On the Thom–Sebastiani Property of Quasi-Homogeneous Isolated Hypersurface Singularities

Epure, Raul

Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 539-547.

Let $(V,0)\subset ({ℂ}^n,0)$ be a quasi-homogeneous isolated hypersurface singularity. In this paper we prove under certain weight conditions a relation between the property of $(V,0)$ being of Thom–Sebastiani type and the dimension of toral Lie subalgebras contained in the Yau algebra $L\left(V\right).$

Classification : 32S25

[1] Chen, Bingyi; Hussain, Naveed; Yau, Stephen S.-T.; Zuo, Huaiqing Variation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularities, J. Differ. Geom., Volume 115 (2020) no. 3, pp. 437-473 | DOI | MR | Zbl

[2] Decker, Wolfram; Greuel, Gert-Martin; Pfister, Gerhard; Schönemann, Hans Singular 4-2-1 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2021

[3] Epure, Raul-Paul Explicit and effective Mather–Yau correspondence in view of analytic gradings, doctoralthesis, Technische Universität Kaiserslautern (2020), II, 170, V pages http://nbn-resolving.de/urn:nbn:de:hbz:386-kluedo-61500

[4] Epure, Raul-Paul; Schulze, Mathias Hypersurface singularities with monomial Jacobian ideal, Bull. Lond. Math. Soc. (2022) (https://doi.org/10.1112/blms.12614) | DOI

[5] Hauser, Herwig; Müller, Gerd On the Lie algebra $\Theta$ (X) of vector fields on a singularity, J. Math. Sci., Tokyo, Volume 1 (1994), pp. 239-250 | MR | Zbl

[6] Humphreys, James E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer, 1975, xiv+247 pages | MR

[7] Hussain, Naveed; Yau, Stephen S.-T.; Zuo, Huaiqing On the new $k$-th Yau algebras of isolated hypersurface singularities, Math. Z., Volume 294 (2020), pp. 1-28 | DOI | MR | Zbl

[8] Hussain, Naveed; Yau, Stephen S.-T.; Zuo, Huaiqing $k$-th Yau number of isolated hypersurface singularities and An Inequality Conjecture, J. Aust. Math. Soc., Volume 110 (2021), 1650, 15 pages | DOI | MR

[9] Hussain, Naveed; Yau, Stephen S.-T.; Zuo, Huaiqing On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities, Mathematics, Volume 9 (2021) no. 14 | DOI

[10] de Jong, Theo; Pfister, Gerhard Local analytic geometry. Basic theory and applications, Advanced Lectures in Mathematics (ALM), Vieweg & Sohn, 2000, xii+382 pages | DOI | MR | Zbl

[11] Kreuzer, Martin; Robbiano, Lorenzo Computational Commutative Algebra. II, Springer, 2005 | Zbl

[12] Mather, John N.; Yau, Stephen S.-T. Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math., Volume 69 (1982) no. 2, pp. 243-251 | DOI | MR | Zbl

[13] Roozemond, Dan Algorithms for Lie algebras of algebraic groups, Ph. D. Thesis, Technische Universiteit Eindhoven (2010)

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

[15] Scheja, Günter; Wiebe, Hartmut Über Derivationen von lokalen analytischen Algebren, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), Academic Press Inc., 1973, pp. 161-192 | MR | Zbl

[16] Xu, Yi-Jing; Yau, Stephen S.-T. Micro-local characterization of quasi-homogeneous singularities, Am. J. Math., Volume 118 (1996) no. 2, pp. 389-399 | MR | Zbl

[17] Yau, Stephen S.-T. A necessary and sufficient condition for a local commutative algebra to be a moduli algebra: weighted homogeneous case, Complex analytic singularities (Advanced Studies in Pure Mathematics), Volume 8, North-Holland, 1987, pp. 687-697 | DOI | MR | Zbl

[18] Yau, Stephen S.-T.; Zuo, Huaiqing Derivations of the moduli algebras of weighted homogeneous hypersurface singularities, J. Algebra, Volume 457 (2016), pp. 18-25 | DOI | MR | Zbl

[19] Yau, Stephen S.-T.; Zuo, Huaiqing Sharp upper estimate conjecture for the Yau number of a weighted homogeneous isolated hypersurface singularity, Pure Appl. Math. Q., Volume 12 (2016), pp. 165-181 | DOI | MR | Zbl