Free Divisors Research Articles

Linear free divisors and Frobenius manifolds

AbstractWe study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauß–Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.

Oscillations of a given size of some arithmetic error terms

A general method of estimating the number of oscillations of a given size of arithmetic error terms is developed. Special attention is paid to the remainder terms in the prime number formula, in the Dirichlet prime number theorem for primes in arithmetic progressions and to the remainder term in the asymptotic formula for the number of square free divisors of an integer.

Linear free divisors and the global logarithmic comparison theorem

A complex hypersurface D in C-n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. By analogy with Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of C-n D. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive. For n at most 4, we show that the GLCT holds for all LFDs. We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT. As a by-product we obtain a topological proof of a theorem of V. Kac on the number of irreducible components of such discriminants.

On the logarithmic comparison theorem for integrable logarithmic connections

Let X be a complex analytic manifold, D ⊂ X a free divisor with jacobian ideal of linear type (for example, a locally quasi-homogeneous free divisor), j: U = X − D ↪ X the corresponding open inclusion, ε an integrable logarithmic connection with respect to D and ℒ the local system of the horizontal sections of ε on U. In this paper we prove that the canonical morphisms Ω X • ( log D ) ( ε ( k D ) ) → R j * L , j ! L → Ω X • ( log D ) ( ε ( − k D ) ) are isomorphisms in the derived category of sheaves of complex vector spaces for k ≫ 0 (locally on X).

Low-dimensional singularities with free divisors as discriminants

We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C 5 \mathbb {C}^5 . It is shown that such free divisors often admit a “fast normalization”, obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C 5 \mathbb {C}^5 , namely the simple elliptic singularity of type A ~ 4 \widetilde A_{4} , we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j j -invariant is the cone over the dual variety of an elliptic curve.

On the k-free divisor problem (II)

Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.$

A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give a sufficient criterion for the property, a simple proof of F.J. Calderón-Moreno’s theorem that free divisors have the property, a proof that divisors in dimension $3$ with only isolated quasi-homogeneous singularities have the property, an example of a nonfree divisor with nonisolated singularity having the property, an example of a divisor not having the property, and an algorithm to compute the V-filtration along a divisor up to a given order.

Singularities

The Oberwolfach workshop Singularity Theory was held in the week September 10th–September 15th, 2006. It was organised by J. Steenbrink (Nijmegen), D. van Straten (Mainz) and V. Vassiliev (Moskow) and continued the sequence of workshops Singularitäten that take place regularly at Oberwolfach. The morning sessions consisted of three talks, which were complemented with two talks in the afternoon. So by taking the traditional Oberwolfach hike on wednesday into account, a total of 23 full hour talks was presented during the workshop. A broad variety of topics was covered by the speakers. Normal surface singularities continue to pose some big questions which were adressed in the talks of A. Nemethi and A. Melle. The theory of motivic integration has brought the arc structure of singularities into the focus of current interest. A. Reguera and P. Popescu-Pampu reported on recent progress related to the original Nash-problem. F. Loeser reported on the first steps in non-archimedian local analytic geometry, A. Parushinski talked on conormal geometry. There has been an increased activity around global aspects of singularity theory, characteristic classes, Thom-polynomials and their applications. These topics were adressed by M. Kazarian, M. Weiss J. Schürmann and V. Sedykh. Applications of ideas from singularity theory to the geometry of the fundamental group were presented in the talks by A. Dimca, V. Kulikov and A. Libgober. S. Shadrin presented a conjectural method to produce tautological relations in the cohomology of the moduli space \mathcal{M}_{g,n} and O. Tommasi reported on the rational cohomology of \mathcal{M}_{4} . A. Gorinov talked about bounds for the size of the automorphism group of complete intersections that arise from the computation of the cohomology of the discriminant-complement. The recent interest in the Hodge/Twistor structures associated to isolated hypersurface germs and tame polynomials on affine varieties, reported on by C. Sevenheck and C. Hertling, stems partly from the interpretation of quantum cohomology via mirror symmetry in terms of so-called Landau–Ginsburg models. D. Mond talked about his idea to relate these to the theory of functions on a free divisor. J. Christophersen showed us a Calabi–Yau space appearing in the deformation theory of a degenerate abelian surface. A. Gabrielov talked about the maximally inflected rational curves and its relation to the Bethe Ansatz. A. Takahasi presented his results for exceptional collections in the category maximal Cohen–Macaulay modules on hypersurface singularities and its mirror interpretation in the case of the exceptional singularities. An informal concert by J. Steenbrink on piano und M. Weiss on flute formed the musical coda of the workshop. The organisers would like to use this occasion to thank the Oberwolfach staff for their efficient handling of boundary conditions, which created the unique Oberwolfach athmosphere that helped to make the workshop a success.

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 k-free divisor problem

Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.$

The depth of the Jacobian ring of a homogeneous polynomial in three variables

The question as to whether the Jacobian ideal of an irreducible projective plane curve always admits an irrelevant component has been going around for some years. One shows that a curve will satisfy this if it has only ordinary nodes or cusps, while an example is given of a family of sextic curves whose respective Jacobian ideals are saturated. The connection between this problem and the theory of homogeneous free divisors in three variables is also pointed out, so the example gives a family of Koszul-free divisors.

The Index of Vector Fields and Logarithmic Differential Forms

We introduce the notion of logarithmic index of a vector field on a hypersurface and prove that the homological index can be expressed via the logarithmic index. Then both invariants are described in terms of logarithmic differential forms for Saito free divisors, which are hypersurfaces with nonisolated singularities, and all contracting homology groups of the complex of regular holomorphic forms on such a hypersurface are computed. In conclusion, we consider the case of normal hypersurfaces, including the case of an isolated singularity, and describe the contracting homology of the complex of regular meromorphic forms with the help of the residue of logarithmic forms.

Dualité et comparaison pour les complexes de de Rham logarithmiques par rapport aux diviseurs libres

Let X be a complex analytic manifold and D⊂X a free divisor. Integrable logarithmic connections along D can be seen as locally free 𝒪 X -modules endowed with a (left) module structure over the ring of logarithmic differential operators 𝒟 X (logD). In this paper we study two related results: the relationship between the duals of any integrable logarithmic connection over the base rings 𝒟 X and 𝒟 X (logD), and a differential criterion for the logarithmic comparison theorem. We also generalize a formula of Esnault-Viehweg in the normal crossing case for the Verdier dual of a logarithmic de Rham complex.

Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Let D , x D,x be a free divisor germ in a complex manifold X X of dimension n > 2 n>2 . It is an open problem to find out which are the properties required for D , x D,x to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D , x D,x . We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.

Testing the Logarithmic Comparison Theorem for Free Divisors

We propose in this work a computational criterion to test if a free divisor D ⊂ Cn verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of D in Cn . For Spencer free divisors D ≡ (f = 0), we solve a conjecture about the generators of the annihilating ideal of 1/f and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.

On the Legacy of Free Divisors II: Free* Divisors and Complete Intersections

On the Legacy of Free Divisors II: Free* Divisors and Complete Intersections

Logarithmic cohomology of the complement of a plane curve

Let D, x be a plane curve germ. We prove that the complex \( \Omega^\bullet(\log D)_x \) computes the cohomology of the complement of D, x only if D is quasihomogeneous. This is a partial converse to a theorem of [5], which asserts that this complex does compute the cohomology of the complement, whenever D is a locally weighted homogeneous free divisor (and so in particular when D is a quasihomogeneous plane curve germ). We also give an example of a free divisor \( D\subset \mathbb {C}^3 \) which is not locally weighted homogeneous, but for which this (second) assertion continues to hold.

On the freeness of equisingular deformations of plane curve singularities

We consider surface singularities in C 3 arising as the total space of an equisingular deformation of an isolated curve singularity of the form f( x, y)+ zg( x, y) with f and g weighted homogeneous. We give a criterion that such a surface is a free divisor in the sense of Saito. We deduce that the Hessian deformation defines a free divisor for nonsimple weighted homogeneous singularities, and that the failure of this property “almost” characterizes the simple singularities. The criterion also yields distinct deformations of the same curve singularity, exactly one of which is free, showing that freeness is not a topological property.

The Module Df s for Locally Quasi-homogeneous Free Divisors

We find explicit free resolutions for the D-modules Dfs and D[s] fs/ D[s] fs+1, where f is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper.

Milnor Number Equals Tjurina Number for Functions on Space Curves

The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor.