Search

As we have seen along this book, for a singular variety V , there are several definitions of Chern classes, the Mather class, the Schwartz. MacPherson class, the Fulton.Johnson class and so forth. They are in the homology of V and, if V is nonsingular, they all reduce to the Poincaré dual of the Chern class c*(TV ) of the tangent bundle TV of V . On the other hand, for a coherent sheaf F on V , the (cohomology) Chern character ch*(F) or the Chern class c*(F) makes sense if either V is nonsingular or F is locally free. In this chapter, we propose a definition of the homology Chern character ch.(F) or the Chern class c.(F) for a coherent sheaf F on a possibly singular variety V . In this direction, the homology Chern character or the Chern class is defined in [140] (see also [100]) using the Nash type modification of V relative to the linear space associated to the coherent sheaf F. Also, the homology Todd class τ(F) is introduced in [15] to describe their Riemann-Roch theorem. Our class is closely related to the latter. The variety V we consider in this chapter is a local complete intersection defined by a section of a holomorphic vector bundle over the ambient complex manifold M. If F is a locally free sheaf on V , then the class ch*(F) coincides with the image of ch*(F) by the Poincaré homomorphism ]> . This fact follows from the Riemann–Roch theorem for the embedding of V into M, which we prove at the level of čech–de Rham cocycles. We also compute the Chern character and the Chern class of the tangent sheaf of V , in the case V has only isolated singularities. [ABSTRACT FROM AUTHOR]

Both Schwartz–MacPherson and Fulton–Johnson classes generalize Chern classes to the case of singular varieties. It is known that for local complete intersections with isolated singularities, the 0-degree SM and FJ classes differ by the local Milnor numbers [149] and all other classes coincide [155]. As we explain in the sequel, is V has nonisolated singularities, the difference ]> of the SM and FJ classes is, for each i, a homology class with support in the homology H2i(Sing(V)) of the singular set of V. That is the reason for which their difference was called in [30,31] the Milnor class of degree i. These classes have been also considered, from different viewpoints, by other authors, most notably by P. Aluffi, T. Ohmoto, A. Parusiński, P. Pragacz, J. Schürmann, S. Yokura. In this chapter we introduce the Milnor classes of a local complete intersection V of dimension n ≥ 1 in a complex manifold M, defined by a regular section s of a holomorphic bundle N over M. The aim of this chapter is to show that, as mentioned above, the Milnor classes are localized at the connected components of the singular set of V : If S is such a component then one has Milnor classes μi of V at S in degrees i = 0, … , dim S. The 0–degree class coincides with the generalized Milnor number of V at S, introduced by Parusiński in [127] (if V is a hypersurface in M). The sum of all the Milnor classes over the connected components of Sing(V) gives the global Milnor classes studied in [8, 126, 131, 169]. See [28] for another presentation. The method we use for constructing the localized Milnor classes comes from [31] and uses Chern–Weil theory. The idea is to use stratified frames to localize at the singular set the Schwartz–MacPherson and the FultonJohnson classes, in such a way that the difference of these localizations canonical. [ABSTRACT FROM AUTHOR]

The constructions described in the previous chapter, mostly based on [31, 33, 139], provide geometric insights of the Schwartz–MacPherson classes via obstruction theory and localization. These approaches are useful for understanding what the classes measure from the viewpoint of indices of vector fields and frames. The Fulton–Johnson classes [59,60] provide another way of generalizing the Chern class of complex manifolds to the case of singular varieties. In the context we consider, they coincide with the virtual classes (see Sect. 11.1). In this chapter we define and study the virtual classes from a viewpoint similar to the one we used in the previous chapter for the SchwartzMacPherson classes. This is based on our articles [31, 34], joint work with D. Lehmann. If the variety V is globally defined by a function on M, the virtual classes can be localized topologically and one can interpret them as ˵weighted″ Schwartz classes. That is explained in Sect. 11.3 where we prove the Proportionality theorem of [34] for this index. This theorem is analogous to, and inspired by, the similar theorem of [33] for the Schwartz index, proved in the previous chapter. In Sect. 11.4, the localization of virtual classes is performed using the differential geometric method of [31], i.e., Chern–Weil theory, and using stratified frames. In that context, we construct localized ˵Fulton–Johnson classes″ at the singular set of the given frames. While Sects. 11.1–11.3 are of a local nature, Sect. 11.4 is global. [ABSTRACT FROM AUTHOR]

As mentioned before, the first generalization of Chern classes to singular varieties is due to M.-H. Schwartz, using obstruction theory and radial frames. These classes are the primary obstructions to constructing a special type of stratified frames on V that she called radial frames. To avoid possible misunderstandings, here we prefer to call them frames constructed by radial extension, as in the case of vector fields. We refer to [28, 33] for details of the construction and we content ourselves with summarizing here their main properties. It was shown in [33] that these classes correspond, by Alexander isomorphism, to the MacPherson classes, that we discuss briefly in the last section of this chapter. In this chapter we provide a viewpoint for studying Schwartz–MacPherson classes which is particularly close to the theory of indices of vector fields that we develop in this book, both from the topological and the differential geometric sides. In the first three sections, we discuss the Schwartz index of frames and a method for defining the Schwartz classes of singular varieties using arbitrary stratified frames, not necessarily constructed by radial extension. As a corollary we obtain that the Schwartz classes are the primary obstruction to constructing a stratified frame (any frame, not necessarily radial) on the skeleton of the appropriate dimension and for an appropriate cellular decomposition: if such a frame exists, then the corresponding Schwartz class vanishes (the converse is false in general). In Sect. 4, we use the methods of [31], joint work with D. Lehmann, for constructing localized Schwartz classes in both the topological and differential geometric contexts, via Chern–Weil theory and using stratified frames. The last section discusses briefly MacPherson and Mather classes (see [28, 117]). [ABSTRACT FROM AUTHOR]

When considering smooth (real) manifolds, the tangent and cotangent bundles are isomorphic and it does not make much difference to consider either vector fields or 1-forms in order to define their indices and their relations with characteristic classes. When the ambient space is a complex manifold, this is no longer the case, but there are still ways for comparing indices of vector fields and 1-forms, and to use these to study Chern classes of manifolds. To some extent this is also true for singular varieties, but there are however important differences and each of the two settings has its own advantages. In this chapter we briefly review the various indices of 1-forms on singular varieties through the light of the indices of vector fields discussed earlier. We define in that way the Schwartz index, the radial index, the GSV index, the homological index and the local Euler obstruction, and we study some of their relations and properties. In this short presentation we include work done by various authors, particularly by W. Ebeling and S. Gusein-Zade, as well as ourselves in [36]. In the last section we discuss briefly the ˵indices of collections of 1-forms″ introduced by W. Ebeling and S. Gusein-Zade: just as the index of a 1-form corresponds to the ˵top Chern class″ (of a manifold or of a singular variety, in a sense that will be made precise in later chapters), so too the indices of collections of 1-forms correspond to other Chern numbers. Let us mention that in his book [138], J. Schürmann introduces methods to studying singular varieties via micro-local analysis, and part of what we say below can also be considered in that framework. [ABSTRACT FROM AUTHOR]

We have already defined and studied several indices of vector fields on singular varieties, each of them being related to some property of the index of Poincaré–Hopf, or to some extension of the tangent bundle to the case of singular varieties. There is another line of research with remarkable works by various authors, that originates in the well-known fact (cf. Example 1.6.2) that for a holomorphic vector field v in Cn with an isolated singularity at 0, the local Poincaré–Hopf index satisfies: ]> where (a1, … , an) is the ideal generated by the components of v. In the real analytic setting, the equivalent statement is given by the formula of EisenbudLevin–Khimshiashvili, expressing the local Poincaré–Hopf index through the signature of a certain quadratic form. These facts motivated the search for algebraic formulas for indices of vector fields on singular varieties. A major contribution in this direction was given by V. I. Arnold for gradient vector fields. There are also significant contributions by various authors, such as X. Gòmez–Mont, S. Gusein–Zade, W. Ebeling and others. In this chapter we give a glance of some of the research in this direction, and we refer to the literature for more on that topic. We discuss first the homological index for holomorphic vector fields, introduced by X. Gómez–Mont and further studied by himself in collaboration with Ch. Bonatti, P. Mardešić, L. Giraldo, H.-C. G. von Bothmer and W. Ebeling. In the last section of this chapter we discuss briefly the Eisenbud–Levin–Khimshiashvili formula for the index of real analytic vector fields, and its generalization to singular varieties. The homological index has the important property of being defined for holomorphic vector fields on arbitrary complex analytic isolated singularity germs (V, 0). When the germ (V, 0) is a complete intersection, the homological index coincides with the GSV–index, by [17, 68]. If we now let V be a compact complex variety with isolated singularities, one has a well–defined notion of the total homological index for holomorphic vector fields on V with isolated singularities, defined in the usual way. This total index ought to be independent of the choice of vector field, being therefore an invariant of V . This is the case if V is a local complete intersection, and the corresponding global invariant is the 0–degree Fulton–Johnson class of V , as we will see in Chap. 11. It would be interesting to know what the homological index measures for singular germs and varieties which are not local ICIS. This is related with extending the notion of Milnor number to isolated singularity germs which are not complete intersections (see Chap. 9). [ABSTRACT FROM AUTHOR]

The local Euler obstruction was first introduced by R. MacPherson in [117] as an ingredient for the construction of characteristic classes of singular complex algebraic varieties. An equivalent definition was given by J.-P. Brasselet and M.-H. Schwartz in [33] using vector fields. Their viewpoint brings the local Euler obstruction into the framework of ˵indices of vector fields on singular varieties,″ though the definition only considers radial vector fields. This approach is most convenient for our study which is based on [29, 32] and shows relations with other indices. There are various other definitions and interpretations, in particular due to Gonzalez-Sprinberg, Verdier, Lê-Teissier and others. The survey [27] provides an overview on the subject. Section 1 below is devoted to the definition of the local Euler obstruction and some of its main properties. The behavior of the local Euler obstruction relatively to hyperplane sections is described in Sect. 2, following [29]. In Sect. 3 and the thereafter we study a generalization of the local Euler obstruction introduced in [32] and called the Euler obstruction of the function, or also the ˵Euler defect″; this is an invariant associated to map-germs on singular varieties. MacPherson΄s local Euler obstruction corresponds to the square of the function distance to the given point. It is shown in [150], and explained in the last section of this chapter, that this invariant can be expressed in terms of the number of critical points in the regular part of a Morsification of the function. [ABSTRACT FROM AUTHOR]

We have seen that for vector fields, there are indices such as the Poincaré–Hopf index and the virtual index, that arise from localizations of certain Chern classes. If the vector field is holomorphic, the localization theory becomes richer because of the Bott vanishing theorem, and this produces further interesting residues. This theory can be developed for general singular foliations on certain singular varieties. We consider here the case of holomorphic vector fields and the slightly more general case of one dimensional singular foliations. We refer to [156] for a systematical treatment of the general case. Here we have three types of residues: (1) Baum–Bott residues and generalizations to singular varieties, (2) Camacho–Sad index and various generalizations, (3) Variations and generalizations. In all the above cases the residues arise from a Bott type vanishing theorem, which in turn comes from an action of the vector field or the foliation on some vector bundle or virtual bundle. The residues of type (1) were first introduced by R. Baum and P. Bott in [13,14]. In general these arise from the action of the foliation on the normal sheaf of the foliation. The Camacho–Sad index (2) was introduced in [42] and was effectively used to prove the existence of a separatrix at a singular point of a holomorphic vector field on the complex plane. Nowadays there are many generalizations of this index, see Remark 6.3.3 below. These residues arise from the action of the foliation on the normal bundle of an invariant subvariety. The residues of type (3) were first introduced by B. Khanedani and T. Suwa in [93] and generalized in [113]; see also the related articles [39] and [40] by M. Brunella. These type of residues arise from the action of the foliation on the ambient tangent bundle. These three types of residues are listed above in historical order, but they are explained below in the reversed order, for logical reasons. In each case, the residue at an isolated singularity can be expressed in terms of a Grothendieck residue on singular variety. [ABSTRACT FROM AUTHOR]

The virtual index was first introduced in [111] by D. Lehmann, M. Soares and T. Suwa for holomorphic vector fields; the extension to continuous vector fields is immediate and has been done in [30, 31, 149]. If the variety has only isolated singularities, the virtual index and the GSV index coincide. The virtual index has several interesting features, as for instance that it is relatively easy to compute when the vector field we deal with is holomorphic, and also that it is defined for vector fields with singular set a compact set of arbitrary dimension. In this chapter we introduce the virtual index in the context of singular varieties V which are local complete intersections defined by a section of a holomorphic vector bundle N over a complex manifold M (see Sect. 5.1 below). The virtual tangent bundle is then defined as (TM − N)|V , where TM denotes the holomorphic tangent bundle of M. One can think of the virtual index as being a localization of the top dimensional Chern class of the virtual tangent bundle, called virtual class, just as the local index of Poincaré–Hopf is a localization of the top Chern class of a manifold. The virtual index is in fact a residue which is the local contribution, relatively to a vector field v, of the top virtual class. In Sect. 2, we show that Chern–Weil theory is very well adapted to this situation, in the framework of čech–de Rham cohomology, and Sect. 3 is devoted to the study of residues in this context. The properties of the virtual index are detailed in the last sections of the chapter, in particular we prove an integral formula for the virtual index. [ABSTRACT FROM AUTHOR]

In the previous chapters we focused on indices of vector fields on complex analytic varieties. The real analytic setting also has its own interest, and that is the subject of this chapter. The following presentation follows the discussion by M. Aguilar, J. Seade and A. Verjovsky in [6] (see also [49]). We describe indices analogous to the GSV and Schwartz indices for vector fields on real analytic singular varieties. In this setting the GSV index is an integer if the singular variety V is odd-dimensional, but it is defined only modulo 2 if the dimension of V is even. The Schwartz and the GSV indices are defined, respectively, in Sects. 1 and 2; there we show that the Schwartz index classifies the homotopy classes of vector fields near an isolated singularity. Section 3 provides a geometric interpretation of the GSV index in the real analytic setting. The information we get is related to previous work by M. Kervaire about the curvatura integra of manifolds, and this is the subject we explore in Sect. 4. Finally, in Sect. 5 we look at the relation of these indices with other invariants of real analytic singularity germs studied previously by C. T. C. Wall and others. This yields to an extension of the concept of Milnor number for real analytic map-germs with isolated singularities which may not be algebraically isolated. We note that there are some related works such as [9, 10, 49]. Since in this chapter we consider only real analytic varieties and functions, for simplicity, we will denote the dimensions here by m, n... instead of m΄, n΄…, as in the rest of the book. [ABSTRACT FROM AUTHOR]

One of the basic properties of the local Poincaré – Hopf index is stability under perturbations. In other words, if a vector field has an isolated singularity on an open set in ℝn and if we perturb it slightly, then the singularity may split into several singular points, with the property that the sum of the indices of the perturbed vector field at these singular points equals the index of the original vector field at its singularity. If we now consider an analytic variety V defined by a holomorphic function ]> with an isolated critical point at 0, and if v is a vector field on V, with an isolated singularity at 0, then one may like ˵the index″ of v at 0 to be stable under small perturbations of both, the function f and the vector field v. This leads naturally to another concept of index, called the GSV index, introduced by X. Góomez-Mont, J. Seade and A. Verjovsky in [71, 144] for hypersurface germs, and extended in [149] to complete intersections. In this chapter we define this index and we study some of its basic properties. We first do it when the ambient space is an isolated complete intersection singularity (ICIS for short), then we explain the recent generalization in [34] to the case where the ambient variety has nonisolated singularities; this relies on a proportionality theorem similar to the one proved in [33] for the local Euler obstruction, that is discussed later in the text. In the following chapters we will study other related indices: the GSV index can be interpreted via Chern.Weil theory as the virtual index introduced by D. Lehmann, M. Soares and T. Suwa in [111], that we study in Chap. 5. And if the vector field v is holomorphic, then the GSV index also coincides with the homological index of Góomez-Mont [68], that we describe in Chap. 7. There is also a recently defined logarithmic index in [7], which coincides with the homological index and therefore, for ICIS, with the GSV index. [ABSTRACT FROM AUTHOR]

In this chapter we review briefly some of the fundamental results of the classical theory of indices of vector fields and characteristic classes of smooth manifolds. These were first defined in terms of obstructions to the construction of vector fields and frames. In the case of a vector field the Poincarée – Hopf Theorem says that Euler.Poincarée characteristic is the obstruction to constructing a nonzero vector field tangent to a compact manifold. Extension of this result to frames yields to the definition of Chern classes from the viewpoint of obstruction theory. There is another important point of view for defining characteristic classes on the differential geometry side, this is the Chern – Weil theory. Sections 3 and 4 provide an introduction to that theory and the corresponding definition of Chern classes. Finally, Sect. 5 sets up one of the key features of this monograph: the interplay between localization via obstruction theory, which yields to the classical relative characteristic classes, and localization via Chern – Weil theory, which yields to the theory of residues. This is one way of thinking of the Poincarée – Hopf Theorem and its generalizations. Throughout the book, M will denote either a complex manifold of (complex) dimension m, or a C∞ manifold of (real) dimension m΄. [ABSTRACT FROM AUTHOR]

The index of a tangent vector field in a singular point is well-defined on manifolds, as described in the previous chapter. When working with singular analytic varieties, it is necessary to give a sense to the notion of ˵tangent″ vector field and, once this is done, it is natural to ask what should be the notion of ˵the index″ at a singularity of the suitable vector field. Indices of vector fields on singular varieties were first considered by M.-H. Schwartz in [139,141] (see also [33,142]) in her study of the Poincarè–Hopf Theorem and Chern classes for singular varieties. For her purpose there was no point in considering vector fields in general, but only a special class of vector fields that she called ˵radial,″ which are obtained by the important process of radial extension. In this chapter we explain the definition of the corresponding index as it was defined by M.-H. Schwartz for vector fields constructed by radial extension. Complete description and constructions will be found in [28]. We define a natural extension of this index for arbitrary (stratified) vector fields on singular varieties. This index is sometimes called ˵radial index″ in the literature, but we prefer to call it here the Schwartz index. The Schwartz index for arbitrary stratified vector fields was first defined by H. King and D. Trotman in [96], and later independently in [6, 49, 149]. In [30, 31] this index was interpreted in differential-geometric terms and this was used to study its relations with various characteristic classes for singular varieties. This is discussed in [28] and in Chap. 10 below. [ABSTRACT FROM AUTHOR]

In deformations of polynomial functions one may encounter "singularity exchange at infinity" when singular points disappear from the space and produce "virtual" singularities which have an influence on the topology of the limit polynomial. We find several rules of this exchange phenomenon, in which the total quantity of singularity turns out to be not conserved in general. [ABSTRACT FROM AUTHOR]

In this paper we prove the following theorem: Let M be the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein surface singularity. Then M is diffeomorphic to a coset space $$ \tilde \Gamma _1 \backslash \tilde G/\tilde \Gamma _2 $$, where $$ \tilde G $$ is the 3-dimensional Lie group $$ \widetilde{PSL}(2,\mathbb{R}) $$, while $$ \tilde \Gamma _1 $$ and $$ \tilde \Gamma _2 $$ are discrete subgroups of $$ \tilde G $$, the subgroup $$ \tilde \Gamma _1 $$ is co-compact and $$ \tilde \Gamma _2 $$ is cyclic. Conversely, if M is diffeomorphic to a coset space as above, then M is diffeomorphic to the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein singularity. We also prove the following characterisation of quasi-homogeneous ℚ-Gorenstein surface singularities: A quasi-homogeneous surface singularity is ℚ-Gorenstein of index r if and only if for the corresponding automorphy factor (U, Γ, L) some tensor power of the complex line bundle L is Γ-equivariantly isomorphic to the rth tensor power of the tangent bundle of the Riemannian surface U. [ABSTRACT FROM AUTHOR]

MacPherson's Chern class transformation on complex algebraic varieties is a certain unique natural transformation from the constructible function covariant functor to the integral homology covariant functor, and it can be extended to a category of provarieties. In this paper, as further extensions of this we consider natural transformations among Mackey functors on provarieties and also on "indvarieties" and discuss some notions and examples related to these extensions. [ABSTRACT FROM AUTHOR]

We give examples of hypersurfaces of degree d in Pn(ℂ), whose singularities are not versally deformed by the family Hd(n) of all hypersurfaces of degree d in Pn(ℂ), and which are of minimal codimension with this property. In the three cases (n, d) = (2, 6), (3, 4) and (5, 3), such hypersurfaces necessarily have one-parameter symmetry. We list the possibilities. The singularities of these hypersurfaces are not all simple, and they are simultaneously topologically versally deformed by Hd(n). In less degenerate cases the examples we give are hypersurfaces with only simple singularities. The failure of versality can be expected to show itself in the geometry of Hd(n), either because the μ-constant stratum S containing the hypersurface is of codimension less than μ in Hd(n), or because S is not smooth. We will see elsewhere that this is the case for the examples we consider here. In particular, the singularities of these hypersurfaces are not topologically versally deformed by Hd(n). [ABSTRACT FROM AUTHOR]

Let X be a hypersurface of degree d in Pn(ℂ) with isolated singularities, and let f: ℂn+1 → ℂ be a homogeneous equation for X. The singularities of X can be simultaneously versally deformed by deforming the equation f, in an affine chart containing all of the singularities, by the addition of all monomials of degree at most r, for sufficiently large r; it is known (see, e.g., §1) that r≥n(d−2) suffices. Conversely, if the addition in the affine chart of all monomials of degree at most n(d−2)−1−a, a ≥ 0, fails to simultaneously versally deform the singularities of X, then we will say that X is a-non-versal. The first main result of this paper shows that X is a-non-versal if, and only if, there exists a homogeneous polynomial vector field with coefficients of degree a, which annihilates f but is not Hamiltonian for f. Our second main result is a sufficient condition for an a-non-versal hypersurface to be topologically a-versal. [ABSTRACT FROM AUTHOR]

We study how to minimize the number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of any finitely determined holomorphic germ f: (ℂn, 0) → (ℂ3, 0), with n > 3. Gaffney showed in [3] that the invariants for the Whitney equisingularity are the 0-stable invariants and the polar multiplicities of the stable types of the germ. First we describe all stable types which appear in these dimensions. Then we find relationships between the polar multiplicities of the stable types in the singular set and also in the discriminant. When n > 3, for any germ f there is an hypersurface in ℂn, which is of special interest, the closure of the inverse image of the discriminant by f, which possibly is with non isolated singularities. For this hypersurface we apply results of Gaffney and Gassler [6], and Gaffney and Massey [7], to show how the Lê numbers control the polar invariants of the strata in this hypersurface. Gaffney shows that the number of invariants needed is 4n+10. In the corank one case we reduce this number to 2n+2. The polar multiplicities are also an interesting tool to compute the local Euler obstruction of a singular variety, see [12]. Here we apply this result to obtain explicit algebraic formulae to compute the local Euler obstruction of the stable types which appear in the singular set and also for the stable types which appear in the discriminant, of corank one map germs from ℂn to ℂ3 with n ≥ 3. [ABSTRACT FROM AUTHOR]

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension n in terms of differential forms. In the case n = 1 such computations have many applications in differential equations and counting their limit cycles. For n > 3, these computations give us an explicit definition of Hodge cycles. [ABSTRACT FROM AUTHOR]

It is known that Goursat distributions (subbundles in the tangent bundles having the tower of consecutive Lie squares growing in ranks very slowly, always by one) possess, from corank 8 onwards, numerical moduli of the local classification, in both C∞ and real analytic categories. (Whereas up to corank 7 that classification is discrete, as shown in a series of papers, the last in that series being [13].) A natural question, first asked by A.Agrachev in 2000, is whether the moduli of Goursat distributions descend to the level of nilpotent approximations: whether they are stiff enough to survive the passing to the nilpotent level. In the present work we show that it is not the case for the first modulus appearing in corank 8 (and the only one known to-date in that corank). [ABSTRACT FROM AUTHOR]

Unimodular functions have a μ-constant line in their miniversal unfoldings. Their miniversal deformations on the other hand contain a nontrivial τ-constant stratum only for the three cases of elliptic singularities. In computer experiments we found six sub-series of the T-series, which have a modular line in the their miniversal deformations. The singular locus of the family restricted to such a line splits into an elliptic singularity and another one of Ak-type, such that the deformation is τ-constant along the modular line. Each modular line can be patched together with the modular line of the associated elliptic singularity, completing it at infinity. All computations are based on the author's algorithm for computing modular spaces as flatness stratum of the relative cotangent cohomology inside a deformation. [ABSTRACT FROM AUTHOR]

Consider a periodic function f of two variables with symmetry Γ and let ℒ ⊂ Γ be the subgroup of translations. The Fourier expansion of a periodic function is a sum over ℒ*, the dual of the set ℒ of all the periods of f. After projecting f, some of its original symmetry remains. We describe the symmetries of the projected function, starting from Γ and from the structure of ℒ*. [ABSTRACT FROM AUTHOR]

A classical theorem of Briançon, Speder and Teissier states that a family of isolated hypersurface singularities is Whitney equisingular if, and only if, the μ*-sequence for a hypersurface is constant in the family. This paper shows that the constancy of relative polar multiplicities and the Euler characteristic of the Milnor fibres of certain families of non-isolated singularities is equivalent to the Whitney equisingularity of a family of corank 1 maps from n-space to n + 1-space. The number of invariants needed is 4n − 2, which greatly improves previous general estimates. [ABSTRACT FROM AUTHOR]

These are notes of the introductory courses on the subject we lectured in Trieste in 2003 and Luminy in 2004. The lectures contain basic notions and fundamental theorems of the local theory. [ABSTRACT FROM AUTHOR]

This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a submodule of finite length of a free module F over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of M is not equal to F, is the multiplicity of M. Specifically, we apply the multiplicity polar theorem to the study of stratification conditions on families of hypersurfaces, obtaining the first set of invariants giving necessary and sufficient conditions for the Af condition for hypersurfaces with non-isolated singularities. [ABSTRACT FROM AUTHOR]