We prove that rational and -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.
@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}, pages = {413--426}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {2}, year = {2011}, mrnumber = {2856154}, zbl = {1241.32017}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_413_0/} }
TY - JOUR AU - Greb, Daniel TI - Rational singularities and quotients by holomorphic group actions JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 413 EP - 426 VL - 10 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_2_413_0/ LA - en ID - ASNSP_2011_5_10_2_413_0 ER -
%0 Journal Article %A Greb, Daniel %T Rational singularities and quotients by holomorphic group actions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 413-426 %V 10 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_2_413_0/ %G en %F 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, Série 5, Tome 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. | EuDML | Numdam | MR | Zbl
[2] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65–68. | EuDML | MR | Zbl
[3] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Études Sci. Publ. Math. 5 (1960), 5–64. | EuDML | Numdam | MR | Zbl
[4] D. Greb, 1-rational singularities and quotients by reductive groups, 2009. arxiv:0901.3539. | Numdam | MR | Zbl
[5] D. Greb, Projectivity of analytic Hilbert and Kähler quotients, Trans. Amer. Math. Soc. 362 (2010), 3243–3271. | MR | Zbl
[6] R. Hartshorne, “Local Cohomology”, Lecture Notes in Mathematics, Vol. 862, Springer-Verlag, Berlin, 1967. | MR
[7] R. Hartshorne, “Ample Subvarieties of Algebraic Varieties”, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin, 1970. | EuDML | MR | Zbl
[8] P. Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), 631–662. | EuDML | MR | Zbl
[9] P. Heinzner and F. Loose, Reduction of complex Hamiltonian -spaces, Geom. Funct. Anal. 4 (1994), 288–297. | EuDML | MR | Zbl
[10] P. Heinzner, L. Migliorini and M. Polito, Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 233–248. | EuDML | Numdam | MR | Zbl
[11] R. Hartshorne and A. Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415–437. | MR | Zbl
[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 | Zbl
[13] U. Karras, Local cohomology along exceptional sets, Math. Ann. 275 (1986), 673–682. | EuDML | MR | Zbl
[14] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), 533–703. | MR | Zbl
[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 | Zbl
[16] S. J. Kovács, Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink, Compos. Math. 118 (1999), 123–133. | MR | Zbl
[17] M. Manaresi, Permanence of local properties under hyperplane sections, In: “Singularities (Warsaw, 1985)”, Banach Center Publ., Vol. 20, PWN, Warsaw, 1988, 291–297. | EuDML | MR | Zbl
[18] Y. Namikawa, Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Internat. J. Math. 13 (2002), 125–135. | MR | Zbl
[19] A. Silva, Relative vanishing theorems. I. Applications to ample divisors, Comment. Math. Helv. 52 (1977), 483–489. | EuDML | MR | Zbl
[20] D. M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), 79–97. | EuDML | MR | Zbl
[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 | Zbl
[22] G. Trautmann, Ein Endlichkeitssatz in der analytischen Geometrie, Invent. Math. 8 (1969), 143–174. | EuDML | MR | Zbl
[23] C. A. Weibel, “An Introduction to Homological Algebra”, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994. | MR | Zbl