Riemann-Hilbert Correspondence Research Articles

MIXED HODGE STRUCTURES OF THE MODULI SPACES OF PARABOLIC CONNECTIONS

In this paper, we investigate the mixed Hodge structures of the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable parabolic Higgs bundles and the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable regular singular parabolic connections. We show that the mixed Hodge polynomials are independent of the choice of generic eigenvalues and the mixed Hodge structures of these moduli spaces are pure. Moreover, by the Riemann–Hilbert correspondence, the Poincaré polynomials of character varieties are independent of the choice of generic eigenvalues.

Period integrals and the Riemann–Hilbert correspondence

A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

We construct a functor from the category of p-adic étale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection on its “base change to $${\mathrm {B}}_{{\text {dR}}}$$ ”, which can be regarded as a first step towards the sought-after p-adic Riemann–Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties.

Representations of Atiyah Algebroids and Logarithmic Connections

In this paper, we investigate representations of $\operatorname{At}(N)$, the Atiyah algebroids of a holomorphic line bundles $N$ over a complex manifold $Y$. In particular, we relate $\operatorname{At}(N)$-modules with logarithmic connections through two functors. On the one hand, we use these functors to the define invariants (monodromy) for representations of Atiyah algebroids. On the other hand, this opens the way to use the theory of Lie algebroids to study problems about logarithmic connections; we will give an example of this by showing that the existence of Deligne's extensions of flat connections and the Riemann-Hilbert correspondence for regular flat meromorphic connections may be obtained as pull-back of similar results for $\operatorname{At}(N)$-modules, and, at this level, these results are a direct consequence of the second theorem of Lie.

Riemann–Hilbert correspondence for irregular holonomic $${\mathscr{D}}$$ D -modules

This is a survey paper on the Riemann–Hilbert correspondence on (irregular) holonomic $${\mathscr{D}}$$ -modules, based on the 16th Takagi Lectures (2015/11/28). In this paper, we use subanalytic sheaves, an analogous notion to the one of indsheaves.

Chain integral solutions to tautological systems

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ, arising from the universal family of Calabi–Yau hypersurfaces Y_a in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in H_n (X−Y_a, ∪ D−Y_a) bounded by the union of G-invariant divisors ∪D in X to the solution sheaf of τ, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in Y_a, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of H_n (X−Y_a,∪ D−Y_a). In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann–Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

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.

Harmonic Maps to Buildings and Singular Perturbation Theory

The notion of a (uni)versal building associated with a point in the Hitchin base is introduced. The universal building is a building equipped with a harmonic map from the universal cover of the given Riemann surface that is initial among harmonic maps which induce the given cameral cover of the Riemann surface. The notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object. In the rank one case, the universal building is the leaf space of the quadratic differential defining the point in the Hitchin base. The main conjectures of this paper are: (1) a versal building always exists; (2) the asymptotics of the Riemann–Hilbert correspondence and the non-abelian Hodge correspondence are controlled by the harmonic map associated to any versal building; (3) spectral networks arise as inverse images of singularities of the versal building; and (4) versal buildings encode the data of a 3d Calabi-Yau category whose space of stability conditions has a connected component that contains the Hitchin base. The main theorem establishes the existence of the universal building, conjecture (3), as well as the Riemann–Hilbert part of conjecture (2), in the case of the rank two example introduced in the seminal work of Berk–Nevins–Roberts on higher order Stokes phenomena. It is also shown that the asymptotics of the Riemann–Hilbert correspondence are always controlled by a harmonic map to a certain building, which is constructed as the asymptotic cone of a symmetric space.

Log Hodge Theoretic Formulation of Mirror Symmetry for Calabi–Yau Threefolds

We hope to understand the Hodge theoretic aspect of mirror symmetry in the framework of the fundamental diagram of log mixed Hodge theory. We give a formulation of mirror conjecture for Calabi–Yau threefolds as the coincidence of log period maps with specified sections under the mirror map. Since a variation of Hodge structure with unipotent monodromies on a product of punctured discs uniquely extends over the puncture to a log Hodge structure, we can work on the boundary point over which the log Riemann–Hilbert correspondence exists, and we can observe clearly in high resolution the behavior of Z-structure over the boundary point (cf. notes in Introduction below). This is an advantage of log Hodge theory.

The higher Riemann–Hilbert correspondence

We construct an A∞-quasi-equivalence of dg-categories between PA — the category of prefect A0-modules with flat Z-connection, associated to the de Rham dga A• of a compact manifold M — and the dg-category of infinity-local systems on M — homotopy-coherent representations of the smooth singular simplicial set of M. We understand this as a generalization of the classical Riemann–Hilbert correspondence to Z-connections (Z-graded superconnections in some circles). This theory makes crucial use of Igusaʼs notion of higher holonomy transport for Z-connections which is a derivative of Chenʼs idea of generalized holonomy.

A Riemann–Hilbert approach to Painlevé IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ℙ1 are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.

Reidemeister torsion for flat superconnections

We use higher parallel transport—more precisely, the integration \(\mathsf{A }_\infty \)-functor constructed in Arias Abad and Schatz (The \(\mathsf{A}_\infty \) de Rham theorem and the integration of representations up to homotopy. Int Math Res Not, 2013) and Block and Smith (A Riemann–Hilbert correspondence for infinity local systems. arXiv:0908.2843, 2012)—to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger–Muller theorem, namely that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu (Contemp Math 546, 199–212, 2011).

The V-filtration for tame unit $$F$$ F -crystals

Let \(X\) be a smooth variety over an algebraically closed field of characteristic \(p > 0, Z\) a smooth divisor, and \(j: U=X {\setminus } Z \rightarrow X\) the natural inclusion. We introduce in an axiomatic way the notion of a \(V\)-filtration on unit \(F\)-crystals and prove such axioms determine a unique filtration. It is shown that if \(\mathcal M \) is a tame unit \(F\)-crystal on \(U\), then such a \(V\)-filtration along \(Z\) exists on \(j_*\mathcal M \). The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton–Kisin Riemann–Hilbert correspondence. A few applications to \(\mathbb A ^1\) and gluing are then discussed.

Algebraic and analytic parallel transport on p-adic curves

We relate two different partial p-adic analogues of the classical Riemann-Hilbert correspondence on curves. The first one comes from Deninger-Werner and Faltings and is of algebraic nature. The second one comes from Andre and Berkovich and is defined on Berkovich analytic spaces.

Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence

Let ( C , t ) (C,\mathbf {t}) ( t = ( t 1 , … , t n ) \mathbf {t}=(t_1,\ldots ,t_n) ) be an n n -pointed smooth projective curve of genus g g and take an element λ = ( λ j ( i ) ) ∈ C n r \boldsymbol {\lambda }=(\lambda ^{(i)}_j)\in \mathbf {C}^{nr} such that − ∑ i , j λ j ( i ) = d ∈ Z -\sum _{i,j}\lambda ^{(i)}_j=d\in \mathbf {Z} . For a weight α \boldsymbol {\alpha } , let M C α ( t , λ ) M_C^{\boldsymbol {\alpha }}(\mathbf {t},\boldsymbol {\lambda }) be the moduli space of α \boldsymbol {\alpha } -stable ( t , λ ) (\mathbf {t},\boldsymbol {\lambda }) -parabolic connections on C C and let R P r ( C , t ) a RP_r(C,\mathbf {t})_{\mathbf {a}} be the moduli space of representations of the fundamental group π 1 ( C ∖ { t 1 , … , t n } , ∗ ) \pi _1(C\setminus \{t_1,\ldots ,t_n\},*) with the local monodromy data a \mathbf {a} for a certain a ∈ C n r \mathbf {a}\in \mathbf {C}^{nr} . Then we prove that the morphism R H : M C α ( t , λ ) → R P r ( C , t ) a \mathbf {RH}:M_C^{\boldsymbol {\alpha }}(\mathbf {t},\boldsymbol {\lambda })\rightarrow RP_r(C,\mathbf {t})_{\mathbf {a}} determined by the Riemann-Hilbert correspondence is a proper surjective bimeromorphic morphism. As a corollary, we prove the geometric Painlevé property of the isomonodromic deformation defined on the moduli space of parabolic connections.

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit geometric Painlev\'e property as in the regular singular cases proved generally in [8].

Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

The starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2 × 2 isomonodromic Fuchsian systems associated to the Painlevé VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painlevé VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard’s solutions of Painlevé VI.

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.

Tree algebras: An algebraic axiomatization of intertwining vertex operators

We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a tree algebra. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\mathbb{C}$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\mathbb{Q}$.

Regular holonomic $\mathscr{D}[[\hbar]]$-modules

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