$\mu$-constant monodromy groups and marked singularities

Hertling, Claus

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2643-2680.

Nous considérons d’un point de vue global les familles $\mu$-constantes de germes de fonctions holomorphes à singularités isolées. Tout d’abord, nous étudions un groupe de monodromie des familles contenant une singularité fixée. Ce groupe est constitué d’automorphismes du réseau de Milnor qui respectent non seulement la forme d’intersection, mais aussi la forme de Seifert et la monodromie. Nous conjecturons qu’il contient tous les automorphismes de ce type, modulo $\pm id$. Ensuite, nous définissons les singularités marquées et construisons leurs espaces de modules globaux pour leurs classes d’équivalence à droite. La conjecture sur le groupe impliquerait que ces espaces de modules sont connexes. Nous discutons de la relation avec les problèmes de type Torelli et nous formulons une nouvelle conjecture de type Torelli global pour les singularités marquées. Toutes ces conjectures sont montrées pour les singularités simples et pour 22 des 28 singularités exceptionnelles.

$\mu$-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo $\pm id$. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these moduli spaces are connected. The relation with Torelli type problems is discussed and a new global Torelli type conjecture for marked singularities is formulated. All conjectures are proved for the simple and $22$ of the $28$ exceptional singularities.

Classification : 32S15, 32S40, 14D22, 58K70
Keywords: $\mu $-constant deformation, monodromy group, marked singularity, moduli space, Torelli type problem, symmetries of singularities
Mot clés : déformations $\mu $-constantes, groupe de monodromie, singularité marquée, espace des modules, problème de type Torelli, symétries de singularités

[1] A’Campo, Norbert La fonction zêta d’une monodromie, Comment. Math. Helv., Volume 54 (1979), pp. 318-327 | MR | Zbl

[2] Arnolʼd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I, Monographs in Mathematics, 82, Birkhäuser Boston Inc., Boston, MA, 1985 (The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds) | MR | Zbl

[3] Arnolʼd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. II, Monographs in Mathematics, 83, Birkhäuser Boston Inc., Boston, MA, 1988 (Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi) | MR | Zbl

[4] Bröcker, Th. Differentiable germs and catastrophes, Cambridge University Press, Cambridge, 1975 (Translated from the German, last chapter and bibliography by L. Lander, London Mathematical Society Lecture Note Series, No. 17) | MR | Zbl

[5] Ebeling, Wolfgang Functions of several complex variables and their singularities, Graduate Studies in Mathematics, 83, American Mathematical Society, Providence, RI, 2007 (Translated from the 2001 German original by Philip G. Spain) | MR | Zbl

[6] Gabrièlov, A. M. Bifurcations, Dynkin diagrams and the modality of isolated singularities, Funkcional. Anal. i Priložen., Volume 8 (1974) no. 2, pp. 7-12 | MR | Zbl

[7] Hertling, Claus Analytische Invarianten bei den unimodularen und bimodularen Hyperflächensingularitäten, Bonner Mathematische Schriften [Bonn Mathematical Publications], 250, Universität Bonn Mathematisches Institut, Bonn, 1993 (Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1992) | MR | Zbl

[8] Hertling, Claus Ein Torellisatz für die unimodalen und bimodularen Hyperflächensingularitäten, Math. Ann., Volume 302 (1995), pp. 359-394 | MR | Zbl

[9] Hertling, Claus Brieskorn lattices and Torelli type theorems for cubics in ${\mathbf{P}}^3$ and for Brieskorn-Pham singularities with coprime exponents, Singularities (Oberwolfach, 1996) (Progr. Math.), Volume 162, Birkhäuser, Basel, 1998, pp. 167-194 | MR | Zbl

[10] Hertling, Claus Classifying spaces for polarized mixed Hodge structures and for Brieskorn lattices, Compositio Math., Volume 116 (1999) no. 1, pp. 1-37 | DOI | MR | Zbl

[11] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, 151, Cambridge University Press, Cambridge, 2002 | DOI | MR | Zbl

[12] Hertling, Claus Generic Torelli for semiquasihomogeneous singularities, Trends in singularities (Trends Math.), Birkhäuser, Basel, 2002, pp. 115-140 | MR | Zbl

[13] Hertling, Claus; Manin, Yu. Weak Frobenius manifolds, Internat. Math. Res. Notices (1999) no. 6, pp. 277-286 | DOI | MR | Zbl

[14] Holmann, Harald Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann., Volume 150 (1963), pp. 327-360 | EuDML | MR | Zbl

[15] Kulikov, Valentine S. Mixed Hodge structures and singularities, Cambridge Tracts in Mathematics, 132, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[16] Malgrange, Bernard Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. (4), Volume 7 (1974), p. 405-430 (1975) | EuDML | Numdam | MR | Zbl

[17] Mather, John N. Stability of $C^{\infty }$ mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. (1968) no. 35, pp. 279-308 | EuDML | Numdam | Zbl

[18] Michel, F.; Weber, C. Sur le rôle de la monodromie entière dans la topologie des singularités, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 1, pp. 183-218 | EuDML | Numdam | MR | Zbl

[19] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J., 1968 | Numdam | MR | Zbl

[20] Orlik, Peter On the homology of weighted homogeneous manifolds, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, Springer, Berlin (1972), p. 260-269. Lecture Notes in Math., Vol. 298 | Numdam | Zbl

[21] Saito, Morihiko On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble), Volume 39 (1989) no. 1, pp. 27-72 | EuDML | Numdam | MR | Zbl

[22] Saito, Morihiko Period mapping via Brieskorn modules, Bull. Soc. Math. France, Volume 119 (1991) no. 2, pp. 141-171 | EuDML | Numdam | MR | Zbl

[23] Scherk, J.; Steenbrink, J. H. M. On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann., Volume 271 (1985) no. 4, pp. 641-665 | DOI | EuDML | MR | Zbl

[24] Slodowy, Peter Einige Bemerkungen zur Entfaltung symmetrischer Funktionen, Math. Z., Volume 158 (1978) no. 2, pp. 157-170 | EuDML | Numdam | MR | Zbl

[25] Teissier, Bernard Déformations à type topologique constant, Quelques problèmes de modules (Sém. de Géométrie Analytique, École Norm. Sup., Paris, 1971–1972), Soc. Math. France, Paris, 1974, p. 215-249. Astérisque, No. 16 | Numdam | MR | Zbl

[26] Tráng, Lê Dũng; Ramanujam, C. P. The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math., Volume 98 (1976) no. 1, pp. 67-78 | MR | Zbl

[27] Varchenko, A. N. The asymptotics of holomorphic forms determine a mixed Hodge structure, Sov. Math. Dokl., Volume 22 (1980), pp. 772-775 | MR | Zbl

[28] Wall, C. T. C. A note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 3, pp. 169-175 | DOI | MR | Zbl

[29] Wall, C. T. C. A second note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 5, pp. 347-354 | DOI | Zbl

[30] Yu, Jian Ming Kombinatorische Geometrie der Stokesregionen, Bonner Mathematische Schriften [Bonn Mathematical Publications], 212, Universität Bonn Mathematisches Institut, Bonn, 1990 (Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1990) | MR | Zbl

[31] Yu, Jian Ming Galois group of Lyashko-Looijenga mapping, Math. Z., Volume 232 (1999), pp. 321-330 | MR | Zbl