Rational singularities and quotients by holomorphic group actions

Greb, Daniel

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 2, pp. 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.

Classification : 32M05, 32S05, 32C36, 14L30

[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 $G$-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