×

A Saito criterion for holonomic divisors. (English) Zbl 1457.32085

In the classic framework of K. Saito [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265–291 (1980; Zbl 0496.32007)], the authors consider a reduced hypersurface singularity germ in a complex manifold, with its module of logarithmic derivations.
The main result shows that a holonomic divisor is free if and only if applying all logarithmic derivations to a generic function with isolated critical point yields a complete intersection Artin algebra.
Compare with the results in [T. Abe et al., J. Math. Soc. Japan 71, No. 4, 1027–1047 (2019; Zbl 1437.32004)].

MSC:

32S65 Singularities of holomorphic vector fields and foliations
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13N15 Derivations and commutative rings

Software:

SINGULAR
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abe, T., Horiguchi, T., Masuda, M., Murai, S., Sato, T.: Hessenberg varieties and hyperplane arrangements, arXiv:1611.00269, (2016) · Zbl 1510.14033
[2] Abe, T., Maeno, T., Murai, S., Numata, Y.: Solomon—Terao algebra of hyperplane arrangements, arXiv:1802.04056, (2018) · Zbl 1437.32004
[3] Bruns, W., Herzog, J.: Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993) · Zbl 0788.13005
[4] Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-1-1—a computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2018)
[5] Flenner, H.: Die Sätze von Bertini für lokale Ringe. Math. Ann. 229(2), 97-111 (1977) · Zbl 0398.13013 · doi:10.1007/BF01351596
[6] Grauert, H., Remmert, R.: Analytische Stellenalgebren, Springer-Verlag, Berlin, Unter Mitarbeit von O. Riemenschneider, Die Grundlehren der mathematischen Wissenschaften, Band 176 (1971) · Zbl 0231.32001
[7] Grothendieck, A.: Techniques de construction en géométrie analytique.VI. Étude locale des morphismes: germes d’espaces analytiques, platitude, morphismes simples, Séminaire Henri Cartan, 13ième année, 1960/61 (Henri Cartan, ed.), Secrétariat mathématique, Paris, pp. 1-13 (1962)
[8] Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967) · Zbl 0153.22301
[9] Houzel, C.: Géométrie analytique locale, II. Théorie des morphismes finis, Séminaire Henri Cartan, 13ième année, 1960/61 (Henri Cartan, ed.), Secrétariat mathématique, Paris, pp. 1-22 (1962)
[10] Houzel, C.: Géométrie analytique locale, III. Séminaire Henri Cartan, 13ième année, 1960/61 (Henri Cartan, ed.), Secrétariat mathématique, Paris, pp. 1-25 (1962)
[11] Kunz, E.: Kähler Differentials, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1986) · Zbl 0587.13014 · doi:10.1007/978-3-663-14074-0
[12] Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2), 265-291 (1980) · Zbl 0496.32007
[13] Scheja, G.: Differentialmoduln lokaler analytischer Algebren, Schriftenreihe Math. Inst. Univ. Fribourg, no. 2, Univ. Fribourg, Switzerland, (1969/70) · Zbl 0199.36003
[14] Scheja, G., Storch, U.: Differentielle Eigenschaften der Lokalisierungen analytischer Algebren. Math. Ann. 197, 137-170 (1972) · Zbl 0223.14002 · doi:10.1007/BF01419591
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.