We show how some of our recent results clarify the relationship between the Lê numbers and the cohomology of the Milnor fiber of a non-isolated hypersurface singularity. The Lê numbers are actually the ranks of the free Abelian groups -- the Lê modules -- appearing in a complex whose cohomology is that of the Milnor fiber. Moreover, the Milnor monodromy acts on the Lê module complex, and we describe the traces of these monodromy actions in terms of the topology of the critical locus.
We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising Déglise's thesis. As a corollary, we recover a theorem of Ananyevskiy and Neshitov and we prove that the Milnor-Witt K-theory groups are birational invariants.
We generalize Rost's theory of cycle modules using Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups. The usual constructions are developed: proper pushfoward, (essentially) smooth pullback, long exact sequences, (coniveau) spectral sequences and products, as well as the homotopy invariance property; in addition, Gysin morphisms for lci maps are constructed. We prove an adjunction theorem linking our theory to Rost's.
The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R…
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…
We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson--Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity of the Hilbert scheme as the degeneracy locus of the Pfaffian function.
In the paper we use the theory of framed correpondences to construct Milnor-Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $\mathbb{A}^1$-homotopy sheaves of smooth varieties with the zeroth homology of corresponding MW-motivic complexes and prove that the hearts of homotopy $t$-structures on the stable $\mathbb{A}^1$-derived category and the category of Milnor-Witt motives are equivalent.
We compute the perverse delta-homotopy heart of the motivic stable homotopy category over a base scheme with a dimension function delta, rationally or after inverting the exponential characteristic in the equicharacteristic case. In order to do that, we define the notion of homological Milnor-Witt cycle modules and construct a homotopy-invariant Rost-Schmid cycle complex. Moreover, we define the category of cohomological Milnor-Witt cycle modules and show a duality result in the smooth case.
We prove some sufficient conditions in order that a root of the Bernstein-Sato polynomial contributes to a difference between certain D-modules generated by rational powers of a holomorphic function; for instance, this holds in the case of isolated singularities with semisimple Milnor monodromies. We then construct an example where a root does not contribute to a difference. This also solves an old open problem about the relation between the Milnor monodromy and the exponential of the residue of the Gauss-Manin connection on the saturation of the Brieskorn lattice. This shows that the structur…
We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, th…
In previous work jointly with Geske, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of 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 ge…
In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an ext…
Given a cycle module M with a ring structure we show that the cycle complex with coefficients in M of a smooth scheme of finite type over a field has a A-infinity algebra structure. In the case of Milnor K-theory this gives a homotopy model for the classical intersection theory of algebraic cycles.
We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $Γ$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection groups of rank two, we establish some general results which relate the isotypic componenents of the monodromy on the cohomology, to the Hodge structure and to the cohomology degree. Using eigenspace theory for reflection groups, we prove some sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler c…
We compute the invariants so of Weyl groups in mod 2 Milnor K-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the coefficient module. All basis elements are induced either by Stiefel-Whitney classes or specific invariants in the Witt ring. The proof is based on Serre's splitting principle that guarantees detection of invariants on elementary abelian 2-subgroups generated by reflections.
We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, ins…
Let $\mathcal{V} \subset M$ denote any of the varieties of singular $m \times m$ complex matrices which may be general, symmetric, or skew-symmetric ($m$ even), or $m \times p$ matrices, in the corresponding space $M$ of such matrices. A "matrix singularity", $\mathcal{V}_0$ of "type $\mathcal{V}$", for any of the $\mathcal{V} \subset M$ is defined as $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$ by a germ $f_0 : \mathbb C^n, 0 \to M, 0$ (appropriately transverse to $\mathcal{V}$). In part I of this paper we introduced the notion of characteristic cohomology for a singularity $\mathcal{V}_0$ of type…
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 fibers of the restriction of the function on t…
We establish a trace formula for rigid varieties $X$ over a complete discretely valued field, which relates the set of unramified points on $X$ to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring $R$, and we introduce the Weil generating series of a regular formal $R$-scheme $\mathfrak{X}$ of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When $\mathfr…
Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pa…
The aim of this Note is to specify the links between the three kinds of Lisbon integrals, trace functions and trace forms with the corresponding D--modules.
For any finitely generated module $M$ with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant $\operatorname{h}(M)$ based on a partial trace ideal of $M$. We study its properties and explore relations between this invariant and the colength of the conductor. Finally we apply this to the universally finite module of differentials $Ω_{R/k}$, where $R$ is a complete $k$-algebra with $k$ any perfect field, to study a long-standing conjecture due to R. W. Berger.
We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive gr…
We study the interplay of the homotopy coniveau tower, the Rost-Schmid complex of a strictly homotopy invariant sheaf, and homotopy modules. For a strictly homotopy invariant sheaf $M$, smooth $k$-scheme $X$ and $q \geqslant 0$ we construct a novel cycle complex $C^*(X, M, q)$ and we prove that in favorable cases, $C^*(X, M, q)$ is equivalent to the homotopy coniveau tower $M^{(q)}(X)$. To do so we establish moving lemmas for the Rost-Schmid complex. As an application we deduce a cycle complex model for Milnor-Witt motivic cohomology. Furthermore we prove that if $M$ is a strictly homotopy inv…
To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e., distributions invariant under Hamiltonian flows. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. As an application, we deduce that noncommutative filtered algebras whose associated graded algebra…
Using his deep and beautiful idea of cutting with a Hyperplane, Lefschetz explained how the homology groups of a projective smooth variety could be constructed from basic pieces, that he called primitive homology. This idea can be applied every time we have a vector space with a biliner form (i.e. homology with cup product) and a $\pm$-symmetric nilpotent operator (i.e. cutting with a hyperplane). We will illustrate this in the context of Singularity Theory: A germ of an isolated singular point of a hypersurface defined by $f=0$. We begin with the algebraic setting in the Jacobian (or Milnor) …
We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities C^{2n-2}/S_n associated to the type A Weyl group S_n and its reflection representation C^{n-1}. We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional ir…
We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.
In this paper we consider reduced (non-normal) commutative noetherian rings $R$. With the help of conductor ideals and trace ideals of certain $R$-modules we deduce a criterion for a reflexive $R$-module to be closed under multiplication with scalars in an integral extension of $R$. Using results of Greuel and Knörrer this yields a characterization of plane curves of finite Cohen--Macaulay type in terms of trace ideals. Further, we study one-dimensional local rings $(R,\mathfrak{m})$ such that that their normalization is isomorphic to the endomorphism ring $\mathrm{End}_R(\mathfrak{m})$: we gi…
