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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.

14B07 Deformations of singularities
13D10 Deformations and infinitesimal methods in commutative ring theory
14F40 de Rham cohomology and algebraic geometry
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[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
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