Related papers: On the comparison of nearby cycles via $b$-functio…

We survey nearby and vanishing cycles for both perverse sheaves and D-modules under analytic setting. Following ideas of A. Beilinson, M. Kashiwara and M. Saito, we explain in detail the proof of the comparison theorem between them in the…

We give an introduction to the theory of $V$-filtrations of Malgrange and Kashiwara. After discussing the basic properties of this construction (in the case of a smooth hypersurface and, later, in the general case), we describe the…

We show the compatibility between the moderate or nearby cycle functor for regular holonomic $\mathcal{D}$-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara.

This paper studies the sliced nearby cycle functor and its commutation with duality. Over a Henselian discrete valuation ring, we show that this commutation holds, confirming a prediction of Deligne. As an application we give a new proof of…

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we…

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Ginsburg, Le and Sabbah (after a conjecture of Deligne). It is also a…

In this paper we generalize and study a notion of (unipotent) nearby cycles over a higher dimensional base based on Be\u{i}linson's description of unipotent nearby cycles, following an idea of Gaitsgory. This generalization, in the setting…

The goal of this article is to prove the comparison theorem between algebraic and topological nearby cycles of a morphism without slopes. We prove in particular that for a family of holomorphic functions without slopes, if we iterate…

We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $\Psi$-factorizability, the constructions of unipotent nearby cycles over an affine…

We present a microlocal criterion for the equivalence of iterated nearby cycles along different flags of subspaces in a higher-dimensional base. The result is motivated by its application to Hitchin systems in the context of Betti geometric…

A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step…

We give a proof of Artin's vanishing theorem in characteristic zero, based on Deligne's Riemann-Hilbert correspondence. Just as a curiosity.

We establish a relative Riemann-Hilbert correspondence for Alexander complexes (also known as Sabbah specialization complexes) by using relative regular holonomic $\mathscr D$-modules in an equivariant way, which particularly gives a…

The discrete-time approximation for nonlinear filtering problems is related to both of strong and weak approximations of stochastic differential equations. In this paper, we propose a new method of proof for the convergence of approximate…

Motivated by the work of Kontsevich-Soibelman on the comparison of isomorphisms conjecture for closed algebraic $1$-forms, we establish a Riemann-Hilbert correspondence of Deligne-Malgrange type. As an application, we prove a variant of the…

Let $f: X \to \mathbb{A}^1$ be a regular function on a smooth complex algebraic variety $X$. We formulate and prove an equivalence between the algebraic formal twisted de Rham complex of $f$ and the vanishing cycles with respect to $f$ as…

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower…

We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the…

We compute the motivic nearby cycles of functions obtained by composition with a polynomial which is non-degenerate with respect to its Newton polyhedron. Our result involves new convolution operators and generalized nearby cycles.

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles…