Milnor Fiber Research Articles

Indices of vector fields for mixed functions

A mixed function is a real analytic map f:Cn→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {C}^n\\rightarrow \\mathbb {C}$$\\end{document} in the complex variables z1,⋯,zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_1,\\dots ,z_n$$\\end{document} and their conjugates z¯1,⋯,z¯n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{z}_1,\\dots ,\\bar{z}_n$$\\end{document}. In this article, we define an integer valued index for vector fields v with isolated singularity at 0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{0}$$\\end{document} on real analytic varieties Vf:=f-1(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_f:=f^{-1}(0)$$\\end{document} defined by mixed functions f with isolated critical point at 0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{0}$$\\end{document}. We call this index the mixed GSV index and it generalizes the classical GSV index defined by Gomez-Mont, Seade and Verjovsky in (Math Ann 291(4):737–751, 1991), i.e., if the function f is holomorphic, then the mixed GSV index coincides with the GSV index. Furthermore, the mixed GSV index is a lifting to Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document} of the Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_2$$\\end{document}-valued real GSV index defined by Aguilar, Seade and Verjovsky in Aguilar et al. (J Reine Angew Math 504:159–176, 1998). As applications, we prove that the mixed GSV index is equal to the Poincaré–Hopf index of v on a Milnor fiber. If f also satisfies the strong Milnor condition, i. e., for every ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} (small enough), the map f‖f‖:Sϵ\\Lf→S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{f}{\\Vert f\\Vert }:\\mathbb {S}_\\epsilon {\\setminus } L_f \\rightarrow \\mathbb {S}^1$$\\end{document} is a fiber bundle, we prove that the mixed GSV index is equal to the curvatura integra of f defined by Cisneros-Molina, Grulha and Seade in (Int J Math 25(7): 1450069, 2014) based on the curvatura integra defined by Kervaire in (Courbure integrale generalisee et homotopie., Mathematische Annalen, pp. 219–252, 1956).

Exotic families of symplectic manifolds with Milnor fibers of ADE-type

Exotic families of symplectic manifolds with Milnor fibers of ADE-type

Characteristic cohomology II: Matrix singularities

AbstractFor a germ of a variety , a singularity of “type ” is given by a germ , which is transverse to in an appropriate sense, such that . In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for a hypersurface), and complement and link (for the general case). It captures the cohomology of inherited from and is given by subalgebras of the cohomology for for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences for Milnor fibers and for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.In this paper, we apply these methods in the case denotes any of the varieties of singular complex matrices, which may be either general, symmetric, or skew‐symmetric (with even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds” for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general complex matrices.Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size ” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ containing such a kite map germ of size . Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.

On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

Milnor fibration theorem for differentiable maps

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps f:(Rn,0)→(Rk,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:({\\mathbb {R}}^n,0) \\rightarrow ({\\mathbb {R}}^k,0)$$\\end{document}, where n≥k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge k\\ge 2$$\\end{document}, with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class Cℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\ell }$$\\end{document}, ℓ≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell \\ge 2$$\\end{document}, with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class Cℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\ell }$$\\end{document} between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.

Fibration theorems à la Milnor for analytic maps with non-isolated singularities

AbstractWe study the topology of real analytic maps in a neighborhood of a (possibly non-isolated) critical point. We prove fibration theorems à la Milnor for real analytic maps with non-isolated critical values. Here we study the situation for maps with arbitrary critical set. We use the concept of d-regularity introduced in an earlier paper for maps with an isolated critical value. We prove that this is the key point for the existence of a Milnor fibration on the sphere in the general setting. Plenty of examples are discussed along the text, particularly the interesting family of functions $$(f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2$$ ( f , g ) : R n → R 2 of the type $$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p, \sum _{i=1}^n b_i x_i^q \right) , \end{aligned}$$ ( f , g ) = ∑ i = 1 n a i x i p , ∑ i = 1 n b i x i q , where $$a_i, b_i \in {\mathbb {R}}$$ a i , b i ∈ R are constants in generic position and $$p,q \ge 2$$ p , q ≥ 2 are integers.

ℤ-local system cohomology of hyperplane arrangements and a Cohen–Dimca–Orlik type theorem

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering spaces. One of the important theorems is the vanishing theorem for generic [Formula: see text]-local systems which goes back to Aomoto’s work. Later, Cohen, Dimca, and Orlik proved a stronger version of the vanishing theorem. In this paper, we prove a Cohen–Dimca–Orlik type theorem for [Formula: see text]-local systems.

Motivic zeta functions of hyperplane arrangements

AbstractFor each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$ . We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.

Harmonic morphisms and their Milnor fibrations

Harmonic morphisms and their Milnor fibrations

On the homology of the noncrossing partition lattice and the Milnor fibre

Let L be the noncrossing partition lattice associated to a finite Coxeter group W. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of L. We define a multiplicative structure on the Whitney homology of L in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of W, the Milnor fibre of the discriminant of W, the hyperplane complement of W and the Artin group of type W, respectively. We also tabulate some computational results on the integral homology of the Milnor fibres.

Topology of first integrals via Milnor fibrations

Topology of first integrals via Milnor fibrations

Homological mirror symmetry for log Calabi–Yau surfaces

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.

On the Topology of the Milnor Fibration

In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.

ON THE MILNOR FIBRATION OF CERTAIN NEWTON DEGENERATE FUNCTIONS

AbstractIt is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots , f^{k_0}$ if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ is a nondegenerate complete intersection variety for any $k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$ .

Asymptotic lines on real Milnor links of a family that realizes any genus

We describe the global foliation of asymptotic lines on compact surfaces of genus greater than one, defined in R4 as the links of the real part of the Milnor fibration of a family of polynomials. Through this analysis, we present a description of the global foliation of lines of curvature of the embeddings, defined as the images of the stereographic projection of these links into R3.

Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Abstract In previous work jointly with Geske, Maxim, and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of (one variable) Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue $1$ can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space. Further, we show a formula relating the Alexander module’s Hodge numbers to those of finite covers of the base space, under some assumptions.Dedicated to the memory of Georgia Benkart

Systolic inequalities, Ginzburg dg algebras and Milnor fibers

Systolic inequalities, Ginzburg dg algebras and Milnor fibers

Homological mirror symmetry for Milnor fibers via moduli of A∞$A_\infty$‐structures

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A ∞ $A_\infty$ -structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n $n$ -dimensional affine hypersurface of degree n + 2 $n+2$ and the double cover of the n $n$ -dimensional affine space branched along a degree 2 n + 2 $2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest.

Vanishing cycle control by the lowest degree stalk cohomology

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the reduced cohomology of the Milnor fiber in lowest possible non-trivial degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant.