007/s002220050089
J. Hausen and P. Heinzner, Actions of compact groups on coherent sheaves, Transform. Groups 4 (1999), no. 1, 25–34. MR 1669182, DOI 10.1007/BF01236660
P. Heinzner, A. T. Huckleberry, and F. Loose, Kählerian extensions of the symplectic reduction, J. Reine Angew. Math. 455 (1994), 123–140. MR 1293876
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
P. Heinzner and F. Loose, Reduction of complex Hamiltonian $G$-spaces, Geom. Funct. Anal. 4 (1994), no. 3, 288–297. MR 1274117, DOI 10.1007/BF01896243
Peter Heinzner and Luca Migliorini, Projectivity of moment map quotients, Osaka J. Math. 38 (2001), no. 1, 167–184. MR 1824905
Peter Heinzner, Luca Migliorini, and Marzia Polito, Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 2, 233–248. MR 1631577
Gerhard P. Hochschild, The structure of Lie groups, Holden-Day Inc., San Francisco, 1965.
Peter Heinzner and Gerald W. Schwarz, Cartan decomposition of the moment map, Math. Ann. 337 (2007), no. 1, 197–232. MR 2262782, DOI 10.1007/s00208-006-0032-8
M. M. Kapranov, Chow quotients of Grassmannians. I, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR 1237834
Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
János Kollár, Quotient spaces modulo algebraic groups, Ann. of Math. (2) 145 (1997), no. 1, 33–79. MR 1432036, DOI 10.2307/2951823
Domingo Luna, Slices étales, Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire 33, Soc. Math. France, Paris, 1973, pp. 81–105 (French). MR 0342523, DOI 10.24033/msmf.110
Domingo Luna, Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, ix, 33–49 (French, with English summary). MR 423398
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
Yoshinori Namikawa, Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Internat. J. Math. 13 (2002), no. 2, 125–135. MR 1891205, DOI 10.1142/S0129167X02001277
Amnon Neeman, Analytic questions in geometric invariant theory, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 11–23. MR 999979, DOI 10.1090/conm/088/999979
Reinhold Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann. 133 (1957), 328–370 (German). MR 92996, DOI 10.1007/BF01342886
Mark Roberts, A note on coherent $G$-sheaves, Math. Ann. 275 (1986), no. 4, 573–582. MR 859331, DOI 10.1007/BF01459138
Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443. MR 82183, DOI 10.2307/2372523
Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175
Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87–129. MR 1314032, DOI 10.2307/2118628
Dennis M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), no. 1, 79–97. MR 656653, DOI 10.1007/BF01456830
Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28. MR 337963, DOI 10.1215/kjm/1250523277
J. H. Sampson and G. Washnitzer, A Künneth formula for coherent algebraic sheaves, Illinois J. Math. 3 (1959), 389–402. MR 106906
Jean Varouchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), no. 1, 13–52. MR 973802, DOI 10.1007/BF01457500