Holonomic D-modules Research Articles

On planar algebraic curves and holonomic 𝒟-modules in positive characteristic

In this paper we study a correspondence between cyclic modules over the first Weyl algebra and planar algebraic curves in positive characteristic. In particular, we show that any such curve has a preimage under a morphism of certain ind-schemes. This property might pave the way to an indirect proof of existence of a canonical isomorphism between the group of algebra automorphisms of the first Weyl algebra over the field complex numbers and the group of polynomial symplectomorphisms of [Formula: see text].

Erratum to: Holonomic D-modules on abelian varieties

Proof. — Since f is smooth, f ∗N = OA ⊗f −1OB f −1N is a holonomic DA-module, and so there is a surjective morphism f ∗N →M to a nontrivial simple holonomic DAmodule M. We will prove the assertion by showing that it is an isomorphism. The support X = SuppN is an irreducible subvariety of B. As N is holonomic, there is a dense Zariski-open subset U ⊆ B such that X ∩ U is nonsingular and such that the restriction N |U is the direct image of a holomorphic vector bundle with integrable connection (E ,∇) on X ∩ U [HTT08, Proposition 3.1.6]. This means that f ∗N is supported on f −1(X), and that its restriction to f −1(U) is the direct image of (f ∗E , f ∗∇). We observe that, on the fibers of f over points of X ∩ U, the latter is a trivial bundle of rank n = rkE . Since M is a quotient of f ∗N , its restriction to f −1(U) is also the direct image of a holomorphic vector bundle with integrable connection on f −1(X ∩ U); as a quotient of (f ∗E , f ∗∇), the restriction of this bundle to the fibers of f must be trivial of some rank k ≤ n. Now let r = dim A−dim B be the relative dimension of f . By adjunction [HTT08, Corollary 3.2.15], the surjective morphism

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.

Hamiltonian reduction and nearby cycles for mirabolic [formula omitted]-modules

We study holonomic D-modules on SLn(C)×Cn, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus.We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D-modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

On Involutivity ofp-Support

The p-support of a holonomic D-module was introduced by Kontsevich. Thomas Bitoun in his PhD thesis proved several properties of p-support conjectured by Kontsevich. In this note we give an alternative proof for involutivity by reducing it to a slight extension of Gabber's theorem on the integrability of the characteristic variety. For the benefit of the reader we review how this extension follows from Kaledin's proof of Gabber's theorem.

Holonomic D-modules on abelian varieties

We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complexes in this t-structure. This supports the conjecture that Fourier-Mukai transforms of holonomic D-modules are “hyperkähler perverse sheaves”.

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at innity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor product with a certain sheaf of formal microlocal dierential operators instead of the formal completion.

Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs

Let g be a complex reductive Lie algebra with Cartan algebra t. Hotta and Kashiwara defined a holonomic D-module M, on g×t, called the Harish-Chandra module. We relate grM, an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g×g×t×t which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of grM with the structure sheaf of Xnorm, the normalization of the isospectral commuting variety. We deduce, using Saito’s theory of Hodge D-modules, that the scheme Xnorm is Cohen–Macaulay and Gorenstein. This confirms a conjecture of M. Haiman. Associated with any principal nilpotent pair in g there is a finite subscheme of Xnorm. The corresponding coordinate ring is a bigraded finite-dimensional Gorenstein algebra that affords the regular representation of the Weyl group. The socle of that algebra is a 1-dimensional space generated by a remarkable W-harmonic polynomial on t×t. In the special case where g=gln the above algebras are closely related to the n!-theorem of Haiman, and our W-harmonic polynomial reduces to the Garsia–Haiman polynomial. Furthermore, in the gln-case, the sheaf grM gives rise to a vector bundle on the Hilbert scheme of n points in C2 that turns out to be isomorphic to the Procesi bundle. Our results were used by I. Gordon to obtain a new proof of positivity of the Kostka–Macdonald polynomials established earlier by Haiman.

Nahm transform and parabolic minimal Laplace transform

We prove that Nahm transform for integrable connections with a finite number of regular singularities and an irregular singularity of rank 1 on the Riemann sphere is equivalent–up to considering integrable connections as holonomic D-modules–to minimal Laplace transform. We assume semi-simplicity and resonance-freeness conditions, and we work in the framework of objects with a parabolic structure. In particular, we describe the definition of the parabolic version of Laplace transform due to C. Sabbah. The proof of the main result relies on the study of a twisted de Rham complex.

Period integrals of CY and general type complete intersections

We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.

On the classification of regular holonomic [formula omitted]-modules on skew-symmetric matrices

We give a classification of regular holonomic D-modules on 2m×2m-skew-symmetric matrices related to the action of the general linear group GL(2m,C). Actually we establish one more case of the conjecture by T. Levasseur (Levasseur, 2009 [12, p. 508, Conjecture 5.17]).

The principle of stationary phase for the Fourier transform ofD-modules

We show that the formal germ at the infinity of the Fourier transform of a holonomic D-module depends only on the formal germ of the D-module at its singular points and at the infinity.

Holonomic D-modules of Fuchsian and logarithmic types

The article deals with holonomic systems of differential equations with regular singularities. The main result is formulated as follows: any integrable Fuchsian D-module is meromorphically isomorphic to integrable logarithmic D-module. As a consequence we obtain a classification of integrable Fuchsian D-modules at non-singular points of their singular loci. A complete description of regular singular integrable systems with one unknown is another important result. In particular, it delivers an explicit representation of closed logarithmic forms in terms of eigenvalues of the monodromy associated with the corresponding logarithmic holonomic D-module. A method of computation of eigenvalues of the monodromy, the most important invariants of a regular singular holonomic integrable system, is also described. In conclusion, several examples for a clear interpretation of the obtained results are constructed and discussed in detail.

On Mirabolic -modules

Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D,G) where D is a sheaf of twisted differential operators on X, we form a left ideal D.g in D generated by the Lie algebra g, of G. Then, D/D.g is a holonomic D-module and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/D.g is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case where the group G=SL(n) acts diagonally on X = F \times F \times P^{n-1}, a triple product where F is the flag manifold for SL(n) and P^{n-1} is the projective space. We further relate D-modules on F \times F \times P^{n-1} to D-modules on the space SL(n) \times P^{n-1} via a pair, CH, HC, of adjoint functors, analogous to those used in Lusztig's theory of character sheaves. A second important result of the paper provides an explicit description of these functors showing that the functor HC gives an exact functor on the abelian category of mirabolic D-modules.

Quantum curves and 𝒟-modules

In this article we continue our study of chiral fermions on a quantum curve. This system is embedded in string theory as an I-brane configuration, which consists of D4 and D6-branes intersecting along a holomorphic curve in a complex surface, together with a B-field. Mathematically, it is described by a holonomic D-module. Here we focus on spectral curves, which play a prominent role in the theory of (quantum) integrable hierarchies. We show how to associate a quantum state to the I-brane system, and subsequently how to compute quantum invariants. As a first example, this yields an insightful formulation of (double scaled as well as general Hermitian) matrix models. Secondly, we formulate c=1 string theory in this language. Finally, our formalism elegantly reconstructs the complete dual Nekrasov-Okounkov partition function from a quantum Seiberg-Witten curve.

On holonomic D-modules attached to Reiffen’s hypersurface isolated singularities

A family of hypersurface isolated singularities, called Reiffen’s examples, is considered in the context of holonomic D-modules. Algebraic local cohomology classes attached to Reiffen’s singularities and their annihilating differential operators are studied. The algebraic local cohomology solution space to the first-order holonomic D-module is determined explicitly. As an application, it is shown that the multiplicity of the holonomic D-module can be described in terms of classical invariants of singularities.

D-Branes and D-Modules

We study of chiral fermions on a quantum curve by embedding them in string theory as an intersecting D-brane configuration together with a B-field. Mathematically, this system is described by a holonomic D-module. Here we focus on spectral curves, which play a prominant role in the theory of (quantum) integrable hierarchies. We show how to associate a quantum state to the system, and subsequently how to compute quantum invariants. The material of this paper is treated at greater length in the paper [R. Dijkgraaf, L. Hollands and P. Sulkowski, arXiv:0810.4157] that also contains many examples.

Supersymmetric gauge theories, intersecting branes and free fermions

We show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-commutative. Including all string loop corrections the free fermion theory is elegantly formulated in terms of holonomic D-modules that replace the classical holomorphic curve in the quantum case.

On the Dimension Formula for the Hyperfunction Solutions of Some Holonomic D-modules

In this short note we improve a recent dimension formula of Takeuchi for the dimension of the hyperfunction solutions of some holonomic D-modules. Besides the constructibility result and the local index formula of Kashiwara for the holomorphic solution complex, we only use a vanishing theorem of Lebeau together with a simple calculation in terms of constructible functions.

Radon and Fourier transforms for [formula omitted]-modules

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [4] in the framework of holonomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface SCP P of pairs ðx; yÞ; where x is a point in the n-dimensional complex projective space P belonging to the hyperplane yAP : As it turns out, a useful variant is obtained by considering the integral transform associated with the open complement U of S in P P : In the first part of this paper, we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the onedimensional case by the use of homogeneous coordinates. The second part of this paper applies the above result to the quantization of the Radon transform, in the sense of [7]. First we deal with line bundles. More precisely, let P 1⁄4 PðVÞ be the projective space of lines in the vector spaceV; denote by ð Þ 3 D R