Milnor Fiber Research Articles

Homological mirror symmetry for hypersurface cusp singularities

We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface Y_{p,q,r}. By using localization techniques on both sides, we get an isomorphism between the derived wrapped Fukaya category of the Milnor fibre and the derived category of coherent sheaves on a quasi-projective surface given by deleting an anti-canonical divisor D from Y_{p,q,r}. In the cusp case, the pair (Y_{p,q,r}, D) is naturally associated to the dual cusp singularity, tying into Gross, Hacking and Keel’s proof of Looijenga’s conjecture.

Motivic Milnor Fibers of Plane Curve Singularities

We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the Hodge–Steenbrink spectrum of such a singularity in terms of that of a quasi-homogeneous singularity.

Homology graph of real arrangements and monodromy of Milnor fiber

We study the first homology group H1(F,C) of the Milnor fiber F of sharp arrangements A‾ in PR2. Our work relies on the minimal complex C⁎(S(A)) of the deconing arrangement A and its boundary map. We describe an algorithm which computes possible eigenvalues of the monodromy operator h1 of H1(F,C). We prove that, if a condition on some intersection points of lines in A is satisfied, then the only possible nontrivial eigenvalues of h1 are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.

Formulas for monodromy

Given a family X of complex varieties degenerating over a punctured disk, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of X. Our first main result is that the motivic nearby fiber of X can be computed by first stratifying X into locally closed subvarieties that are nondegenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when X is a family of nondegenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.

The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

Given any connected topological space X, assume that there exists an epimorphism \(\phi {:}\; \pi _1(X) \rightarrow {\mathbb {Z}}\). The deck transformation group \({\mathbb {Z}}\) acts on the associated infinite cyclic cover \(X^\phi \) of X, hence on the homology group \(H_i(X^\phi , {\mathbb {C}})\). This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring \({\mathbb {C}}[t,t^{-1}]\), which is a finite dimensional \({\mathbb {C}}\)-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact Kahler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration.

On mixed polynomials of bidegree (n,1)

Specifying the bidegrees (n,m) of mixed polynomials P(z,z¯) of the single complex variable z, with complex coefficients, allows to investigate interesting roots structures and counting; intermediate between complex and real algebra. Multivariate mixed polynomials appeared in recent papers dealing with Milnor fibrations, but in this paper we focus on the univariate case and m=1, which is closely related to the important subject of harmonic maps. Here we adapt, to this setting, two algorithms of computer algebra: Vandermonde interpolation and a bissection-exclusion method for root isolation. Implemented in Maple, they are used to explore some interesting classes of examples.

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

A central question in arrangement theory is to determine whether the characteristic polynomial Δ q of the algebraic monodromy acting on the homology group H q ( F ( A ) , C ) of the Milnor fiber of a complex hyperplane arrangement A is determined by the intersection lattice L ( A ) . Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ 1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers β p ( A ) , which in turn are extracted from L ( A ) . When A defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of A to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over C) for matroids, and estimate the number of essential components in the first complex resonance variety of A. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of SL 2 ( C ) -representation varieties, which are governed by the Maurer–Cartan equation.

Motivic Milnor fibre of cyclic L ∞-algebras

We define the motivic Milnor fiber of cyclic L∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the etale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.

Homology computations for complex braid groups II

We complete the computation of the integral homology of the generalized braid group B associated to an arbitrary irreducible complex reflection group W of exceptional type. In order to do this we explicitly computed the recursively-defined differential of a resolution of Z as a ZB-module, using parallel computing. We also deduce from this general computation the rational homology of the Milnor fiber of the singularity attached to most of these reflection groups.

Modular Equalities for Complex Reflection Arrangements

We compute the combinatorial Aomoto-Betti numbers $\beta_p(\mathcal{A})$ of a complex reflection arrangement. When $\mathcal{A}$ has rank at least $3$, we find that $\beta_p(\mathcal{A})\le 2$, for all primes $p$. Moreover, $\beta_p(\mathcal{A})=0$ if $p>3$, and $\beta_2(\mathcal{A})\ne 0$ if and only if $\mathcal{A}$ is the Hesse arrangement. We deduce that the multiplicity $e_d(\mathcal{A})$ of an order $d$ eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto-Betti number, when $d$ is prime. We give a uniform combinatorial characterization of the property $e_d(\mathcal{A})\ne 0$, for $2\le d\le 4$. We completely describe the monodromy action for full monomial arrangements of rank $3$ and $4$. We relate $e_d(\mathcal{A})$ and $\beta_p(\mathcal{A})$ to multinets, on an arbitrary arrangement.

Equisingular and equinormalizable deformations of isolated non-normal singularities

We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using $\delta$ and $\mu$. Moreover, we determine the number of connected components of the Milnor fibre of an arbitrary INNS. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families. Finally we state some open problems and conjectures. In addition we give a survey of classical results about equisingularity and equinormalizability so that the article may be useful as a reference source.

Motives of rigid analytic tubes and nearby motivic sheaves

Let k be a field of characteristic zero, R D k[[t]] the ring of formal power series and K D k((t)) its fraction field. Let X be a finite type R-scheme with smooth generic fiber. Let H be the t-adic completion of X and Hη the generic fiber of H. Let Z ⊂ Xσ bea locally closed subset of the special fiber of X. In this article, we establish a relation between the rigid motive of [Z] (the tube of Z in Hη) and the restriction to Z of the nearby motivic sheaf associated with the R-scheme X. Our main result, Theorem 7.1, can be interpreted as a motivic analog of a theorem of Berkovich. As an application, given a rational point x ∈ Xσ, we obtain an equality, in a suitable Grothendieck ring of motives, between the motivic Milnor fiber of Denef-Loeser at x and the class of the rigid motive of the analytic Milnor fiber of Nicaise-Sebag at x.

On the vanishing of Hochster’s 𝜃 invariant

Hochster's theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochster's theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation. In this paper, we give purely algebraic versions of some of these results. In particular, we establish the vanishing of the theta invariant for isolated hypersurface singularities of odd dimension in characteristic p > 0, under some mild extra assumptions. This confirms, in a large number of cases, a conjecture of Hailong Dao.

Perverse Results on Milnor Fibers inside Parameterized Hypersurfaces

We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. We also discuss applications to stable unfoldings of finite maps with isolated instabilities.

Vanishing polyhedron and collapsing map

In this paper we give a detailed proof of the fact that the Milnor fiber $$X_t$$ of an analytic complex isolated singularity function defined on a reduced n-equidimensional analytic complex space X is a regular neighborhood of a polyhedron $$P_t \subset X_t$$ of real dimension $$n-1$$ . Moreover, we describe the degeneration of $$X_t$$ onto the special fiber $$X_0$$ , by giving a continuous collapsing map $$\psi _t: X_t \rightarrow X_0$$ which sends $$P_t$$ to $$\{0\}$$ and which restricts to a homeomorphism $$X_t {\backslash } P_t \rightarrow X_0 {\backslash } \{0\}$$ .

Local systems on analytic germ complements

We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.

Arrangements of lines and monodromy of associated Milnor fibers

Consider an arrangement [Formula: see text] of homogeneous hyperplanes in [Formula: see text] with complement [Formula: see text]. The (co)homology of [Formula: see text] with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto [Formula: see text] endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of [Formula: see text] by the commutator of its length zero subgroup, which didn’t appear in the literature before. In Sec. 2, we state a conjecture of [Formula: see text]-monodromicity for the first homology, which is of a different nature with respect to the known results. Let [Formula: see text] be the graph of double points of [Formula: see text] we conjecture that if [Formula: see text] is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of [Formula: see text]. We show the truth of the conjecture under some stronger hypotheses.

A computational approach to Milnor fiber cohomology

Abstract In this paper we consider the Milnor fiber F associated to a reduced projective plane curve C. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of F, also known as the Alexander polynomial of the curve C, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

For a germ $f$ on a complex manifold $X$, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the $D$-module generated by $f^s$ to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of $f^s$ is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of $1/f$; we confirm in many cases a corresponding conjecture on the annihilator of $f^s$ but we disprove it in general; we show that the Bernstein--Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of $f^s$ is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.

On the Lê–Milnor fibration for real analytic maps

In this paper, we study the topology of real analytic map‐germs with isolated critical value , with . We compare the topology of f with the topology of the compositions , where are the projections , for . As a main result, we give necessary and sufficient conditions for f to have a Lê–Milnor fibration in the tube.