Milnor Fiber Research Articles

Motivic Milnor fibers of a rational function

In this paper, following the approach of Denef and Loeser, we define motivic Milnor fibers of a rational function corresponding to its topological Milnor fibers considered by Gusein-Zade, Luengo and Melle–Hernández.

Local Invariants on Quotient Singularities and a Genus Formula for Weighted Plane Curves

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 {\delta}-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 {\delta}-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 defined on a weighted projective plane in terms of its degree and the local type of its singularities.

Lê cycles and Milnor classes

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Le cycles of Z; and vice versa: The Le cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.

ERRATUM: Milnor Fibrations and the Thom Property for maps f \bar g

ERRATUM: Milnor Fibrations and the Thom Property for maps f \bar g

Milnor fibers of real line arrangements

We study Milnor fibers of complexified real line arrangements. We give a new algorithm computing monodromy eigenspaces of the first cohomology. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enables us to generalize a vanishing result of Libgober, give new upper-bounds and characterize the $A_3$-arrangement in terms of non-triviality of Milnor monodromy.

Lagrangian homology spheres in (Am) Milnor fibres via ℂ∗–equivariantA∞–modules

We establish restrictions on Lagrangian embeddings of rational homology spheres into certain open symplectic manifolds, namely the (A_m) Milnor fibres of odd complex dimension. This relies on general considerations about equivariant objects in module categories (which may be applicable in other situations as well), as well as results of Ishii-Uehara and Ishii-Ueda-Uehara concerning the derived categories of coherent sheaves on the resolutions of (A_m) surface singularities.

Nearby motives and motivic nearby cycles

We prove that the construction of motivic nearby cycles, introduced by Jan Denef and Francois Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let $$k$$ be an arbitrary field of characteristic zero, and let $$X$$ be a smooth quasi-projective $$k$$ -scheme. Precisely, we show that, in the Grothendieck group of constructible etale motives, the image of the nearby motive associated with a flat morphism of $$k$$ -schemes $$f:X\rightarrow \mathbb A ^1_k$$ , in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with $$f$$ . In particular, it provides a realization of the motivic Milnor fiber of $$f$$ in the “non-virtual” motivic world.

DEFORMATION OF SINGULARITIES AND THE HOMOLOGY OF INTERSECTION SPACES

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface. Regardless of monodromy, the middle degree homology of intersection spaces is always a subspace of the homology of the deformation, yet itself contains the middle intersection homology group, the ordinary homology of the singular space, and the ordinary homology of the regular part.

On the Milnor fibre and Lê numbers of semi-weighted homogeneous arrangements

Inthispaperwestudygermsofpolynomialsformedbytheproductofsemi-weighted homogeneous polynomials of the same type, which we call semi-weighted homogeneous arrangements. It is shown how the Le numbers of such polynomials are computed using only their weights and degree of homogeneity. A key point of the main theorem is to find the number called polar ratio of this polynomial class. An important consequence is the description of the Euler characteristic of the Milnor fibre of such arrangements only depending on their weights and degree of homogeneity.

Duality of Gauß–Manin systems associated to linear free divisors

We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these \({\mathcal{D}}\) -modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Motivic Milnor fiber of a quasi-ordinary hypersurface

Abstract. Let f : ( 𝐂 d + 1 , 0 ) → ( 𝐂 , 0 ) $f \hspace{-1.0pt}:\hspace{-1.0pt}(\mathbf {C}^{d+1}, 0) \hspace{-1.0pt}\rightarrow \hspace{-1.0pt}(\mathbf {C},0)$ be a germ of complex analytic function such that its zero level defines an irreducible germ of quasi-ordinary hypersurface ( S , 0 ) ⊂ ( 𝐂 d + 1 , 0 ) $(S, 0) \subset (\mathbf {C}^{d+1}, 0)$ . We describe the motivic Igusa zeta function, the motivic Milnor fibre and the Hodge–Steenbrink spectrum of f at 0 in terms of topological invariants of ( S , 0 ) ⊂ ( 𝐂 d + 1 , 0 ) $(S, 0) \subset (\mathbf {C}^{d+1}, 0)$ .

Motivic Milnor fibers of a rational function

Let P and Q be two complex polynomials and f be the induced rational function. In this Note we define a motivic Milnor fiber of the germ of f at an indeterminacy point x for a value a, a motivic Milnor fiber of f for a value a and finally motivic bifurcation sets.

Equivalence of real Milnor fibrations for quasi-homogeneous singularities

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in order to do that, we explicitly characterize the critical points of projection $\frac{f}{\|f\|}:S_{\epsilon}^{m}\setminus K_{\epsilon}\to S^{1}$, where $K_{\epsilon}$ is the link of singularity.

A Note on Exceptional Unimodal Singularities and K3 Surfaces

This is a short note on the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived category of K3 surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by Ebeling and Ploog (arXiv:0809.2738).

Monodromy at Infinity of Polynomial Maps and Newton Polyhedra (with an Appendix by C. Sabbah)

By introducing motivic Milnor fibers at infinity of polynomial maps, we propose some methods for the study of nilpotent parts of monodromies at infinity. The numbers of Jordan blocks in the monodromy at infinity will be described by the Newton polyhedron at infinity of the polynomial.

A remark on vanishing cycles with two strata

Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a space with an isolated singular point at the origin 0, and that the Milnor number of f restricted to normal slices of Σ−{0} is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of f at 0, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.

Orbifold Euler Characteristics for Dual Invertible Polynomials

AbstractTo construct mirror symmetric Landau–Ginzburg models, P. Berg-lund, T. Hu¨bsch and M. Henningson considered a pair (f,G) consistingof an invertible polynomial f and an abelian group Gof its symmetriestogether with a dual pair (f,e Ge). Here we study the reduced orbifoldEuler characteristics of the Milnor fibres of f and fewith the actions ofthe groups G and Ge respectively and show that they coincide up to asign. Introduction P. Berglund and T. Hu¨bsch [4] proposed a method to construct some mirrorsymmetric pairs of manifolds. Their construction involves a polynomial f ofa special form, a so called (non-degenerate) invertibleone, and its Berglund–Hu¨bschtransposefe. In their paper, these polynomials appeared as potentialsof Landau–Ginzburg models. The construction of [4] was generalized in [3]to orbifold Landau–Ginzburg models described by pairs (f,G), where f is a(non-degenerate) invertible polynomial and G is a (finite) abelian group ofsymmetries of f. For a pair (f,G) one defines the dual pair (f,e Ge). In [3, 10],there were described some symmetries between invariants of the pairs (f,G)and (f,e Ge) corresponding to the orbifolds defined by the equations f = 0 andfe= 0 in weighted projective spaces. Moreover, in [7, 8, 14], it was observedthat the singularities defined by f and fehave some duality properties.One can consider the Milnor fibre V

Invariant Symmetries of Unimodal Function Singularities

We classify finite order symmetries g of the 14 exceptional unimodal function singularities f in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of f should not be contained in one eigenspace of g?. We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising. One of the most famous classical results in singularity theory is the Arnold and Brieskorn discovery of the close relationship between simple function singularities and Weyl groups Aμ, Dμ, Eμ [1, 6]. A few years after it, Arnold extended the relationship to simple singularities with the Z2 reflection symmetry and Weyl groups Bμ, Cμ, F4 [2] (see also Slodowy’s book [24]). Consideration of Zm symmetries of simple functions led in [10, 11, 12, 25] to the appearance of Shephard-Todd groups within function singularities. The emphasis there was on realisations of the complex reflection groups as equivariant monodromy groups acting on the appropriate character subspaces in the homology of invariant Milnor fibres, and on the diffeomorphisms between the discriminants of the reflection groups and of the Zm-equivariant functions. A further series of papers [13, 14, 15], on cyclic symmetries of the parabolic functions, brought in similar singularity realisations of certain complex crystallographic groups [22]. In this paper, we are naturally expanding the programme to cyclic symmetries of the 14 exceptional unimodal function singularities on one hand, and complex hyperbolic reflection groups on the other. The basic idea is as follows. In the 3-variable case, the intersection form on the vanishing homology of an exceptional unimodal function f is non-degenerate and has positive signature 2. Assume g is an automorphism of C of finite order m, and our function is g-invariant. Then g acts on the second homology of the Milnor fibre f−1(e), and decomposes it into a direct sum of the character subspaces Hχ, χ = 1, on which g acts as multiplication by χ. Assume the rank 2 positive subspace of the intersection form splits between two character summands. Then the monodromy within a g-invariant versal deformation of f acts as a complex hyperbolic reflection group on each of them. Developing further the technique introduced in papers on cyclically symmetric functions [10, 11, 12], we construct vanishing bases in the hyperbolic summands and obtain the generating reflections as the corresponding Picard-Lefschetz operators. The main result of the paper is a complete classification of the invariant symmetries of the 14 singularities, which split the positive subspace in the vanishing homology, and the description – via constructing the corresponding Dynkin diagrams – of the complex hyperbolic groups arising. All the rank 2 reflection groups obtained projectivise to the triangle groups of the Poincare disk. The task of identification of higher dimensional groups is left for a future paper, along with the consideration of the equivariant symmetry setting. It should be noted that it is the first time when complex hyperbolic reflection groups are appearing in a singularity theory context. The approach introduced may be useful for constructing new complex hyperbolic lattices (cf. [8, 20]). The paper is organised as follows. Section 1 introduces the notion of singularities with symmetry, recalls the definitions and constructions given in [10, 11, 12]. Section 2 contains classification of splitting invariant symmetries of the 14 singularities. In Section 3.3 we construct Dynkin diagrams of the hyperbolic monodromy groups associated with the symmetric functions. Projectivisations of the rank 2 monodromy groups are considered in Section 4. More details of the constructions may be found in [16].