Motivic Milnor Fiber Research Articles

Motivic integration and Milnor fiber

We put forward a uniform narrative that weaves together several variants of Hrushovski– Kazhdan style integral, and describe how it can facilitate the understanding of the Denef–Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called “nonarchimedean Milnor fiber” that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski–Loeser construction. Then, applying T -convex integration after descent, matching the Euler characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski–Loeser proof, but is also free of other sophisticated algebro-geometric machineries. Finally, we also establish, in a much more intuitive manner, a new Thom–Sebastiani formula, which can be specialized to the one given by Guibert–Loeser–Merle.

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.

A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory

We present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semialgebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson–Thomas theory.

The Thom–Sebastiani theorem for the Euler characteristic of cyclic L∞-algebras

Let L be a cyclic L∞-algebra of dimension 3 with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic L∞-algebra structure on the cohomology H⁎(L). The inner product plus the higher products of the cyclic L∞-algebra defines a superpotential function f on H1(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 formula for the Euler characteristic of cyclic L∞-algebras. As applications we prove the Joyce–Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi–Yau threefolds. A motivic Thom–Sebastiani type formula and a conjectural motivic Joyce–Song formula for the motivic Milnor fiber of cyclic L∞-algebras are also discussed.

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.

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.

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.

Motivic infinite cyclic covers

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (under certain finiteness conditions) an element in the Grothendieck ring K0(VarCμˆ), which we call motivic infinite cyclic cover, and show its birational invariance. Our construction provides a unifying approach for the Denef–Loeser motivic Milnor fiber of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.

Monodromies at infinity of non-tame polynomials

AR Abstract. We consider the monodromy at infinity and the monodromies around the bifurcation points of polynomial functions f : C n −→ C which are not tame and might have non-isolated singularities. Our description of their Jordan blocks in terms of the Newton polyhedra and the motivic Milnor fibers relies on two new issues: the non- atypical eigenvalues of the monodromies and the corresponding concentration results for their generalized eigenspaces.

The motivic Thom–Sebastiani theorem for regular and formal functions

Abstract Thanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular functions. Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag, Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants.

Proofs of the integral identity conjecture over algebraically closed fields

Recently, it is well known that the conjectural integral identity is of crucial importance in the motivic Donaldson-Thomas invariants theory for non-commutative Calabi-Yau threefolds. The purpose of this article is to consider different versions of the identity, for regular functions and formal functions, and to give them the positive answer for the ground field algebraically closed. Technically, the result on motivic Milnor fiber by Hrushovski-Loeser using Hrushovski-Kazhdan's motivic integration and Nicaise's computations on motivic integrals on special formal schemes are main tools.

Monodromy and the Lefschetz fixed point formula

We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milnor fiber.

Motivic Milnor Fibers and Jordan Normal Forms of Milnor Monodromies

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.

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.

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.

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.

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.

Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré

Let k be a field of characteristic zero and P be a Laurent polynomial in d variables, with coefficients in k and non degenerate for its Newton polyhedron at infinity. Let (f l ) be d non constant functions with separated variables and defined on smooth varieties. As Guibert, Loeser and Merle in the local case, we compute in this article the motivic Milnor fiber at infinity of P(f) in terms of the Newton polyhedron at infinity of P. For P equal to the sum x 1 +x 2 , we obtained a Thom-Sebastiani formula. Then we can introduce a notion of motivic vanishing cycles of a function g for the infinite value denoted by S g∞,U Φ , and which verified, as in the local case, a convolution formula. In particular if g is the polynomial x 1 +...+x n +1/x 1 ...x n , we show that the spectrum S g,∞,U Φ is 1+t+..+t n which coincides with the spectrum at infinity of g considered by Douai and Sabbah.

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

We study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of by using the theory of motivic Milnor fibers. The results will be explicitly described by the mixed volumes of the faces of Newton polyhedrons.