μ-constant monodromy groups and marked singularities
Annales de l'Institut Fourier, Volume 61 (2011) no. 7, p. 2643-2680
μ-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 ±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.
Nous considérons d’un point de vue global les familles μ-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 ±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.
DOI : https://doi.org/10.5802/aif.2789
Classification:  32S15,  32S40,  14D22,  58K70
Keywords: μ-constant deformation, monodromy group, marked singularity, moduli space, Torelli type problem, symmetries of singularities
@article{AIF_2011__61_7_2643_0,
     author = {Hertling, Claus},
     title = {$\mu $-constant monodromy groups and marked singularities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     pages = {2643-2680},
     doi = {10.5802/aif.2789},
     zbl = {1279.32021},
     mrnumber = {3112503},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2011__61_7_2643_0}
}
Hertling, Claus. $\mu $-constant monodromy groups and marked singularities. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2643-2680. doi : 10.5802/aif.2789. https://aif.centre-mersenne.org/item/AIF_2011__61_7_2643_0/

[1] A’Campo, Norbert La fonction zêta d’une monodromie, Comment. Math. Helv., Tome 54 (1979), pp. 318-327 | MR 535062 | Zbl 0441.32004

[2] ArnolʼD, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I, Birkhäuser Boston Inc., Boston, MA, Monographs in Mathematics, Tome 82 (1985) (The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds) | MR 516034 | Zbl 0304.57018

[3] ArnolʼD, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. II, Birkhäuser Boston Inc., Boston, MA, Monographs in Mathematics, Tome 83 (1988) (Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi) | MR 777682 | Zbl 0554.58001

[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 966191 | Zbl 0659.58002

[5] Ebeling, Wolfgang Functions of several complex variables and their singularities, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 83 (2007) (Translated from the 2001 German original by Philip G. Spain) | MR 494220 | Zbl 0302.58006

[6] Gabrièlov, A. M. Bifurcations, Dynkin diagrams and the modality of isolated singularities, Funkcional. Anal. i Priložen., Tome 8 (1974) no. 2, pp. 7-12 | MR 2319634 | Zbl 1188.32001

[7] Hertling, Claus Analytische Invarianten bei den unimodularen und bimodularen Hyperflächensingularitäten, Universität Bonn Mathematisches Institut, Bonn, Bonner Mathematische Schriften [Bonn Mathematical Publications], 250 (1993) (Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1992) | MR 430286 | Zbl 0343.32002

[8] Hertling, Claus Ein Torellisatz für die unimodalen und bimodularen Hyperflächensingularitäten, Math. Ann., Tome 302 (1995), pp. 359-394 | MR 440066 | Zbl 0344.32007

[9] Hertling, Claus Brieskorn lattices and Torelli type theorems for cubics in P 3 and for Brieskorn-Pham singularities with coprime exponents, Singularities (Oberwolfach, 1996), Birkhäuser, Basel (Progr. Math.) Tome 162 (1998), pp. 167-194 | MR 1286737 | Zbl 0833.32006

[10] Hertling, Claus Classifying spaces for polarized mixed Hodge structures and for Brieskorn lattices, Compositio Math., Tome 116 (1999) no. 1, pp. 1-37 | Article | MR 1336340 | Zbl 0922.32019 | Zbl 0843.32020

[11] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 151 (2002) | Article | MR 1652473 | Zbl 1023.14018 | Zbl 0915.14023

[12] Hertling, Claus Generic Torelli for semiquasihomogeneous singularities, Trends in singularities, Birkhäuser, Basel (Trends Math.) (2002), pp. 115-140 | MR 1669448 | Zbl 1025.32025 | Zbl 0922.32019

[13] Hertling, Claus; Manin, Yu. Weak Frobenius manifolds, Internat. Math. Res. Notices (1999) no. 6, pp. 277-286 | Article | MR 1900783 | Zbl 0960.58003 | Zbl 1025.32025

[14] Holmann, Harald Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann., Tome 150 (1963), pp. 327-360 | MR 1924259 | Zbl 0156.30603 | Zbl 1023.14018

[15] Kulikov, Valentine S. Mixed Hodge structures and singularities, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 132 (1998) | MR 1680372 | Zbl 0902.14005 | Zbl 0960.58003

[16] Malgrange, Bernard Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. (4), Tome 7 (1974), p. 405-430 (1975) | MR 150789 | Zbl 0305.32008 | Zbl 0156.30603

[17] Mather, John N. Stability of C mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. (1968) no. 35, pp. 279-308 | MR 1621831 | Zbl 0159.25001 | Zbl 0902.14005

[18] Michel, F.; Weber, C. Sur le rôle de la monodromie entière dans la topologie des singularités, Ann. Inst. Fourier (Grenoble), Tome 36 (1986) no. 1, pp. 183-218 | MR 399088 | Zbl 0557.57017 | Zbl 0351.32009

[19] Milnor, John Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., Annals of Mathematics Studies, No. 61 (1968) | Numdam | MR 372243 | Zbl 0184.48405 | Zbl 0305.32008

[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 0249.57029 | Zbl 0159.25001

[21] Saito, Morihiko On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble), Tome 39 (1989) no. 1, pp. 27-72 | Numdam | MR 840719 | Zbl 0644.32005 | Zbl 0557.57017

[22] Saito, Morihiko Period mapping via Brieskorn modules, Bull. Soc. Math. France, Tome 119 (1991) no. 2, pp. 141-171 | MR 239612 | Zbl 0760.32009 | Zbl 0184.48405

[23] Scherk, J.; Steenbrink, J. H. M. On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann., Tome 271 (1985) no. 4, pp. 641-665 | Article | MR 430307 | Zbl 0618.14002

[24] Slodowy, Peter Einige Bemerkungen zur Entfaltung symmetrischer Funktionen, Math. Z., Tome 158 (1978) no. 2, pp. 157-170 | Numdam | MR 1011977 | Zbl 0352.58009 | Zbl 0644.32005

[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 1116843 | Zbl 0301.32013 | Zbl 0760.32009

[26] Tráng, Lê Dũng; Ramanujam, C. P. The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math., Tome 98 (1976) no. 1, pp. 67-78 | MR 790119 | Zbl 0351.32009 | Zbl 0618.14002

[27] Varchenko, A. N. The asymptotics of holomorphic forms determine a mixed Hodge structure, Sov. Math. Dokl., Tome 22 (1980), pp. 772-775 | MR 474379 | Zbl 0516.14007 | Zbl 0352.58009

[28] Wall, C. T. C. A note on symmetry of singularities, Bull. London Math. Soc., Tome 12 (1980) no. 3, pp. 169-175 | Article | MR 485870 | Zbl 0427.32010 | Zbl 0373.14007

[29] Wall, C. T. C. A second note on symmetry of singularities, Bull. London Math. Soc., Tome 12 (1980) no. 5, pp. 347-354 | Article | MR 414931 | Zbl 0424.58006 | Zbl 0301.32013

[30] Yu, Jian Ming Kombinatorische Geometrie der Stokesregionen, Universität Bonn Mathematisches Institut, Bonn, Bonner Mathematische Schriften [Bonn Mathematical Publications], 212 (1990) (Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1990) | MR 1072752 | Zbl 0516.14007

[31] Yu, Jian Ming Galois group of Lyashko-Looijenga mapping, Math. Z., Tome 232 (1999), pp. 321-330 | MR 1718705 | MR 572095 | Zbl 0945.58028 | Zbl 0427.32010