Riemann-Hilbert Correspondence Research Articles

Bi-algebraicity in the rank one Riemann-Hilbert correspondence via o-minimality

For a smooth, projective, complex algebraic variety X, the Riemann–Hilbert correspondence establishes a complex analytic isomorphism between the ‘Betti moduli space’ of rank n local systems on Xan and the ‘de Rham moduli space’ of rank n vector bundles with flat connection on X. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are ‘bi-algebraic’ for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We adapt the o-minimal proof to a p-adic setting, namely that of Mumford curves.

Relative regular Riemann–Hilbert correspondence II

We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.

The mod-p Riemann–Hilbert correspondence and the perfect site

The mod $p$ Riemann-Hilbert correspondence (in covariant and contravariant forms) relates $\mathbb{F}_p$-\'etale sheaves on the spectrum of an $\mathbb{F}_p$-algebra $R$ and Frobenius modules over $R$. We give an exposition of these correspondences using Breen's vanishing results on the perfect site.

Note on algebraic irregular Riemann–Hilbert correspondence

The subject of this paper is an algebraic version of the irregular Riemann–Hilbert correspondence which was mentioned in \[Tsukuba J. Math. 44 (2020), 155–201]. In particular, we prove an equivalence of categories between the triangulated category ${\mathbf{D}}^{\mathrm{b}}{\mathrm{hol}} (\mathcal{D}X)$ of holonomic $\mathcal{D}$-modules on a smooth algebraic variety $X$ over $\mathbb{C}$ and the triangulated category $\mathbf{E}^{\mathrm{b}}{\operatorname{\mathbb{C}-c}}( \mathrm{I}\mathbb{C}{X\_\infty})$ of algebraic $\mathbb{C}$-constructible enhanced ind-sheaves on a bordered space $X^\mathrm{an}\infty$. Moreover, we show that there exists a t-structure on the triangulated category $\mathbf{E}^{\mathrm{b}}{\operatorname{\mathbb{C}-c}}(\mathrm{I}\mathbb{C}{X\infty})$ whose heart is equivalent to the abelian category of holonomic $\mathcal{D}$-modules on $X$. Furthermore, we shall consider simple objects of its heart and minimal extensions of objects of its heart.

Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension ge 2, its analytification is a locally integrable singular foliation on the associated complex manifold X^h. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.

Isotropy quotients of Hopf algebroids and the fundamental groupoid of digraphs

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotropy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotropy quotient of the Hopf algebroid corresponding to flat connections over the digraph.

Goldman form, flat connections and stable vector bundles

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle $\mathscr{A}\to \mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections – local holomorphic sections of $\mathscr{A}$ – that allow to pull back the Liouville symplectic form on $T\*\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann–Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to $\mathscr{N}$ by the Narasimhan–Seshadri connection is the natural symplectic form on $\mathscr{N}$, introduced by Narasimhan and Atiyah & Bott.

Logarithmic Riemann–Hilbert correspondences for rigid varieties

On any smooth algebraic variety over a p p -adic local field, we construct a tensor functor from the category of de Rham p p -adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a p p -adic analogue of Deligne’s classical Riemann–Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this p p -adic Riemann–Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.

Betti Structures of Hypergeometric Equations

Abstract We study Betti structures in the solution complexes of confluent hypergeometric equations. We use the framework of enhanced ind-sheaves and the irregular Riemann–Hilbert correspondence of D’Agnolo–Kashiwara. The main result is a group theoretic criterion that ensures that enhanced solutions of such systems are defined over certain subfields of $\mathds{ C}$. The proof uses a description of the hypergeometric systems as exponentially twisted Gauß–Manin systems of certain Laurent polynomials.

Symplectic aspects of the tt*-Toda equations

We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt*-Toda equations. This constant problem for the sinh–Gordon equation, which is the case n = 1 of the tt*-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297–311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

The higher Riemann-Hilbert correspondence and principal 2-bundles

In this note we show that the holonomies for higher local systems provided by the higher Riemann-Hilbert correspondence coincide with those associated to principal 2-bundles where both formalisms meet.

The space of monodromy data for the Jimbo–Sakai family ofq-difference equations

We formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of $q$-isomonodromy and $q$-isoStokes to $q$-Painleve.

Irregular perverse sheaves

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

Maurer\u2013Cartan Moduli and Theorems of Riemann\u2013Hilbert Type

We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several different contexts, Schlessinger–Stasheff type theorems comparing the notions of homotopy and gauge equivalence for Maurer–Cartan elements as well as their categorified versions. As an application, we re-prove and generalize Block–Smith’s higher Riemann–Hilbert correspondence, and develop its analogue for simplicial complexes and topological spaces.

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.

The geometry of generalized Lamé equation, III: One-to-one of the Riemann–Hilbert correspondence

The geometry of generalized Lamé equation, III: One-to-one of the Riemann–Hilbert correspondence

On the comparison of nearby cycles via b-functions

In this article, we give a simple proof of the comparison of nearby and vanishing cycles in the sense of Riemann-Hilbert correspondence following the idea of Beilinson and Bernstein, without using the Kashiwara-Malgrange $V$-filtrations.

Riemann-Hilbert correspondence and blown up surface defects

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional N = 2 gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in SU(2) theory with Nf = 4 fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with 5 regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.

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.

Flat Gauge Fields and the Riemann-Hilbert Correspondence

The geometric phase that appears in the effects of Aharonov-Bohm type is interpreted in the frame of Deligne's version of the Riemann-Hilbert correspondence. We extend also the concept of flat gauge field to $B$-branes which are coherent sheaves, so that such a field on a sheaf turns it into a holonomic regular $D$-module.