# zbMATH — the first resource for mathematics

Deformations of free and linear free divisors. (Déformations de diviseurs libres et linéaires libres.) (English. French summary) Zbl 1301.14004
A reduced divisor $$D=V(f)\subseteq \mathbb C^n$$ is called free if the sheaf $$\text{Der}(-\log D)$$ of logarithmic vector fields is a locally free $$\mathcal{O}_{\mathbb C^n}$$-module. It is called linear if, furthermore, $$\text{Der}(-\log D)$$ is globally generated by a basis consisting of vector fields whose all of coefficients, with respect to the standard basis $$\partial /\partial x_1, \cdots, \partial/\partial x_n$$ of the space $$\text{Der}_{\mathbb C^n}$$, are linear functions. Free divisors were introduced by K. Saito [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265–291 (1980; Zbl 0496.32007)] and linear free divisors were introduced by R.-O. Buchweitz and D. Mond [Lond. Math. Soc. Lect. Note Ser. 324, 41–77 (2006; Zbl 1101.14013)].
In the paper under review, the author describes the spaces of infinitesimal deformations and obstructions of a germ of a (linear) free divisor and performed calculations for some concrete examples. The main results of this paper can be summarized as follows:
Theorem A. (see Corollary 3.45). Let $$(D,0) \subseteq (\mathbb C^n,0)$$ be a germ of a reductive linear free divisor $$D$$. Then it is formally rigid.
This is equivalent to say that for a germ of a reductive linear free divisor, there are no non-trivial families, at least on the level of formal power series.
Theorem B. (see Corollary 3.50) Let $$(D,0) \subseteq (\mathbb C^n,0)$$ be a germ of a free divisor generated by a weighted homogeneous polynomial. Then the functor $$\mathbf{FD}_D$$ has a hull, i.e. $$(D,0)$$ has a formally versal deformation.
Here the functor $$\mathbf{FD}_D$$ associates each local Artin ring $$A$$ to the set of isomorphism classes of admissible deformations of $$(D,0)$$ over $$\text{Spec} A$$.
Theorem C. (see Corollary 4.24) Let $$(D,0) \subseteq (\mathbb C^n,0)$$ be a germ of a Koszul free divisor. If we can put a logarithmic connection on $$\text{Der}_{\mathbb C^n}$$ and $$\text{Der}(-\log D)$$, then $$\mathbf{FD}_D$$ has a hull.

##### MSC:
 14B07 Deformations of singularities 13D10 Deformations and infinitesimal methods in commutative ring theory 14F40 de Rham cohomology and algebraic geometry
Full Text:
##### References:
 [1] Abad, Camilo Arias; Crainic, Marius, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math., 663, 91-126, (2012) · Zbl 1238.58010 [2] Buchweitz, R. O.; Mond, D., Linear free divisors and quiver representations, Singularities and computer algebra, 324, 41 pp., (2006) · Zbl 1101.14013 [3] Calderón Moreno, F. J., Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor, Ann. Sci. École Norm. Sup. 4e série, 32, 701-714, (1999) · Zbl 0955.14013 [4] Calderón Moreno, F. J.; Narváez Macarro, L., Locally quasi-homogeneous free divisors are Koszul free, Proceedings of the Steklov Institute of Mathematics-Interperiodica Translation, 238, 72-76, (2002) · Zbl 1031.32006 [5] Calderón Moreno, F. J.; Narváez Macarro, L., Dualité et comparaison sur LES complexes de de Rham logarithmiques par rapport aux diviseurs libres, Ann. Inst. Fourier (Grenoble), 55, 1, 47-75, (2005) · Zbl 1089.32003 [6] De Gregorio, I.; Mond, D.; Sevenheck, C., Linear free divisors and Frobenius manifolds, Compos. Math, 145, 1305-1350, (2009) · Zbl 1238.32022 [7] Dixmier, Jacques, Enveloping algebras, xvi+375 pp., (1977), North-Holland Publishing Co., Amsterdam · Zbl 0346.17010 [8] Granger, M.; Mond, D.; Nieto-Reyes, A.; Schulze, M., Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble), 59, 2, 811-850, (2009) · Zbl 1163.32014 [9] Greuel, G.-M.; Lossen, C.; Shustin, E., Introduction to singularities and deformations, xii+471 pp., (2007), Springer, Berlin · Zbl 1125.32013 [10] Hartshorne, Robin, Deformation theory, 257, viii+234 pp., (2010), Springer, New York · Zbl 1186.14004 [11] Hochschild, G.; Serre, J.-P., Cohomology of Lie algebras, Annals of Mathematics, 57, 3, 591-603, (1953) · Zbl 0053.01402 [12] Orlik, P.; Terao, H., Arrangements of hyperplanes, 300, xviii+325 pp., (1992), Springer-Verlag, Berlin · Zbl 0757.55001 [13] Rinehart, G. S., Differential forms on general commutative algebras, Trans. Amer. Math. Soc., 108, 195-222, (1963) · Zbl 0113.26204 [14] 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 [15] Schlessinger, M., Functors of Artin rings, Trans. Amer. Math. Soc., 130, 208-222, (1968) · Zbl 0167.49503 [16] Sevenheck, C., Lagrange-singularitäten und ihre deformationen, (1999), Heinrich-Heine Universität, Düsseldorf [17] Sevenheck, C., Lagrangian singularities, x+190 pp., (2003), Cuvillier Verlag, Göttingen · Zbl 1059.14006 [18] Sevenheck, C.; van Straten, D., Deformation of singular Lagrangian subvarieties, Math. Ann., 327, 1, 79-102, (2003) · Zbl 1051.14006 [19] Torielli, M., Free divisors and their deformations, (2012) [20] Weibel, C. A., An introduction to homological algebra, 38, xiv+450 pp., (1994), Cambridge University Press, Cambridge · Zbl 0797.18001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.