no. 6
A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface
[La réciproque d’un théorème sur les formes normales des formes volumes par rapport à une hypersurface]
Kourliouros, Konstantinos1
1 Imperial College London Dept. of mathematics Huxley Building 180 Queen’s Gate London, SW7 (UK)
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2437-2447.

Nous donnons ici une réponse positive à une question posée par Y. Colin de Verdière concernant la réciproque du théorème suivant, dû à A. N. Varchenko : deux germes de formes volumes sont équivalents modulo difféomorphismes préservant un germe d’hypersurface à singularités isolées, si leur différence est la différentielle d’une forme dont la restriction sur la partie lisse de l’hypersurface est exacte.

We give here a positive answer to a question asked by Y. Colin de Verdière concerning the converse of the following theorem, due to A. N. Varchenko: two germs of volume forms are equivalent with respect to diffeomorphisms preserving a germ of an isolated hypersurface singularity, if their difference is the differential of a form whose restriction on the smooth part of the hypersurface is exact.

DOI : 10.5802/aif.2992
Classification : 10X99, 14A12, 11L05
Keywords: Isolated Singularities, De Rham Cohomology, Volume Forms, Normal Forms
Mot clés : Singularités Isolées, Cohomologie de de Rham, Formes Volumes, Formes Normales

Kourliouros, Konstantinos 1

1 Imperial College London Dept. of mathematics Huxley Building 180 Queen’s Gate London, SW7 (UK) 
@article{AIF_2015__65_6_2437_0,
     author = {Kourliouros, Konstantinos},
     title = {A {Converse} to a {Theorem} on {Normal} {Forms} of {Volume} {Forms} with {Respect} to a {Hypersurface}},
     journal = {Annales de l'Institut Fourier},
     pages = {2437--2447},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2992},
     zbl = {1336.32029},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2992/}
}
TY  - JOUR
AU  - Kourliouros, Konstantinos
TI  - A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 2437
EP  - 2447
VL  - 65
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2992/
DO  - 10.5802/aif.2992
LA  - en
ID  - AIF_2015__65_6_2437_0
ER  - 
%0 Journal Article
%A Kourliouros, Konstantinos
%T A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface
%J Annales de l'Institut Fourier
%D 2015
%P 2437-2447
%V 65
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2992/
%R 10.5802/aif.2992
%G en
%F AIF_2015__65_6_2437_0
Kourliouros, Konstantinos. A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2437-2447. doi : 10.5802/aif.2992. https://aif.centre-mersenne.org/articles/10.5802/aif.2992/

[1] Brieskorn, Egbert Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., Volume 2 (1970), pp. 103-161 | MR | Zbl

[2] Colin De Verdière, Yves Singular Lagrangian manifolds and semiclassical analysis, Duke Math. J., Volume 116 (2003) no. 2, pp. 263-298 | DOI | MR | Zbl

[3] Ferrari, Aldo Cohomology and holomorphic differential forms on complex analytic spaces, Ann. Scuola Norm. Sup. Pisa (3), Volume 24 (1970), pp. 65-77 | Numdam | MR | Zbl

[4] Françoise, J.-P. Relative cohomology and volume forms, Singularities (Warsaw, 1985) (Banach Center Publ.), Volume 20, PWN, Warsaw, 1988, pp. 207-222 | MR | Zbl

[5] Giventalʼ, A. B. Singular Lagrangian manifolds and their Lagrangian mappings, Current problems in mathematics. Newest results, Vol. 33 (Russian) (Itogi Nauki i Tekhniki), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, p. 55-112, 236 Translated in J. Soviet Math. 52 (1990), no. 4, 3246–3278 | MR | Zbl

[6] Greuel, G.-M. Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann., Volume 214 (1975), pp. 235-266 | MR | Zbl

[7] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, 151, Cambridge University Press, Cambridge, 2002, pp. x+270 | DOI | MR | Zbl

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

[9] Saito, Kyoji Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 14 (1971), pp. 123-142 | MR | Zbl

[10] Sebastiani, Marcos Preuve d’une conjecture de Brieskorn, Manuscripta Math., Volume 2 (1970), pp. 301-308 | MR | Zbl

[11] Sevenheck, C. Singularités Lagrangiennes, École Polytechnique (France) (2003) (Ph. D. Thesis)

[12] Sternberg, Shlomo Infinite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., Volume 10 (1961), pp. 451-474 | MR | Zbl

[13] Varchenko, A. N. Local classification of volume forms in the presence of a hypersurface, Funktsional. Anal. i Prilozhen., Volume 19 (1985) no. 4, p. 23-31, 95 | MR | Zbl

Cité par Sources :