Related papers: The strong monodromy conjecture for hyperplane arr…

This paper studies Bernstein--Sato polynomials $b_{f,0}$ for homogeneous polynomials $f$ of degree $d$ with $n$ variables. It is open to know when $-{n\over d}$ is a root of $b_{f,0}$. For essential indecomposable hyperplane arrangements,…

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2\pi i c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture…

In this work, we develop a new theory of multivariate V-filtration on D-modules along a simple normal crossing divisor and relate it with Sabbah's multi-filtration. We establish several new structural results and relate them with the Hodge…

A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which…

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated…

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…

The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the…

Using recent results by A. Macinic, S. Papadima and R. Popescu, and a refinement of an older construction of ours, we determine the monodromy action on $H^1(F(G),C)$, where $F(G)$ denotes the Milnor fiber of a hyperplane arrangement…

We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more generally for semi-quasihomogeneous hypersurfaces. We also give a strange…

A conjecture of Denef-Jacobs-Veys relates motivic principal value integrals of multivalued rational top-forms with cohomology support loci of rank one local systems. We give a stronger positive answer to this conjecture for hyperplane…

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

An arrangement of hyperplanes is strongly inscribable if it has an inscribed (or ideal hyperbolic) zonotope. We characterize inscribed zonotopes and prove that the family of strongly inscribable arrangements is closed under restriction and…

Let A be an n by d matrix having full rank n. An orthogonal dual A^{\perp} of A is a (d-n) by d matrix of rank (d-n) such that every row of A^{\perp} is orthogonal (under the usual dot product) to every row of A. We define the orthogonal…

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…

We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights.…

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…

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous…

In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-K\"ahler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-K\"ahler varieties…

In the recent articles by Alper, Eastwood and Isaev, it was conjectured that all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of…