Given a complex analytic function with a one-dimensional critical locus at the origin, we examine the monodromy action on the integral cohomology of the Milnor fiber. We relate this monodromy to that of a generic hyperplane slice through the origin, and to that of a generic hyperplane slice near the origin. We thereby obtain number-theoretic restrictions on the monodromy and on the cohomology of the original Milnor fiber.
We show that there are obstructions to the existence of certain types of invariant subspaces of the Milnor monodromy; this places restrictions on the cohomology of Milnor fibres of non-isolated hypersurface singularities.
Monodromy of Calabi-Yau threefold flops via grade restriction rule and their quantum Kahler moduli
Korea Institute for Advanced Study（韩国高等研究院）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学交叉科学中心）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉科学研究院）
We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve $\cal Y$ with bi-degree (2,2) in a product of projective lines ${\Bbb P}^{1} \times {\Bbb P}^{1}$. We calculate two differenent monodromy representations of period integrals for the affine variety ${\cal X}^{(2,2)}$ obtained by the dual polyhedron mirror variety construction from $\cal Y$. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analyt…
Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial mod p^N. The proof involves an analysis of the action of the Galois group of a finitely generated field on the etale fundamental group of X. We also prove many arithmetic statements about fundamental groups which are of independent interest, and give several applications.
Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with res…
We prove several results on monodromies associated to Macdonald integrals, that were used in our previous work on the finite field analogue of a conjecture of Macdonald. We also give a new proof of our formula expressing recursively the zeta function of the local monodromy at the origin of the discriminant of a finite Coxeter group in terms of the degrees of the group.
We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the $\integ/\ell$ monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group $\sp_{2g}(\integ/\ell)$. We prove that the $\integ/\ell$ monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group $\su_{(r,s)}(\integ/\ell\tensor\integ[ζ_3])$.
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.
Using Mellin-Barnes integrals we give a method to compute a relevant subgroup of the monodromy group of an A-hypergeometric system of differential equations. Presumably this group is the full monodromy group of the system. This article is a major rewrite of an article posted 2 years ago.
We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If ther…
We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen to be equivalent to $X$ being a homology manifold. Given an analytic function $f$ on an intersection cohomology manifold, we describe a simple relation between $V(f)$ being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of $f$. We then describe how the Sebastiani-Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Fin…
The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of smooth curve of genus 2 or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on the fundamental groups of Kodaira surfaces. In the specific case of a Kodaira surface, these results are in the form of restrictions on the monodromy representation into the mapping class group. When the monodromy is composed with certain standard representations, the images are Zariski dense in a semisimple group of Hermitian type.
Let $\Cal U$ be an open neighborhood of the origin in $\Bbb C^{n+1}$ and let $f:(\Cal U, \bold 0)\to(\Bbb C, 0)$ be complex analytic. Let $z_0$ be a generic linear form on $\Bbb C^{n+1}$. If the relative polar curve $Γ^1_{f, z_0}$ at the origin is irreducible and the intersection number $\big(Γ^1_{f, z_0}\cdot V(f))_\bold 0$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $\big(Γ^1_{f, z_0}\cdot V(f))_\bold 0$ is not prime.
We establish a braid monodromy representation of functions satisfying an algebraic equation containing three terms (trinomial equation). We follow global analytic continuation of the roots to a trinomial algebraic equation that are expressed by Mellin-Barnes integral representations. We depict braids that arise from the monodromy around all branching points. The global braid monodromy is described in terms of rational twists of strands that yield a classical Artin braid representation of algebraic functions. As a corollary, we get a precise description of the Galois group of a trinomial algebr…
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…
Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms of multiplication in an associated wreath group algebra. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers. In case G is trivial, the formula specializes to the well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge integrals over the moduli space of curves in terms of single Hurwitz…
Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equi…
We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space Γ_X. We give a geometric interpretation of the cohomology of Γ_X in terms of the action of a monodromy operato…
Let $F_n$ denote the Fermat curve given by $x^n+y^n=z^n$ and let $μ_n$ denote the Galois module of $n$th roots of unity. It is known that the integral homology group $H_1(F_n,\Z)$ is a cyclic $\Z[μ_n\times μ_n]$ module. In this paper, we prove this result using modular symbols and the modular description of Fermat curves; moreover we find a basis for the integral homology group $H_1(F_n,\Z)$. We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.
Geometric monodromy of double covers of $\mathbb{P}^n$ branched along hyperplane arrangements
Beijing Institute of Mathematical Sciences and Applications（北京数学与应用科学研究所）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the $K3$ surface. As an application we establish the higher rank Segre-Verlinde correspondence for $K3$ surfaces as conjectured by Göttsche and Kool.
We study the Zariski closure of the monodromy group $\mathbf{Mon}$ of Lauricella's hypergeometric function $F_C$. If the identity component $\mathbf{Mon}^0$ acts irreducibly, then $\overline{\mathbf{Mon}} \cap \mathbf{SL}_{2^n}(\mathbb{C})$ must be one of classical groups $\mathbf{SL}_{2^n}(\mathbb{C}), \mathbf{SO}_{2^n}(\mathbb{C})$ and $\mathbf{Sp}_{2^n}(\mathbb{C})$. We also study Calabi-Yau varieties arising from integral representations of $F_C$.
The main objective of this article is to study the topology of the fibers of a generic rational function of the type $F^p/G^q$ in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle $δ$ generates the first homology group of a generic fiber of $F^p/G^q$. In particular, we will prove that for any two Lefschetz vanishing cycles $δ_0$ and $δ_1$ in a regular compact fiber of $F^p/G^q$, there exists a monodromy $h$ such that $h(δ_0)=\pm δ_1$.
We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete discrete valuation ring $K$ of mixed characteristic. For this we introduce log rigid cohomology and generalize the so called Hyodo-Kato isomorphism to versions for non-proper $Y$, for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of $Y$. We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel'd's …
The volume function defined by a domain in Euclidean space $\mathbb{R}^n$ is the function on the space of affine hyperplanes equal to volumes cut by these hyperplanes from the domain. The study of these functions originates from the works of Archimedes and Newton and is closely related to the theory of lacunas of hyperbolic partial differential equations.The volume functions are regular at the hyperplanes of general position with respect to the boundary of the cut domain. We study their behavior at the non-regular hyperplanes, which are either tangent to the boundary of the domain at its finit…
Let $f$ be a Laurent polynomial in two variables, whose Newton polygon strictly contains the origin and whose vertices are primitive lattice points, and let $L_f$ be the minimal-order differential operator that annihilates the period integral of $f$. We prove several results about $f$ and $L_f$ in terms of the Newton polygon of $f$ and the combinatorial operation of *mutation*, in particular we give an in principle complete description of the monodromy of $L_f$ around the origin. Special attention is given to the class of *maximally mutable* Laurent polynomials, which has applications to the c…
The Riemann-Wirtinger integral is an analogue of the hypergeometric integral, which is defined as an integral on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral. We derive explicit formulas of some intersection numbers, and apply them to study the monodromy representation, connection problems, and contiguity relations.
We study the vanishing cycle complex $φ_fA_X$ for a holomorphic function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the monodromy eigenvalue decomposition may hold after a localization of $A$). Assuming the perversity of the shifted constant sheaf $A_X[d_X]$, we show that the lowest possibly-non-zero vanishing cohomology at $0\in X$ can be calculated by the restriction of $φ_fA_X$ to an appropriate nearby curve in the singular locus $Y$ of $f$, which is given by intersecting $Y$ with …
