We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function $f$ whose restriction $f|(X,0)$ to an arbitrarily singular reduced complex analytic space $(X,0) \subset (\mathbb C^n,0)$ has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, $μ_f(α;X,0)$, is the number of Morse critical points in a stratum $\mathscr S_α$ of $(X,0)$ in a morsification of $f|(X,0)$. Our formula expresses $μ_f(α;X,0)$ as a homological index based on the derived geometry of the Nash modification of the closure of the stratum $\mathscr S_α$. While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers $μ_f(α;X,0)$ by means of our formula in the case where the closure $\overline{ \mathscr S_α} \subset X$ of the stratum in question is a hypersurface.
We derive a formula for the Milnor class of scheme-theoretic global complete intersections (with arbitrary singularities) in a smooth variety in terms of the Segre class of its singular scheme. In codimension one the formula recovers a formula of Aluffi for the Milnor class of a hypersurface.
We find a recursive formula for the motivic Milnor fiber of an irreducible plane curve, using the notions of a truncation and derived curve. We then apply natural transformations to obtain a similar recursion for the Hodge-theoretic spectrum.
We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.
We give formulas for the image Milnor number of a weighted-homogeneous map-germ $(\mathbb{C}^n,0)\to(\mathbb{C}^{n+1},0)$, for $n=4$ and $5$, in terms of weights and degrees. Our expressions are obtained by a purely interpolative method, applied to a result by Ohmoto. We use our approach to recover the formulas for $n=2$ and $3$ due to Mond and Ohmoto, respectively. For $n\geq 6$, the method is valid as long as certain multi-singularity conjecture holds.
We show a combinatorial formula for a lower bound of the dimension of the non-unipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also prove a non-combinatorial formula for the dimension of the non-unipotent part of the first Milnor cohomology, which apparently depends on the position of the singular points. The latter generalizes a formula previously obtained by the second named author.
By calculating the equivariant mixed Hodge numbers of motivic Milnor fibers introduced by Denef-Loeser, we obtain explicit formulas for the Jordan normal forms of Milnor monodromies. The numbers of the Jordan blocks will be described by the Newton polyhedron of the polynomial.
We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows to express a class as a Z[1/2]- linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincaré polynomial for basic semialgebraic formulas. We then define …
The Milnor formula $μ=2δ-r+1$ relates the Milnor number $μ$, the double point number $δ$ and the number $r$ of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality $μ\geq 2δ-r+1$ in arbitrary characteristic and showed that the equality $μ=2δ-r+1$ characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic $p$. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. (2010)…
In this paper, we prove that the conductor formula of Bloch implies the conjecture of Deligne on Milnor numbers of isolated singularities. In particular, thanks to the work of Bloch on his conjecture, our result implies that this so-called Deligne-Milnor's conjecture is equally true in the unequal characteristic case.
Let $f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g:(\mathbb{C}^{n-1},0)\rightarrow (\mathbb{C}^{n},0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers $μ_I(f)+μ_I(g)$ is equal to the number of critical points of the stratified Morse function $p|_{X_s}:X_s\to\mathbb{C}$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).
We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch-Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hy…
The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)$. We give a formula for the jump in some class of surface singularities in the case deformations are non-degenerate.
Given a nonconstant polynomial map over the reals having an isolated critical point in the origin and with zero locus of positive dimension, we establish a formula for the singular homology groups of a Milnor fibre relative to its boundary.
We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms.
We determine the relation between the local Euler obstruction $Eu_f$ of a holomorphic function $f$ and different generalizations of the Milnor number for functions on singular spaces.
In this note we give a simple proof of the following relative analog of the well known Milnor-Palamodov theorem: the Bruce-Roberts number of a function relative to an isolated hypersurface singularity is equal to its topological Milnor number (the rank of a certain relative (co)homology group) if and only if the hypersurface singularity is quasihomogeneous. The proof relies on an interpretation of the Bruce-Roberts number in terms of differential forms and the Lê-Greuel formula.
Let $(X,0) \subset (\mathbb{R}^n,0)$ be the germ of a closed subanalytic set and let $f$ and $g : (X,0) \rightarrow (\mathbb{R},0)$ be two subanalytic functions. Under some conditions, we relate the critical points of $g$ on the real Milnor fibre $X \cap f^{-1}(δ) \cap B_ε$, $0 <| δ| \ll ε\ll 1$, to the topology of this fibre and other related subanalytic sets. As an application, when $g$ is a generic linear function, we obtain an "asymptotic" Gauss-Bonnet formula for the real Milnor fibre of $f$. From this Gauss-Bonnet formula, we deduce "infinitesimal" linear kinematic formulas.
We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz-MacPherson and Fulton-Johnson classes, $c^{SM}(X)$ and $c^{FJ}(X)$. Their difference (up to sign) is the total Milnor class ${\mathcal M}(X)$, a generalization of the Milnor number for varieties with arbitrary singular set. We get first Verdier-Riemann-Roch type formulae for the total classes $c^{SM}(X)$ and $c^{FJ}(X)$, and use these to prove a surprisingly simple formula for the total Milnor class when $X$ is defined by a finite number of local complete inter…
We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.
We use Altmann's toric fan description of P-resolutions to formulate a new description of deformation theory invariants for two-dimensional cyclic quotient singularities. In particular, we show how to calculate the dimensions of the (reduced) versal base space components as well as Milnor numbers of smoothings over them.
The jump of the Milnor number of an isolated singularity $f_{0}$ is the minimal non-zero difference between the Milnor numbers of $f_{0}$ and one of its deformation $(f_{s}).$ In the case $f_{s}$ are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of $f_{0}$ in the case $f_{0}$ is a convenient singularity with only one $(n-1)$-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn-Pham singularities.
We analyze the monodromy action, over the rationals, on the first homology group of the Milnor fiber, for arbitrary subarrangements of Coxeter arrangements. We propose a combinatorial formula for the monodromy action, involving Aomoto complexes in positive characteristic. We verify the formula, in cases A, B and D.
In this work we study algebraic, geometric and topological properties of the Milnor classes of local complete intersections with arbitrary singularities. We describe first the Milnor class of the intersection of a finite number of hypersurfaces, under certain conditions of transversality, in terms of the Milnor classes of the hypersurfaces. Using this description we obtain a Parusiński-Pragacz type formula, an Aluffi type formula and a description of the Milnor class of the local complete intersection in terms of the global Lê cycles of the hypersurfaces that define it. We consider next the ge…
We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $\{f_t\}$ of analytic maps with $f_0=f, f_1=g$ which has a so-called {\it uniform stable radius for the Milnor fibration}. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is…
We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…
We give a concrete formula for the total Milnor number of the weighted-Le-Yomdin-at-infinity polynomial in most of the interesting cases. As an application, we give a description of the monodromy fibration at infinity for such kind of polynomial.
For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…
We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We prove in dimension $3$ a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing previous results by J. Snoussi in dimension $2$. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Lê D. T. concerning constancy of the Milnor number.
In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local δ-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical δ-invariant in terms of a Q-resolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve def…
Let $L$ be a cyclic $L_\infty$-algebra of dimension $3$ with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic $L_\infty$-algebra structure on the cohomology $H^*(L)$. The inner product plus the higher products of the cyclic $L_\infty$-algebra defines a superpotential function $f$ on $H^1(L)$. We associate with an analytic Milnor fiber for the formal function $f$ and define the Euler characteristic of $L$ is to be the Euler characteristic of the étale cohomology of the analytic Milnor fiber. In this paper we prove a Thom-Sebastiani type form…
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the comparison, a perverse sheaf naturally associated to any reduced complex analytic space on which the constant sheaf $\Q_X^\bullet[\dim X]$ is perverse. This generalizes the classical formula for the Milnor number of a plane curve in terms of double point…
We develop recursive formulas for the horizontal and vertical monodromies of a quasi-ordinary surface. These are monodromies associated to the Milnor fiber of a slice transverse to a component of the singular locus. In the course of working out these recursions, we have discovered what appears to be a new way to express the monodromy associated to the Milnor fibration of a singular plane curve.
We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.
Let f and g be holomorphic function-germs vanishing at the origin of a complex analytic germ of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ given by the multiplication of f by the conjugate of g has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ 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 given by the multiplication…
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 ext…
This article and its successor concern the topology of real isolated hypersurface singularities. We prove that after attaching a certain number of handles the real Milnor fibres become contractible, with each handle corresponding to a critical point of a morsification; in particular one recovers the classical formula of Khimshiashvili for the Euler characteristic of the Milnor fibres. We then give sufficient conditions for having that the integer homology groups of the real Milnor fibres are isomorphic to the homology groups of a bouquet of spheres. This is followed by a discussion on real van…
The Euler obstruction of a function can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we give a version of the Lê-Greuel formula for two germs of analytic functions with isolated singularity at the origin on a singular space. Using this formula and results of Loeser, we also present an integral formula for the Euler obstruction of a function, generalizing a formula of Kennedy.
Let Y be a hypersurface in projective space having only ordinary double points as singularities. We prove a variant of a conjecture of L. Wotzlaw on an algebraic description of the graded quotients of the Hodge filtration on the top cohomology of the complement of Y except for certain degrees of the graded quotients, as well as its extension to the Milnor cohomology of a defining polynomial of Y for degrees a little bit lower than the middle. These partially generalize theorems of Griffiths and Steenbrink in the Y smooth case, and enable us to determine the structure of the pole order spectral…
Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\geq 0$ we consider the Milnor number $μ(f)$, the delta invariant $δ(f)$ and the number $r(f)$ of its irreducible components. Put $\bar μ(f)=2δ(f)-r(f)+1$. If $p=0$ then $\bar μ(f)=μ(f)$ (the Milnor formula). If $p>0$ then $μ(f)$ is not an invariant and $\bar μ(f)$ plays the role of $μ(f)$. Let $\mathcal N_f$ be the Newton polygon of $f$. We define the numbers $μ(\mathcal N_{f})$ and $r(\mathcal N_{f})$ which can be computed by explicit formulas. The aim of this note is to give a simple proof of the ine…
In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and …
Differently from the Milnor number, no formula relating the Tjurina number of a reducible algebroid curve to invariants of its branches was known. The aim of this work is to provide such a formula for a complete intersection algebroid curve with several branches in terms of invariants of its branches and the maximal points of the set of values of Kähler differentials on components of the curve.
Let $X$ be a nonsingular complex variety and $D$ a reduced effective divisor in $X$. In this paper we study the conditions under which the formula $c_{SM}(1_U)=c(\textup{Der}_X(-\log D))\cap [X]$ is true. We prove that this formula is equivalent to a Riemann-Roch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of $D$. We also show the Rimann-Roch type of formula is true if the Jacobian scheme of $D$ is nonsingular or a complete intersection.
We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of…
We study singularities f in K[[x_1,...,x_n]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positive characteristic. Moreover, we consider different non-degeneracy conditions of Kouchnirenko, Wall and Beelen-Pellikaan in positive characteristic, and we show that planar Newton non-degenerate singularities satisfy Milnor's formula mu=2 delta-r+1. Th…
Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres of a real weighted-homogeneous polynomial and a real version of the L{ê}-Iomdine formula. We have also included some results of the same flavor on the local topology of locally closed definable sets.
