Rational singularities and quotients by holomorphic group actions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 413-426

We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven.

Published online : 2018-08-07
Classification:  32M05,  32S05,  32C36,  14L30
@article{ASNSP_2011_5_10_2_413_0,
     author = {Greb, Daniel},
     title = {Rational singularities and quotients by holomorphic group actions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {2},
     year = {2011},
     pages = {413-426},
     zbl = {1241.32017},
     mrnumber = {2856154},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_413_0}
}
Greb, Daniel. Rational singularities and quotients by holomorphic group actions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 413-426. http://www.numdam.org/item/ASNSP_2011_5_10_2_413_0/

[1] A. Andreotti and A. Kas, Duality on complex spaces, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 187–263. | Numdam | MR 425160 | Zbl 0278.32007

[2] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65–68. | MR 877006 | Zbl 0619.14029

[3] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Études Sci. Publ. Math. 5 (1960), 5–64. | Numdam | MR 121814 | Zbl 0100.08001

[4] D. Greb, 1-rational singularities and quotients by reductive groups, 2009. arxiv:0901.3539. | Numdam | MR 2856154 | Zbl 0619.14029

[5] D. Greb, Projectivity of analytic Hilbert and Kähler quotients, Trans. Amer. Math. Soc. 362 (2010), 3243–3271. | MR 2592955 | Zbl 1216.14045

[6] R. Hartshorne, “Local Cohomology”, Lecture Notes in Mathematics, Vol. 862, Springer-Verlag, Berlin, 1967. | MR 224620

[7] R. Hartshorne, “Ample Subvarieties of Algebraic Varieties”, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin, 1970. | MR 282977 | Zbl 0208.48901

[8] P. Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), 631–662. | MR 1103041 | Zbl 0728.32010

[9] P. Heinzner and F. Loose, Reduction of complex Hamiltonian G-spaces, Geom. Funct. Anal. 4 (1994), 288–297. | MR 1274117 | Zbl 0816.53018

[10] P. Heinzner, L. Migliorini and M. Polito, Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 233–248. | Numdam | MR 1631577 | Zbl 0922.32017

[11] R. Hartshorne and A. Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415–437. | MR 347821 | Zbl 0286.13013

[12] M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math. 13 (1974), 115–175. | MR 347810 | Zbl 0289.14010

[13] U. Karras, Local cohomology along exceptional sets, Math. Ann. 275 (1986), 673–682. | MR 859337 | Zbl 0583.32026

[14] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), 533–703. | MR 1149195 | Zbl 0773.14004

[15] J. Kollár, Singularities of pairs, In: “Algebraic Geometry (Santa Cruz, 1995)”, Proc. Sympos. Pure Math., Vol. 62, Amer. Math. Soc., Providence, RI, 1997, 221–287. | MR 1492525 | Zbl 0905.14002

[16] S. J. Kovács, Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink, Compos. Math. 118 (1999), 123–133. | MR 1713307 | Zbl 0962.14011

[17] M. Manaresi, Permanence of local properties under hyperplane sections, In: “Singularities (Warsaw, 1985)”, Banach Center Publ., Vol. 20, PWN, Warsaw, 1988, 291–297. | MR 1101846 | Zbl 0673.14003

[18] Y. Namikawa, Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Internat. J. Math. 13 (2002), 125–135. | MR 1891205 | Zbl 1055.32015

[19] A. Silva, Relative vanishing theorems. I. Applications to ample divisors, Comment. Math. Helv. 52 (1977), 483–489. | MR 460733 | Zbl 0372.32014

[20] D. M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), 79–97. | MR 656653 | Zbl 0509.32021

[21] Y.-T. Siu and G. Trautmann, “Gap-Sheaves and Extension of Coherent Analytic Subsheaves”, Lecture Notes in Mathematics, Vol. 172, Springer-Verlag, Berlin, 1971. | MR 287033 | Zbl 0208.10403

[22] G. Trautmann, Ein Endlichkeitssatz in der analytischen Geometrie, Invent. Math. 8 (1969), 143–174. | MR 251251 | Zbl 0175.37601

[23] C. A. Weibel, “An Introduction to Homological Algebra”, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994. | MR 1269324 | Zbl 0797.18001