Giraldo Suárez, Luis and Gómez-Mont, X. and Mardešic, P.
(2008)
*Flags in zero dimensional complete intersection algebras
and indices of real vector fields.*
Mathematische Zeitschrift, 260
(1).
pp. 77-91.
ISSN 0025-5874

PDF
Restricted to Repository staff only until 2020. 247kB |

Official URL: http://www.springerlink.com/content/9777668l3584u182/fulltext.pdf

## Abstract

We introduce bilinear forms in a flag in a complete intersection local R-algebra of dimension 0, related to the Eisenbud-Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen's filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable with finite dimensional commutative algebra methods.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Singularities of functions · Local algebra · Bilinear form · Index of vector field |

Subjects: | Sciences > Mathematics > Differential geometry |

ID Code: | 16442 |

References: | Arnold, V.: Index of a singular point of a vector field, the Petrovskii–Oleinik inequality and mixed Hodge structures. Funct. Anal. Appl. 12(1), 1–14 (1978) Bonatti, Ch., Gómez-Mont, X.: The index of a holomorphic vector field on a singular variety, I. Astérisque 222, 9–35 (1994) Eisenbud, D., Levine, H.: An algebraic formula for the degree of a C∞ map germ. Ann. Math. 106, 19–38 (1977) Eisenbud, D.: Commutative Algebra with a View Towards Algebraic Geometry. Springer, Heidelberg (1995) Giraldo, L., Gómez-Mont, X., Mardešic, P.: Computation of topological numbers via linear algebra: Hypersurfaces, vector fields and vector fields on hypersurfaces. Contemp. Math. 240, 175–182 (1999) Gómez-Mont, X., Mardešic, P.: The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant. Ann. Inst. Fourier 47(5), 1523–1539 (1997) Gómez-Mont, X., Mardešic, P.: The index of a vector field tangent to an odd dimensional hypersurface and the signature of the relative Hessian. Funct. Anal. Appl. 33(1), 1–10 (1999) Gómez-Mont, X., Seade, J., Verjovsky, A.: The index of a holomorphic flow with an isolated singularity. Math. Ann. 291, 737–751 (1991) Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0. A Computer Algebra System for Polynomial Computations. Center for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de Gunning, R.C.: Introduction to Holomorphic Functions of Several Complex Variables, vol. II. Local theory. Wadsworth & Brooks/Cole Mathematics series, Wadsworth & Brooks/Cole Pacific Grove, CA (1990) Jantzen, J.C.: Moduln mit einem Höchsten Gewicht. Lecture Notes in Mathematics, vol. 750. Springer, Berlin (1979) Khimshiashvili, G.N.: On the local degree of a smooth map. Trudi Tbilisi math. Inst, pp. 105–124 (1980) Montaldi, J., van Straten, D.: One-forms on singular curves and the topology of real curves singularities. Topology 294, 501–510 (1990) Vogan, D.: Unitarizability of certain series of representations. Ann. Math. (2) 120(1), 141–187 (1984)123 |

Deposited On: | 20 Sep 2012 09:15 |

Last Modified: | 07 Feb 2014 09:29 |

Repository Staff Only: item control page