Non-isolated Singularities Research Articles

Nonisolated forms of rational triple point singularities of surfaces and their resolutions

We give a list of nonisolated hypersurface singularities of which normalizations are the rational triple point singularities of surfaces and construct their minimal resolution graphs by a method introduced by Oka for isolated complete intersection singularities. We also show that nonisolated forms of rational triple point singularities and their normalizations are both Newton non-degenerate.

Real submanifolds of maximum complex tangent space at a CR singular point, I

We study a germ of real analytic n-dimensional submanifold of C n that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions , we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we investigate the existence of a complex submanifold of positive dimension in C n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non-isolated CR singularity.

Topological triviality of linear deformations with constant Lê numbers

Let f(t,z) = f0(z) + tg(z) be a holomorphic function defined in a neighbourhood of the origin in C × Cn. It is well known that if the one-parameter deformation family {ft} defined by the function f is a μ-constant family of isolated singularities, then {ft} is topologically trivial—a result of A. Parusinski. It is also known that Parusinski's result does not extend to families of non-isolated singularities in the sense that the constancy of the Le numbers of ft at 0, as t varies, does not imply the topological triviality of the family ft in general—a result of J. Fernandez de Bobadilla. In this paper, we show that Parusinski's result generalizes all the same to families of non-isolated singularities if the Le numbers of the function f itself are defined and constant along the strata of an analytic stratification of C × (f0−1(0) $\cap$ g−1(0)). Actually, it suffices to consider the strata that contain a critical point of f.

Monodromies at infinity of non-tame polynomials

AR Abstract. We consider the monodromy at infinity and the monodromies around the bifurcation points of polynomial functions f : C n −→ C which are not tame and might have non-isolated singularities. Our description of their Jordan blocks in terms of the Newton polyhedra and the motivic Milnor fibers relies on two new issues: the non- atypical eigenvalues of the monodromies and the corresponding concentration results for their generalized eigenspaces.

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.

Finite determinacy of non-isolated singularities

Finite determinacy of non-isolated singularities

Hermitian vs <i>PT</i>-Symmetric Scalar Yukawa Model

A comparative analysis of a model of complex scalar field φ and real scalar field χ with interaction for the real and purely imaginary values of coupling g in perturbative and non-perturbative regions is provided. In contrast to the usual Hermitian version (real g), which is asymptotically free and energetically unstable, the non-Hermitian PT-symmetric theory (imaginary g) is energetically stable and not asymptotically free. The non-perturbative approach based on Schwinger-Dyson equations reveals new interesting feature of the non-Hermitian model. While in the Hermitian version of theory the phion propagator has the non-physical non-isolated singularity in the Euclidean region of momenta, the non-Hermitian theory substantially free of this drawback, as the singularity moves in the pseudo-Euclidean region.

Topology of Real Milnor Fibrations for Non-Isolated Singularities

We consider a real analytic map $F=(f_1,\ldots,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)$, $2 \le k \le n-1$, that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic $f_I=(f_{i_1},\ldots,f_{i_l}) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^l,0)$, induced from $F$ by projections where $1 \le l \le n-2$ and $I=\{i_1,\ldots,i_l\}$, also satisfies Milnor's conditions (a) and (b). We give several relations between the Euler characteristics of the Milnor fibre of $F$, the Milnor fibres of the maps $f_I$, the link of $F^{-1}(0)$ and the links of $f_I^{-1}(0)$.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

AbstractWe show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

AbstractWe show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families withisolatedsingularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Gorenstein-duality for one-dimensional almost complete intersections – with an application to non-isolated real singularities

AbstractWe give a generalisation of the duality of a zero-dimensional complete intersection for the case of one-dimensional almost complete intersections, which results in a Gorenstein module M = I/J. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity f, with a one-dimensional critical locus, we relate the signature on the Jacobian module I/Jf to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in ℙ2(ℝ) of even degree is given.

Lê's polyhedron for line singularities

We study the topology of line singularities, which are complex hypersurface germs with non-isolated singularity given by a smooth curve. We describe the degeneration of its Milnor fiber to the singular hypersurface by means of a vanishing polyhedron in the Milnor fiber. As a milestone, we also study the topology of the degeneration of a complex isolated singularity hypersurface under a nonlocal point of view.

Carrousel in family and\break non-isolated hypersurface singularities in ℂ3

Abstract We study the boundary L t ${L_{t}}$ of the Milnor fiber for the reduced holomorphic germs f : ( ℂ 3 , 0 ) → ( ℂ , 0 ) ${f:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C},0)}$ having a non-isolated singularity at 0. We prove that L t ${L_{t}}$ is a graph manifold by using a new technique of carrousels depending on one parameter. Our results enable us to compare the topology of L t ${L_{t}}$ and of the link of the normalization of f - 1 ⁢ ( 0 ) ${f^{-1}(0)}$ .

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups. Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix singularities. We compute the Milnor number of isolated Cohen--Macaulay surface singularities of this type in $\mathbb{C}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of this type in $\mathbb{C}^5$.

Big denominators and analytic normal forms

Abstract We study the regular action of an analytic pseudo-group of transformations on the space of germs of various analytic objects of local analysis and local differential geometry. We fix a homogeneous object F 0 and we are interested in an analytic normal form for the whole affine space { F 0 + h.o.t. } ${\lbrace F_0 + \text{h.o.t.}\rbrace }$ . We prove that if the cohomological operator defined by F 0 has the big denominators property and if a formal normal form is well chosen, then this formal normal form holds in analytic category. We also define big denominators in systems of nonlinear PDEs and prove a theorem on local analytic solvability of systems of nonlinear PDEs with big denominators. Moreover, we prove that if the denominators grow “relatively fast”, but not fast enough to satisfy the big denominators property, then we have a normal form, respectively local solvability of PDEs, in a formal Gevrey category. We illustrate our theorems by explanation of known results and by new results in the problems of local classification of singularities of vector fields, non-isolated singularities of functions, tuples of germs of vector fields, local Riemannian metrics and conformal structures.

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

Let $$f$$ and $$g$$ be holomorphic function-germs vanishing at the origin of complex analytic germs of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ $$f\bar{g}$$ has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ $$f\bar{g}$$ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs of the form $$f\bar{g}$$ defined in a complex surface germ, and we prove an A’Campo-type formula for the zeta function of its monodromy.

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of $\operatorname{Spec}R$ , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w = fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.