A log generic hypersurface in $\mathbb{P}^n$ with respect to a birational modification of $\mathbb{P}^n$ is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product $f=f_1\ldots f_p$ of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for the tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log very-generic.
In this note we prove that the crepant transformation conjecture for a crepant birational transformation of Lawrence toric DM stacks studied in \cite{CIJ} implies the monodromy conjecture for the associated wall crossing of the symplectic resolutions of hypertoric stacks, due to Braverman, Maulik and Okounkov.
A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced in [GZLM1]. In this article we define the topological zeta function for meromorphic germs and we study its poles in the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.
This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding s…
The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πi c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every pole is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: -n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement o…
We study Deligne's conjecture on the monodromy weight filtration on the nearby cycles in the mixed characteristic case, and reduce it to the nondegeneracy of certain pairings in the semistable case. We also prove a related conjecture of Rapoport and Zink which uses only the image of the Cech restriction morphism, if Deligne's conjecture holds for a general hyperplane section. In general we show that Deligne's conjecture is true if the standard conjectures hold.
The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of thr…
Deligne's weight-monodromy conjecture gives control over the poles of local factors of L-functions of varieties at places of bad reduction. His proof in characteristic p was a step in his proof of the generalized Weil conjectures. Scholze developed the theory of perfectoid spaces to transfer Deligne's proof to characteristic 0, proving the conjecture for complete intersections in toric varieties. Building on Scholze's techniques, we prove the weight-monodromy conjecture for complete intersections in abelian varieties.
We prove the weight-monodromy conjecture for varieties which are p-adically uniformized by a product of the Drinfeld upper half spaces. It is an easy consequence of Dat's work on the cohomology complex of the Drinfeld upper half space.
The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A'Campo) to compute the 'Verdier monodromy' eigenvalues associated to an ideal. Afterwards we prove a generalized monodromy conjecture for arbitrary ideals in two variabl…
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated surface singularity. We start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f in Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of the complex zero locus of f close to the origin. Essentially the entire paper is…
We prove for abelian varieties a global form of Denef and Loeser's motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety $A$ over a complete discretely valued field, its motivic zeta function has a unique pole at Chai's base change conductor $c(A)$ of $A$, and that the order of this pole equals one plus the potential toric rank of $A$. Moreover, we show that for every embedding of $\Q_\ell$ in $\C$, the value $\exp(2πi c(A))$ is an $\ell$-adic tame monodromy eigenvalue of $A$. The main tool in the paper is Edixhoven's…
Monodromy Eigenvalues of Milnor Fibers for Line Arrangements
Qiuzhen College, Tsinghua University（邱耀学院，清华大学）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉科学研究院）；Tsinghua University（清华大学）
We speculate on the relationship between the solvable monodromy extension (SME) property and log canonical models. A motivating example is the moduli space of smooth curves which, by earlier work, is known to have this SME property. In this case the maximal SME compactification is the moduli space of stable nodal curves which coincides with its log canonical model.
We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an application, we study nearby monodromy eigenvalues -- the eigenvalues of the monodromy operators of singularities, adjacent to a given singularity of a holomorphic function $f$. More precisely, we express some of such values in terms of certain resolutions of $f$. The expression is given it terms of the exceptional divisor strata of arbitrary codimension, general…
Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of $X$. This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, étale and $p$-adic settings). In the specific case in which $k=\mathbb C$ and $X$ is the central fiber of a semistable degeneration over the complex disc, our …
We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.
A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the $d^2$ differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer etale topology. In the case of curves, these data a…
This paper is a sequel to [He7]. There a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. One can consider it as a global $μ$-constant stratum or as a Teichmüller space for singularities. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_Z$. Here $G_Z$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_…
The aim of this paper is to prove the weight-monodromy conjecture (Deligne's conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theor…
In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature conjecture stated and proved by Esnault and Langer in "On a positive equicharacteristic variant of the $p$-curvature conjecture" (Doc. Math. 18 (2013)). The main result is that if the ground field is uncountable then the strong form holds. In the case where the ground field is countable we provide positive and negative answers to possible generalizations.
We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the size of the maximal Jordan block of the corresponding monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the fundamental invariants appearing in the proof.
The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic …
We prove the Morrison--Kawamata cone conjecture for projective primitive symplectic varieties with $\Q$-factorial and terminal singularities with $b_2\geq 5$, from which we derive for instance the finiteness of minimal models of such varieties, up to isomorphisms. To prove the conjecture we establish along the way some results on the monodromy group which may be interesting in their own right, such as the fact that reflections in prime exceptional divisors are integral Hodge monodromy operators which, together with monodromy operators provided by birational transformations, yield a semidirect …
mu-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo {id,-id}. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these modu…
The aim of this paper is to study certain properties of the weight spectral sequences of Rapoport-Zink by a specialization argument. By reducing to the case over finite fields previously treated by Deligne, we prove that the weight filtration and the monodromy filtration defined on the $l$-adic étale cohomology coincide, up to shift, for proper smooth varieties over equal characteristic local fields. We also prove that the weight spectral sequences degenerate at $E_2$ in any characteristic without using log geometry. Moreover, as an application, we give a modulo $p>0$ reduction proof of a Hodg…
In 1982, Yano proposed a conjecture predicting the $b$-exponents of an irreducible plane curve singularity which is generic in its equisingularity class. In this article we prove the conjecture for the case of two Puiseux pairs and monodromy with distinct eigenvalues. The hypothesis on the monodromy implies that the $b$-exponents coincide with the opposite of the roots of the Bernstein polynomial, and we compute the roots of the Bernstein polynomial.
The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with a complex analytic function germ f defined on a normal surface singularity (X,0). The article targets the `right' extension in the case when the link of (X,0) is a homology sphere. As a first step, we prove a spl…
The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${\mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.
We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.
This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Als…
We show that complex local systems with quasi-unipotent monodromy at infinity over a normal complex variety are Zariski dense in their moduli. v2: we waited for feedback and added a consequence of Alexandr Petrov's theorem. 3: we tightened the last section. Final version: appears in Israel Journal of Mathematics. footnote added to Conjecture 1.1: Aaron Landesman and Daniel Litt just made available a preprint showing that there is a lower bound for the rank of geometric local systems with infinite mon-odromy on certain curves, and consequently the conjecture can not be true in this generality.
We investigate an analogue of the Grothendieck $p$-curvature conjecture, where the vanishing of the $p$-curvature is replaced by the stronger condition, that the module with connection mod $p$ underlies a $\mathcal{D}_X$-module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a $\mathbb{Z}$-variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod $p$ Tannakian fundmental groups, which in particular resolves in the …
Let U be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $ρ$ of $π_1(U)$ corresponds to an étale $F$-isocrystal. By work of Tsuzuki and Crew an $F$-isocrystal is overconvergent precisely when $ρ$ has finite monodromy at each $x \in X-U$. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth $F$-isocrystals …
We study $A$-hypergeometric systems $H_A(β)$ in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove first that rank-jumping parameters always correspond to reducible systems, and we show that the property of being reducible is ``invariant modulo the lattice''. In the second part we study a conjecture of Nobuki Takayama which states that the holonomic dual of $H_A(β)$ is of the form $H_A(β')$ for suitable $β'$. We prove the conjecture for all matrices $A$ and generic parameter $β$, exh…
The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom-Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of ${\mathbb Z}$-lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminant…
