Milnor Fiber Research Articles

Homological mirror symmetry of CPn and their products via Morse homotopy

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many examples, the derived category Db(coh(X)) of coherent sheaves on a toric manifold X is compared with the Fukaya–Seidel category of the Milnor fiber of the corresponding Landau–Ginzburg potential. We instead consider the dual torus fibration π: M → B of the complement of the toric divisors in X, where B̄ is the dual polytope of the toric manifold X. A natural formulation of homological mirror symmetry in this setup is to define Fuk(M̄) a variant of the Fukaya category and show the equivalence Db(coh(X))≃Db(Fuk(M̄)). As an intermediate step, we construct the category Mo(P) of weighted Morse homotopy on P≔B̄ as a natural generalization of the weighted Fukaya–Oh category proposed by Kontsevich and Soibelman [Symplectic geometry and mirror symmetry (Seoul, 2000) (World Science Publishing, 2001), p. 203]. We then show that a full subcategory MoE(P) of Mo(P) generates Db(coh(X)) for the cases X is a complex projective space and their products.

Newton transformations and motivic invariants at infinity of plane curves

In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial f in two variables with coefficients in an algebraic closed field of characteristic zero. This expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm at infinity of f without any condition of convenience or non degeneracy. In the complex setting, we compute the Euler characteristic of the generic fiber of f in terms of the area of the surfaces associated to faces of the Newton polygons. Furthermore, if f has isolated singularities, we compute similarly the classical invariants at infinity \(\lambda _{c}(f)\) which measures the non equisingularity at infinity of the fibers of f in \({\mathbb {P}}^2\), and we prove the equality between the topological and the motivic bifurcation sets and give an algorithm to compute them.

Hyperplane Arrangements and Mixed Hodge Numbers of the Milnor Fiber

Abstract For a 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 an 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 any simple normal crossing compactification of $F_{\mathcal{A}}$. As a direct consequence, we obtain a combinatorial formula for the top weight cohomology of $F_{\mathcal{A}}$, recovering a result of Dimca and Lehrer.

On the Milnor fibration for $f(\boldsymbol{z}) \bar{g}(\boldsymbol{z})$ II

We consider a mixed function of type $H(\boldsymbol{z}, \bar{\boldsymbol{z}}) = f(\boldsymbol{z}) \bar{g} (\boldsymbol{z})$ where $f$ and $g$ are holomorphic functions which are non-degenerate with respect to the Newton boundaries. We assume also that the variety $f = g = 0$ is a non-degenerate complete intersection variety. In our previous paper, we considered the case that $f, g$ are convenient so that they have isolated singularities. In this paper we do not assume the convenience of $f$ and $g$. In non-convenient case, two hypersurfaces may have non-isolated singularities at the origin. We will show that $H$ still has both a tubular and a spherical Milnor fibrations under the local tame non-degeneracy and the toric multiplicity condition. We also prove the equivalence of two fibrations.

Generalized chain surgeries and applications

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

Durfee-type inequality for complete intersection surface singularities

We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee's weak conjecture and a partial answer to Kerner--N\'emethi's conjecture.

Vanishing cycles of matrix singularities

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.

Geometrical Conditions for the Existence of a Milnor Vector Field

We introduce several sufficient conditions to guarantee the existence of the Milnor vector field for new classes of singularities of map germs. This special vector field is related with the equivalence problem of the Milnor fibrations for real and complex singularities, if they exit.

Geodesic fibrations for packing diabolic domains

We describe a theory of packing hyperboloid "diabolic" domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts," yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik–Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles.The purpose of this paper is to prove the algebraic counterpart of the Orlik–Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund–Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).

Boundary of the Milnor fiber of a Newton non-degenerate surface singularity

We give an algorithm for the description as a graph manifold of the boundary of the Milnor fiber of a Newton non-degenerate surface singularity embedded in a germ of toric three-dimensional variety, starting from its local Newton polyhedron. This algorithm is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any smoothing of a reduced non-isolated singularity of complex surface.

Relative logarithmic cohomology and Nambu structures of maximal degree

We present local classification results for isolated singularities of functions with respect to a Nambu structure (multi-vector field) of maximal degree, in a neighbourhood of a smooth point of its degeneracy hypersurface. The results depend on a logarithmic version of the Brieskorn-Sebastiani theorem, which guarantees the finiteness and freeness of the corresponding deformation module. This relates the functional moduli of the classification problem with the integrals of logarithmic forms along the vanishing cycles of the complement of the Milnor fibres of the restriction of the function on the degeneracy hypersurface of the Nambu structure, inside the Milnor fibres of the function itself.

Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic

P. Berglund, T. H\"ubsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.

Brasselet Number and Function-Germs with a One-Dimensional Critical Set

The Brasselet number of a function fhnonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, using the Brasselet number, we present several formulas for germs $$f:(X,0)\rightarrow (\mathbb {C},0)$$ and $$g:(X,0)\rightarrow (\mathbb {C},0)$$ in the case where g has a one-dimensional critical locus. We also give applications when f has isolated singularities and when it is a generic linear form.

Topology of smoothings of non-isolated singularities of complex surfaces

We give in this work an explicit combinatorial algorithm for the description of the Milnor fiber of a Newton non degenerate surface singularity as a graph manifold. This is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any reduced non isolated singularity of complex surface.

Results on Milnor Fibrations for Mixed Polynomials with Non-isolated Singularities

In this article we investigate mixed polynomials and present conditions that can be applied on a specific class of polynomials in order to prove the existence of the Milnor Fibration, Milnor–LeFibration and the equivalence between them. We prove for this class the of functions that the Milnor–Le fiber on a regular value is homeomorphic to the Milnor–Le fiber on a critical value. We develop a criterion to verify the transversality property and apply it to a special case of the class of mixed polynomial.

The boundary of the Milnor fiber of the singularity $$f(x,y) + zg(x,y) = 0$$

Let $$f,g\in \mathbb {C}\{x,y\}$$ be germs of functions defining plane curve singularities without common components in and let $$\Phi (x,y,z) = f(x,y) + zg(x,y)$$ . We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of $$\Phi $$ in terms of a common resolution for f and g. We give an example of a choice for f and g yielding a boundary of a Milnor fiber having more than one irreducible component.

Bouquet decomposition for Determinantal Milnor fibers

We provide a bouquet decomposition for the determinantal Milnor fiber of an essentially isolated determinantal singularity of arbitrary type $(m,n,t)$. The building blocks in the decomposition are (suspensions of) hyperplane sections of the associated generic determinantal variety $M_{m,n}^t$ in general position off the origin.

On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hubsch and M. Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and a finite abelian group $G$ of its diagonal symmetries and associated to this pair a dual pair $(\widetilde{f}, \widetilde{G})$. A. Takahashi suggested a generalization of this construction to pairs $(f, G)$ where $G$ is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group $G$: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair $(f,G)$. We show that, for an abelian group $G$, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group $G$ and the dual group $\widetilde{G}$ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

On the topology of non-isolated real singularities

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 Le-Iomdine formula. We have also included some results of the same flavor on the local topology of locally closed definable sets.