Representation theory for log-canonical surface singularities

Gustavsen, Trond Stølen; Ile, Runar

Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 389-416.

Nous considérons la théorie des représentations pour une classe des singularités des surfaces log-canoniques dans le sens de modules réflexifs (ou d’une manière équivalente, modules maximals de Cohen-Macaulay) et dans le sens de représentations de dimension finie du groupe fondamental local. Une classification et une énumération détaillées des modules réflexifs indécomposables sont données, et nous montrons que n’importe quel module réflexif admet une connexion intégrable, et par conséquent est induit par une représentation de dimension finie du groupe fondamental local.

We consider the representation theory for a class of log-canonical surface singularities in the sense of reflexive (or equivalently maximal Cohen-Macaulay) modules and in the sense of finite dimensional representations of the local fundamental group. A detailed classification and enumeration of the indecomposable reflexive modules is given, and we prove that any reflexive module admits an integrable connection and hence is induced from a finite dimensional representation of the local fundamental group.

[1] Atiyah, M. F. Vector bundles over an elliptic curve, Proc. London Math. Soc. (3), Volume 7 (1957), pp. 414-452 | DOI | MR | Zbl

[2] Auslander, Maurice Rational singularities and almost split sequences, Trans. Amer. Math. Soc., Volume 293 (1986) no. 2, pp. 511-531 | DOI | MR | Zbl

[3] Behnke, Kurt On Auslander modules of normal surface singularities, Manuscripta Math., Volume 66 (1989) no. 2, pp. 205-223 | MR | Zbl

[4] Bernšteĭn, I. N.; Gelfand, I. M.; Gelfand, S. I. Differential operators on a cubic cone, Uspehi Mat. Nauk, Volume 27 (1972) no. 1(163), pp. 185-190 | MR | Zbl

[5] Drozd, Yuriy A.; Greuel, Gert-Martin; Kashuba, Irina On Cohen-Macaulay modules on surface singularities, Mosc. Math. J., Volume 3 (2003) no. 2, p. 397-418, 742 (Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday) | MR | Zbl

[6] Esnault, Hélène Reflexive modules on quotient surface singularities, J. Reine Angew. Math., Volume 362 (1985), pp. 63-71 | DOI | MR | Zbl

[7] Grauert, H.; Peternell, Th.; Remmert, R.; Gamkrelidze, R.V. Several complex variables VII. Sheaf-theoretical methods in complex analysis, 74, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 1994 (369 p.) | MR | Zbl

[8] Grauert, Hans Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 | DOI | MR | Zbl

[9] Grothendieck, Alexander Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2), Volume 9 (1957), pp. 119-221 | MR | Zbl

[10] Gustavsen, Trond Stølen; Ile, Runar Reflexive modules on normal surface singularities and representations of the local fundamental group., J. Pure Appl. Algebra, Volume 212 (2008) no. 4, pp. 851-862 | DOI | MR | Zbl

[11] Herzog, Jürgen Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln, Math. Ann., Volume 233 (1978) no. 1, pp. 21-34 | DOI | MR | Zbl

[12] Kahn, Constantin P. M. Reflexive Moduln auf einfach-elliptischen Flächensingularitäten, Bonner Mathematische Schriften [Bonn Mathematical Publications], 188, Universität Bonn Mathematisches Institut, Bonn, 1988 (Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1988) | MR | Zbl

[13] Kahn, Constantin P. M. Reflexive modules on minimally elliptic singularities, Math. Ann., Volume 285 (1989) no. 1, pp. 141-160 | DOI | MR | Zbl

[14] Kawamata, Yujiro Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2), Volume 127 (1988) no. 1, pp. 93-163 | DOI | MR | Zbl

[15] Laufer, Henry B. On minimally elliptic singularities, Amer. J. Math., Volume 99 (1977) no. 6, pp. 1257-1295 | DOI | MR | Zbl

[16] Lehmann, D. Connexions à courbure nulle et $K$-théorie, An. Acad. Brasil. Ci., Volume 40 (1968), pp. 1-6 | MR | Zbl

[17] Lenzing, Helmut; Meltzer, Hagen Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, Representations of algebras (Ottawa, ON, 1992) (CMS Conf. Proc.), Volume 14, Amer. Math. Soc., Providence, RI, 1993, pp. 313-337 | MR | Zbl

[18] Levasseur, Thierry Opérateurs différentiels sur les surfaces munies d’une bonne ${\mathbf{C}}^{*}$-action, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39e Année (Paris, 1987/1988) (Lecture Notes in Math.), Volume 1404, Springer, Berlin, 1989, pp. 269-295 | MR | Zbl

[19] Lipman, Joseph Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 195-279 | DOI | Numdam | MR | Zbl

[20] Matsuki, Kenji Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002 | MR | Zbl

[21] Mumford, David The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961) no. 9, pp. 5-22 | DOI | Numdam | MR | Zbl

[22] Pinkham, H. Normal surface singularities with $C^{*}$ action, Math. Ann., Volume 227 (1977) no. 2, pp. 183-193 | DOI | MR | Zbl

[23] Ploog, David Equivariant autoequivalences for finite group actions, Adv. Math., Volume 216 (2007) no. 1, pp. 62-74 | DOI | MR | Zbl

[24] Polishchuk, A. Holomorphic bundles on 2-dimensional noncommutative toric orbifolds, Noncommutative geometry and number theory (Aspects Math., E37), Vieweg, Wiesbaden, 2006, pp. 341-359 | MR | Zbl

[25] Saito, Kyoji Einfach-elliptische Singularitäten, Invent. Math., Volume 23 (1974), pp. 289-325 | DOI | MR | Zbl

[26] Schlessinger, Michael Rigidity of quotient singularities, Invent. Math., Volume 14 (1971), pp. 17-26 | DOI | MR | Zbl

[27] Wagreich, Philip Singularities of complex surfaces with solvable local fundamental group, Topology, Volume 11 (1971), pp. 51-72 | DOI | MR | Zbl

[28] Wahl, Jonathan M. Equations defining rational singularities, Ann. Sci. École Norm. Sup. (4), Volume 10 (1977) no. 2, pp. 231-263 | Numdam | MR | Zbl