On simple \(\mathbb{Z}_2\)-invariant and corner function germs. (English. Russian original) Zbl 1446.32021
Math. Notes 107, No. 6, 939-945 (2020); translation from Mat. Zametki 107, No. 6, 855-864 (2020).
Let the cyclic group \(\mathbb Z_2=\langle \sigma \rangle\) of order \(2\) act on \((\mathbb C^{m+n},0)\) via \(\sigma(x_1,\ldots,x_m,y_1,\ldots,y_n)=(-x_1,\ldots,-x_m,y_1,\ldots,y_n)\). Let \(\mathbb Z_2^m=\langle \sigma_1,\ldots,\sigma_m\rangle\) act on \((\mathbb C^{m+n},0)\) via \[\sigma_i(x_1,\ldots,x_m,y_1,\ldots,y_n)=(x_1,\ldots,x_{i-1},-x_i,x_{i+1},\ldots,x_m,y_1,\ldots,y_n).\]
Equivariantly simple function germs with respect to these actions (corner singularities) are classified up to stable equivalence. In both cases the property simple is characterized in terms of the intersection form respectively in terms of the monodromy.
Equivariantly simple function germs with respect to these actions (corner singularities) are classified up to stable equivalence. In both cases the property simple is characterized in terms of the intersection form respectively in terms of the monodromy.
Reviewer: Gerhard Pfister (Kaiserslautern)
MSC:
| 32S05 | Local complex singularities |
References:
| [1] | Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M., Singularities of Differentiable Maps, Vol. I: The Classification of Critical Points, Caustics and Wave Fronts (1982), Moscow: Nauka, Moscow · Zbl 1290.58001 |
| [2] | Arnold, V. I., Normal forms for functions near degenerate critical points, the Weyl groups of A_k, D_k, E_k and Lagrangian singularities, Funktsional. Anal. Prilozhen., 6, 4, 3-25 (1972) |
| [3] | Arnold, V. I., Critical points of functions on a manifold with boundary, the simple lie groups B_k, C_k, F_4 and singularities of evolutes, Uspekhi Mat. Nauk, 33, 5203, 91-105 (1978) · Zbl 0408.58009 |
| [4] | Slodowy, P., Einige Bemerkungen zur Entfaltung symmetrischer Funktionen, Math. Z., 158, 2, 157-170 (1978) · Zbl 0352.58009 · doi:10.1007/BF01320865 |
| [5] | Siersma, D., Singularities of functions on boundaries, corners, etc., Quart. J. Math. Oxford Ser., 32, 125, 119-127 (1981) · Zbl 0417.58002 · doi:10.1093/qmath/32.1.119 |
| [6] | Gabrielov, A. M., Intersection matrices for certain singularities, Funktsional. Anal. Prilozhen., 7, 3, 18-32 (1973) · Zbl 0288.32011 |
| [7] | Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M., Singularities of Differentiable Maps, Vol. II: Monodromy and Asymptotics of Integrals (1984), Moscow: Nauka, Moscow · Zbl 0545.58001 |
| [8] | Gusein-Zade, S. M., Intersection matrices for certain singularities of functions of two variables, Funktsional. Anal. Prilozhen., 8, 1, 11-15 (1974) · Zbl 0304.14009 |
| [9] | Wall, C. T C., A note on symmetry of singularities, Bull. London Math. Soc., 12, 3, 169-175 (1980) · Zbl 0427.32010 · doi:10.1112/blms/12.3.169 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
