Michel Granger

Free divisors in prehomogeneous vector spaces

We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric- and representation-theoretic irreducible components coincide. As a consequence, we find that a quiver can only give rise to a linear free divisor if it has no (oriented or unoriented) cycles. We also deduce that the linear free divisors which appear in Sato and Kimuras list of irreducible prehomogeneous vector spaces are the only irreducible reductive linear free divisors. Furthermore, we show that all quiver linear free divisors are strongly Euler homogeneous, that they are locally weakly quasihomogeneous at points whose corresponding representation is not regular, and that all tame quiver linear free divisors are locally weakly quasihomogeneous. In particular, the latter satisfy the logarithmic comparison theorem.

The analytic standard fan of a D-module

Abstract In this paper, we associate with any monogeneous module over the ring D of germs of linear differential operators at the origin of C n , with holomorphic coefficients, a combinatorial object which we call the standard fan of this D -module (see Section 6 for a precise geometric description of this object). The main tool of the proof is the homogenization technique and a convergent division theorem in the homogenization ring D [t]. This last result is the key tool to an extension to the analytic D -module case of our results in the algebraic case of the Weyl algebra (see Assi et al., J. Pure Appl. Algebra, 150 (1) (2000) 27–39.

The Gröbner fan of an An-module

Abstract Let I be a non-zero left ideal of the Weyl algebra A n of order n over a field k and let L: R 2n → R be a linear form defined by L(α,β)=∑ i=1 n e i α i +∑ i=1 n f i β i . If e i +f i ≥0 , then L defines a filtration F • L on A n . Let gr L (I) be the graded ideal associated with the filtration induced by F • L on I . Let finally U denote the set of all linear form L for which e i +f i ≥0 for all 1≤i≤n . The aim of this paper is to study, by using the theory of Grobner bases, the stability of gr L (I) when L varies in U . In a previous paper, we obtained finiteness results for some particular linear forms (used in order to study the regularity of a D -module along a smooth hypersurface). Here we generalize these results by adapting the theory of Grobner fan of Mora-Robbiano to the D -module case. Our main tool is the homogenization technique initiated in our previous paper, and recently clarified in a work by F. Castro-Jimenez and L. Narvaez-Macarro.

On the formal structure of logarithmic vector fields

In this article, we prove that a free divisor in a three dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. F.J. Calderon-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem which describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.

On the Symmetry of b-Functions of Linear Free Divisors

We introduce the concept of a prehomogeneous determinant as a possibly nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the b-function are symmetric about 1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under which this symmetry persists. Combined with Kashiwara’s theorem on the roots of b-functions, our symmetry result shows that 1 is the only integer root of the b-function. This gives a positive answer to a problem posed by Castro-Jim enez and Ucha-Enr quez in the above cases. We study the condition of strong Euler homogeneity in terms of the action of the stabilizers on the normal spaces. As an application of our results, we show that the logarithmic comparison theorem holds for reductive linear Koszul free divisors exactly when they are strongly Euler homogeneous.

Tangent cone algorithm for homogenized differential operators

We extend Moras tangent cone or the ecart division algorithm to a homogenized ring of differential operators. This allows us to compute standard bases of modules over the ring of analytic differential operators with respect to sufficiently general orderings which are needed in the D-module theory.

Normal crossing properties of complex hypersurfaces via logarithmic residues

We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of L\^e and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, describe all free divisors with Gorenstein singular locus.

Quasihomogeneity of isolated hypersurface singularities and logarithmic cohomology

We characterize quasihomogeneity of isolated hypersurface singularities by the injectivity of the map induced by the first differential of the logarithmic differential complex in the top local cohomology supported in the singular point.

Bernstein-Sato polynomials and functional equations

These notes are an expanded version of the lectures given in the frame of the I.C.T.P. School held at Alexandria in Egypt from 12 to 24 November 2007. Our purpose in this course was to give a survey of the various aspects, algebraic, analytic and formal, of the functional equations which are satisfied by the powers fs of a function f and involve a polynomial in one variable bf (s) called the Bernstein-Sato polynomial of f . Since this course is intended to be useful for newcomers to the subject we give enough significant details and examples in the most basic sections, which are sections 1, 2, and also 4. The latter is devoted to the calculation of the Bernstein-Sato polynomial for the basic example of quasi-homogeneous polynomials with isolated singularities. This case undoubtedly served as a guide in the first developments of the theory. We particularly focused our attention on the problem of the meromorphic continuation of the distribution fs + in the real case, which in turn motivated the problem of the existence of these polynomials, without forgetting related questions like the Mellin transform and the division of distributions. See the content of section 3. The question of the analytic continuation property was brought up as early as 1954 at the congress of Amsterdam by I.M. Gelfand. The meromorphic continuation was proved 15 years later independently by Atiyah and I.N. Bernstein-S.I. Gel’fand who used the resolution of singularities. The existence of the functional equations proved by I.N. Bernstein in the polynomial case allowed him to give a simpler proof which does not use the resolution of singularities. His proof establishes at the same time a relationship between the poles of the continuation and the zeros of the Bernstein Sato polynomial. The already known rationality of the poles gave a strong reason for conjecturing the famous result about the rationality of the zeros of the b-function which was proved by Kashiwara and Malgrange. We also want to mention another source of interest for studying functional equations due to Mikio Sato. It concerns the case of the semi-invariants of prehomogeneous actions of an algebraic group, and especially of a reductive group. In the latter case the functional equation is of a very particular type and the name b-function frequently employed as a shortcut for the Bernstein-Sato Polynomial, comes from this theory. We give the central step of the proof of the existence theorem in the reductive case in section 2.4. Let us summarize the contents of the different sections. In section 1, we give the basic definitions and elementary facts about b-functions with emphasis on first hand examples. In section 2 we recall the proof of Bernstein for the existence theorem in the polynomial case. Although also treated by F. Castro in this volume we give it for the sake of completeness and also to make it clear that the case of multivariable Bernstein-Sato polynomials can be solved with the same proof in the algebraic case. In section 3 we give a detailed proof of the analytic continuation property using the functional equation and we make a

Quasihomogeneity of Curves and the Jacobian Endomorphism Ring

We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos’ normalization algorithm. In the process, we give a simplified proof of the Kunz–Ruppert criterion.