arXiv subjects

Andrea D'Agnolo

Publications and source records attributed to Andrea D'Agnolo.

At least 19 recordsLinked to original sources

arXiv2024

On the irregular Riemann-Hilbert correspondence

The original Riemann-Hilbert problem asks to find a Fuchsian ordinary differential equation with prescribed singularities and monodromy in the complex line. In the early 1980's Kashiwara solved a generalized version of the problem, valid on complex manifolds of any dimension. He presented it as a correspondence between regular holonomic D-modules and perverse sheaves. The analogous problem where one drops the regularity condition remained open for about thirty years. We solved it in the paper that received a 2024 Frontiers of Science Award. Our construction requires in particular an enhancement of the category of perverse sheaves. Here, using some examples in dimension one, we wish to convey the gist of the main ingredients used in our work. This is a written account of a talk given by the first named author at the International Congress of Basic Sciences on July 2024 in Beijing.

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arXiv2021

Enhanced nearby and vanishing cycles in dimension one and Fourier transform

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 give a topological proof of the following fact. Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, and denote by ${}^{\mathsf{L}}\mathcal M$ its Fourier-Laplace transform. For a point $a$ on the affine line, denote by $\ell_a$ the corresponding linear function on the dual affine line. Then, the vanishing cycles of $\mathcal M$ at $a$ are isomorphic to the graded component of degree $\ell_a$ of the Stokes filtration of ${}^{\mathsf{L}}\mathcal M$ at infinity.

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arXiv2020

Topological computation of some Stokes phenomena on the affine line

Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform $\widehat{\mathcal M}$, including its Stokes multipliers at infinity, in terms of the quiver of $\mathcal M$. Let $F$ be the perverse sheaf of holomorphic solutions to $\mathcal M$. By the irregular Riemann-Hilbert correspondence, $\widehat{\mathcal M}$ is determined by the enhanced Fourier-Sato transform $F^\curlywedge$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way, by computing $F^\curlywedge$ using Borel-Moore cycles. In this paper, we also consider some irregular $\mathcal M$'s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.

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arXiv2020

On a topological counterpart of regularization for holonomic D-modules

On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $\mathcal{M}\mapsto\mathcal{M}_{\mathrm{reg}}$, called regularization. Recall that $\mathcal{M}_{\mathrm{reg}}$ is reconstructed from the de Rham complex of $\mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.

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arXiv2019

Enhanced specialization and microlocalization

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we show how Sato's specialization and microlocalization functors have a natural enhancement, and discuss some of their properties.

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arXiv2017

A microlocal approach to the enhanced Fourier-Sato transform in dimension one

Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to $\mathcal{M}$ at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of $\mathcal{M}$ are obtained by Legendre transform from the exponential factors of $\mathcal{M}$. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.

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arXiv2015

Enhanced perversities

The Riemann-Hilbert correspondence embeds the triangulated category of (not necessarily regular) holonomic D-modules into that of $\mathbb R$-constructible enhanced ind-sheaves. The source category has a standard t-structure. Here, we provide the target category with a middle perversity t-structure, and prove that the embedding is exact.

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arXiv2015

Riemann-Hilbert correspondence for holonomic D-modules

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.

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arXiv2015

Morita classes of microdifferential algebroids

Projective cotangent bundles of complex manifolds are the local models of complex contact manifolds. Such bundles are quantized by the algebra of microdifferential operators (a localization of the algebra of differential operators on the base manifold). Kashiwara proved that any complex contact manifold $X$ is quantized by a canonical microdifferential algebroid (a linear stack locally equivalent to an algebra of microdifferential operators). Besides the canonical one, there can be other microdifferential algebroids on $X$. Our aim is to classify them. More precisely, let $Y$ be the symplectification of $X$. We prove that Morita (resp. equivalence) classes of microdifferential algebroids on $X$ are described by $H^2(Y;{\mathbb C}^\times)$. We also show that any linear stack locally equivalent to a stack of microdifferential modules is in fact a stack of modules over a microdifferential algebroid. To obtain these results we use techniques of microlocal calculus, non-abelian cohomology and Morita theory for linear stacks.

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arXiv2013

On the Laplace transform for tempered holomorphic functions

In order to discuss the Fourier-Sato transform of not necessarily conic sheaves, we compensate the lack of homogeneity by adding an extra variable. We can then obtain Paley-Wiener type results, using a theorem by Kashiwara and Schapira on the Laplace transform for tempered holomorphic functions. As a key tool in our approach, we introduce the subanalytic sheaf of holomorphic functions with exponential growth, which should be of independent interest.

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arXiv2012

On a reconstruction theorem for holonomic systems

Let X be a complex manifold. The classical Riemann-Hilbert correspondence associates to a regular holonomic system M the C-constructible complex of its holomorphic solutions. Denote by t the affine coordinate in the complex projective line. If M is not necessarily regular, we associate to it the ind-R-constructible complex G of tempered holomorphic solutions to the exterior product of M with the D-module associated with the exponential e^t. We conjecture that this provides a Riemann-Hilbert correspondence for holonomic systems. We discuss the functoriality of this correspondence, we prove that M can be reconstructed from G if X has dimension 1, and we show how the Stokes data are encoded in G.

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arXiv2010

On quantization of complex symplectic manifolds

Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X+1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.

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arXiv2010

A note on quantization of complex symplectic manifolds

To a complex symplectic manifold X we associate a canonical quantization algebroid. Our construction is similar to that of Polesello-Schapira's deformation-quantization algebroid, but the deformation parameter is no longer central. If X is compact, the triangulated category of regular holonomic quantization modules is a linear complex Calabi-Yau category of dimension 1 + dim X.

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arXiv2009

Regular holonomic D[[h]]-modules

We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure related to h-torsion.

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arXiv2006

Quantization of complex Lagrangian submanifolds

Let $Λ$ be a smooth Lagrangian submanifold of a complex symplectic manifold $X$. We construct twisted simple holonomic modules along $Λ$ in the stack of deformation-quantization modules on $X$.

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arXiv2005

Deformation-Quantization of Complex Involutive Submanifolds

The sheaf of rings of WKB operators provides a deformation-quantization of the cotangent bundle to a complex manifold. On a complex symplectic manifold $X$ there may not exist a sheaf of rings locally isomorphic to a ring of WKB operators. The idea is then to consider the whole family of locally defined sheaves of WKB operators as the deformation-quantization of $X$. To state it precisely, one needs the notion of algebroid stack, introduced by Kontsevich. In particular, the stack of WKB modules over $X$ defined in Polesello-Schapira (see also Kashiwara for the contact case) is better understood as the stack of modules over the algebroid stack of deformation-quantization of $X$. Let $V$ be an involutive submanifold of $X$, and assume for simplicity that the quotient of $V$ by its bicharacteristic leaves is isomorphic to a complex symplectic manifold $Z$. The algebra of endomorphisms of a simple WKB module along $V$ is locally (anti-)isomorphic to the pull-back of WKB operators on $Z$. Hence we may say that a simple module provides a deformation-quantization of $V$. Again, since in general there do not exist globally defined simple WKB modules, the idea is to consider the algebroid stack of locally defined simple WKB modules as the deformation-quantization of $V$. In this paper we start by defining what an algebroid stack is, and how it is locally described. We then discuss the algebroid stack of WKB operators on a complex symplectic manifold $X$, and define the deformation-quantization of an involutive submanifold $V$ by means of simple WKB modules along $V$. Finally, we relate this deformation-quantization to that given by WKB operators on the quotient of $V$ by its bicharacteristic leaves.

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arXiv2004

Sheaves and D-modules in integral geometry

Integral geometry deals with those integral transforms which associate to ``functions'' on a manifold their integrals along submanifolds parameterized by another manifold. Basic problems in this context are range characterization--where systems of PDE appear--and inversion formulae. As we pointed out in a series of joint papers with Pierre Schapira, the language of sheaves and D-modules provides both a natural framework and powerful tools for the study of such problems. In particular, it provides a general adjunction formula which is a sort of archetypical theorem in integral geometry. Focusing on range characterization, we illustrate this approach with a discussion of the Radon transform, in some of its manifold manifestations.

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arXiv2004

On twisted microdifferential modules I. Non-existence of twisted wave equations

Using the notion of subprincipal symbol, we give a necessary condition for the existence of twisted D-modules simple along a smooth involutive submanifold of the cotangent bundle to a complex manifold. As an application, we prove that there are no generalized massless field equations with non trivial twist on grassmannians, and in particular that the Penrose transform does not extend to the twisted case.

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