The trivial locus of an analytic map germ
Annales de l'Institut Fourier, Volume 39 (1989) no. 4, p. 831-844

We prove: For a local analytic family {X s } sS of analytic space germs there is a largest subspace T in S such that the family is trivial over T. Moreover the reduction of T equals the germ of those points s in S for which X s is isomorphic to the special fibre X 0 .

On prouve : Toute famille locale analytique {X s } sS de germes d’espaces analytiques admet un plus grand sous-espace T de S au-dessus duquel elle soit triviale. En plus, la réduction de T est égale au germe des points s de S tels que X, soit isomorphe à la fibre spéciale X 0 .

@article{AIF_1989__39_4_831_0,
     author = {Hauser, H. and Muller, G.},
     title = {The trivial locus of an analytic map germ},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {39},
     number = {4},
     year = {1989},
     pages = {831-844},
     doi = {10.5802/aif.1191},
     zbl = {0678.32013},
     mrnumber = {91m:32035},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_4_831_0}
}
Hauser, H.; Muller, G. The trivial locus of an analytic map germ. Annales de l'Institut Fourier, Volume 39 (1989) no. 4, pp. 831-844. doi : 10.5802/aif.1191. http://www.numdam.org/item/AIF_1989__39_4_831_0/

[A] M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. IHES, 36 (1969), 23-58. | Numdam | MR 42 #3087 | Zbl 0181.48802

[D] I. F. Donin, Complete families of deformations of germs of complex spaces, Math. USSR-Sbornik, 18 (1972), 397-406. | MR 48 #11574 | Zbl 0275.32011

[E] R. Ephraim, Isosingular loci and the cartesian product structure of complex analytic singularities, Trans. Am. Math. Soc., 241 (1978), 357-371. | MR 80i:32027 | Zbl 0395.32006

[Fi] G. Fischer, Complex analytic geometry, Springer Lect. Notes, 538, 1976. | MR 55 #3291 | Zbl 0343.32002

[FiG] W. Fischer, H. Grauert, Lokal-triviale Familien kompakter komplexer Mannig-faltigkeiten, Nachr. Akad. Wiss. Göttingen, Math.-Phys. K1. II, 6 (1965), 89-94. | MR 32 #1731 | Zbl 0135.12601

[FIK] H. Flenner, S. Kosarew, On locally trivial deformations, Publ. Res. Inst. Math. Sci., 23 (1987), 627-665. | MR 89c:32055 | Zbl 0636.32010

[GaH] T. Gaffney, H. Hauser, Characterizing singularities of varieties and of mappings, Invent. Math., 81 (1985), 427-447. | MR 87m:32019 | Zbl 0627.14004

[GrK] G.-M. Greuel, U. Karras, Families cf varieties with prescribed singularities, Compos. Math., 69 (1989), 83-110. | Numdam | MR 90d:32037 | Zbl 0684.32015

[H] J. E. Humphreys, Linear algebraic groups, Springer, 1975. | MR 53 #633 | Zbl 0325.20039

[M] J. N. Mather, Stability of C∞-mappings, III : Finitely determined map germs, Publ. Math. IHES, 35 (1968), 127-156. | Numdam | MR 43 #1215a | Zbl 0159.25001

[PfPo] G. Pfister, D. Popescu, Die strenge Approximationseigenschaft lokaler Ringe, Invent. Math., 30 (1975), 145-174. | MR 52 #395 | Zbl 0293.13011

[Se] A. Seidenberg, Analytic products, Am. J. Math., 91 (1969), 577-590. | MR 40 #7261 | Zbl 0185.49304

[Sc] H. W. Schuster, Sur Theorie der Deformationen kompakter komplexer Räume, Invent. Math., 9 (1970), 284-294. | MR 42 #3818 | Zbl 0192.44201

[T] B. Teissier, The hunting of invariants in the geometry of discriminants, In : Real and complex singularities, Oslo 1976, 565-677. Holm, P., (ed.), Sijthoff and Noordhoff, 1977. | Zbl 0388.32010

[V] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice Hall, 1974. | MR 51 #13113 | Zbl 0371.22001

[W] J. J. Wavrik, A theorem on solutions of analytic equations with applications to deformations of complex structures, Math. Ann., 216 (1975), 127-142. | MR 52 #8488 | Zbl 0303.32018