D-Modules and the Riemann-Hilbert Correspondence

D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE

JOSE´ SIMENTAL

Abstract. These are notes for a talk given at the MIT/NEU Seminar on Hodge modules, Fall 2016. We give a quick introduction to the theory of D-modules, following mostly [BCEY, HTT, K].

1. Differential operators

1.1. Setup. Let X be a smooth algebraic variety, and let dX := dim X. We remark that all algebraic varieties here are always assumed to be quasi-projective. By OX we will denote the structure sheaf of X, and VectX denotes the sheaf of vector ﬁelds on X, that is, the sheaf of regular sections of TX. Consider the sheaf EndCX (OX ), its sections on an open set U are HomCX (OU , OU ). Note that we have embeddings OX ,→ EndCX (OX ) and VectX ,→ EndCX (OX ).

Deﬁnition 1.1. The sheaf of diﬀerential operators DX is the sheaf of algebras that is generated by OX , VectX inside of EndCX (OX ).

Let us remark that DX is actually a coherent sheaf of algebras on X. This follows from the corresponding result for OX and results of Subsection 1.2 below, see Corollary 1.4. We also remark that DX is a sheaf of noncommutative algebras. Indeed, if ξ ∈ VectX , f ∈ OX , then [ξ, f] = ξ(f), which is not necessarily zero. In local coordinates x1, . . . , xn, ∂x1, . . . , ∂xn, and element P ∈ DX may be written as

X α P = aα(x)∂x n α∈Z≥0

α1+···+αn n α ∂ where aα = 0 for all but a ﬁnite number of α ∈ Z≥0, and ∂x = α1 αn . ∂x1 ··· ∂xn

Remark 1.2. We remark that a local coordinate system x1, . . . , xn, ∂x1 , . . . , ∂xn does not give an isomorphism n n from a neighborhood U to an aﬃne subvariety of C . Rather, what we have is an ´etalemorphism U → C .

1.2. Order of differential operators. Let P ∈ DX be a differential operator. Assume that, in local P α coordinates, we have P = n aα(x)∂x . We would like to say that P has order at most n if max{|α| : α∈Z≥0 aα(x) 6= 0} ≤ n. Of course, a representation of P in local coordinates is not unique, so this definition does not work as stated. However, we do have a filtration on DX that formalizes this idea. ≤m Definition 1.3. We define a filtration F (DX ) as follows: ≤m (1) F DX = 0 for m < 0. ≤0 (2) F DX = OX . ≤m ≤m−1 (3) F DX = {P ∈ DX :[P, OX ] ∈ F DX } for m ≥ 1. S ≤m ≤m m+1 It is clear, using a computation in local coordinates, that DX = n F DX and that F DX ⊆ F DX ≤m ≤n ≤m+n for every m. By double induction, it is easy to show that F DX · F DX ⊆ F DX , so that this is ≤m ≤n ≤m+n−1 F an algebra filtration, and that [F DX ,F DX ] ⊆ F DX , so that the associated graded gr DX is commutative. Let us describe this graded algebra explicitly. First, it is easy to check that we have F F gr0 DX = OX and gr1 DX = VectX . Note that this implies that we have a surjection:

F (1) gr D Sym Vect = π O ∗ X OX X ∗ T X where π : T ∗X → X is the canonical projection. A computation in local coordinates tells us that this is actually an isomorphism. 1 2 JOSE´ SIMENTAL

Corollary 1.4. The sheaf DX is a noetherian (in particular, coherent) sheaf of algebras on X.

Proof. This follows since gr DX is noetherian. We remark that the isomorphism in (1) is actually an isomorphism of Poisson algebras with respect to the F Poisson algebra structure in gr DX induced by the ﬁltration on DX and the Poisson bracket on π∗OT ∗X induced by the canonical symplectic form on T ∗X. Using local coordinates, this is an easy exercise.

1.3. Analytic setting. When discussing the Riemann-Hilbert correspondence, we need to consider differential operators on analytic, rather than algebraic, manifolds. Most of what we have discussed so far transfers to the analytic setting, although the proofs are slightly more diﬃcult. For example, we still have that gr D = Sym Vect . However, the latter algebra no longer coincides with π O ∗ . The coherence X OX X ∗ T X of DX here follows from the Oka-Cartan theorem, and the Hilbert basis theorem.

2. D-modules

We remark that DX is quasi-coherent as a OX -module. Most of the time below, we will restrict our attention to those DX -modules which are quasi-coherent as OX -modules. We denote by DX -Modqc ⊆ DX -Mod the category of such modules.

2.1. Connections. By a DX -module we mean a left DX -module. Perhaps the easiest example of such a module is given by the sheaf OX , which is a left DX -module by deﬁnition. Note that we have an exact sequence of DX -modules:

(2) DX ⊗OX VectX −→ DX −→ OX −→ 0 where the map DX ⊗OX VectX is P ⊗ ξ 7→ P ξ, and the map DX 7→ OX is P 7→ P · 1. Later, we will extend the sequence (2) to a complex of left DX -modules, see (7).

Now let M be an OX -module. If we want to extend the OX -module structure on M to a DX -module structure, we need a compatible action of vector ﬁelds, that is a map VectX ⊗OX M → M. Taking duals, we get a connection:

1 (3) ∇ : M → M ⊗OX ΩX 1 Where ΩX is the sheaf of 1-forms. It is an easy exercise to check that the fact that this actually defines a 2 2 2 DX -module structure on M means that the connection (3) is flat, i.e. ∇ = 0, where ∇ : M → M ⊗OX ΩX 2 is ∇ = (∇ ⊗ 1 + 1 ⊗ d)∇. In this sense, DX -modules are a generalization of vector bundles on X with a flat connection.

2.2. O-coherent D-modules. Following with the idea of the last paragraph, we will see in this subsection that, if a DX -module is O-coherent, then it is indeed a vector bundle with a ﬂat connection.

Proposition 2.1. Let M be a DX -module. Assume that M is OX -coherent. Then, M is locally free over OX .

Proof. Let x ∈ X be a closed point. It suffices to show that Mx is free over the local ring OX,x. By Nakayama’s lemma, we may find generators m1, . . . , mk ∈ Mx such that m1,..., mk form a basis of Mx/mxMx over the field OX,x/mx. We claim that m1, . . . , mk form a basis of Mx. Indeed, assume the P contrary so there exist f1, . . . , fk ∈ OX,x such that fimi = 0. Define the order of f ∈ OX,x by ord(0) = ∞ ` `+1 and ord(f) = ` if f ∈ m but f 6∈ m . Note that ord(fi) ≥ 0 for every i = 1, . . . , n and ord(fi) > 0 for at least one i. Moreover, let `0 := min{ord(fi): i = 1, . . . , n, ord(fi) > 0}. If the m1, . . . , mk are not lineraly independent over OX,x, we may assume that `0 is minimal among such linear combinations and `0 > 0. Without loss of generality, we may assume that `0 = ord(f1). Note that there exists a vector field ξ defined locally around x such that ξ(f1) 6= 0 and ord(ξ(f1)) < ord(f1) = `0. Now, D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 3

  X X X X (4) 0 = ξ fimi = ξ(fi)mi + fiξ(mi) = ξ(fi) + fjaji mi i j P P where ξ(mi) = j aijmj. Note that ord(ξ(f1) + j fjaj1) is strictly less than `0. We have arrived at a contradiction. Thus, M is locally free over OX .

2.3. Coherent D-modules. The following class of DX -modules will be important for us.

Deﬁnition 2.2. A DX -module M is coherent if it is locally ﬁnitely generated, that is, for every x ∈ X there ⊕N exists a neighborhood U of x and N > 0 such that we have an epimorphism (DX |U ) M|U .

We denote the category of coherent DX -modules by DX -mod. Remark 2.3. In general, the definition of coherence is more complicated. In general, for a sheaf of algebras A, a A-module is coherent if it is locally finitely generated and every locally finitely generated submodule is finitely presented. In view of Corollary 1.4, this agrees with our Definition 2.2 in the case A = DX .

The following result makes sure that coherent DX -modules have nice properties, and it is the key to build a bridge between the non-commutative setting of D-modules and the more familiar commutative setting of algebraic geometry, cf. Subsection 2.4.

Proposition 2.4. Let M be a coherent DX -module. Then, M is globally generated (over DX ) by a coherent OX -submodule. S Proof. We can take a finite, affine open covering X = Ui such that M|Ui is finitely generated over DUi . Thus, we may take a coherent OU -submodule Ni ⊆ M|U that generates M|U as a DU -module. Extend this i P i i i to a coherent OX -submodule Ni ⊆ M. Then, i Ni is OX -coherent and generates M as a DX -module. Remark 2.5. From now on, when we mention ’coherent D-module’ we will always mean coherent in the sense of this subsection, not coherent as a O-module.

2.4. Singular support. From Proposition 2.4 we can construct, for any coherent DX -module M, a global 0 ≤j ≤j 0 filtration on M. To do this, pick a coherent OX -submodule M of M. Then, define F M := F DX M . 0 This is a filtration of M. Since M is OX -coherent, we have that gr M is a finitely generated gr DX - module. Let us call a filtration satisfying this condition good. So we can define a coherent OT ∗X module by −1 gr]M := O ∗ ⊗ −1 π gr M. Of course, gr]M depends on the choice of a good filtration on M. It T X π π∗OT ∗X is an exercise to show that its set-theoretic support does not. ∗ Definition 2.6. Let M be a coherent DX -module. We define the singular support of M, SS(M) ⊆ T X, to be the set-theoretic support of the OT ∗X -module gr]M for any good filtration on M. ∗ × ∗ Note that SS(M) is an algebraic subset of T X, stable under the action of C on T X that rescales the cotangent fibers. The following property, however, is considerably harder th prove.

Theorem 2.7. Let M be a coherent DX -module. Then, its singular support SS(M) is a coisotropic subva- ∗ riety of T X. In particular, dim(SS(M)) ≥ dX . We remark that, here, ‘coisotropic’ means that, for every smooth point x ∈ SS(M), we have that ⊥ ⊥ ∗ (Tx SS(M)) ⊆ Tx SS(M), where • is taken with respect to the bilinear form on TxT X induced by ∗ the√ tautological√ √ symplectic form on T X. We also remark that Theorem 2.7 is equivalent to saying that { I, I} ⊆ I, where I is the annihilator of grF M under some good ﬁltration F . For a proof of Theorem 2.7 see e.g. [BCEY, Theorem 3.30]. ∗ ∗ For an algebraic variety X, we denote by TX X ⊆ T X the zero section of X.

Proposition 2.8. Let M be a coherent DX -module. The following are equivalent.

(1) M is OX -coherent. ∗ (2) SS(M) = TX X. 4 JOSE´ SIMENTAL

Proof. (1) ⇒ (2) is easy and it is an exercise. Let us do (2) ⇒ (1). Since the problem is local, we may assume ∗ that X is an affine algebraic variety with a coordinate system x1, . . . , xn, ∂x1 , . . . , ∂xn . That SS(M) = TX X means that we can find a good filtration F on M such that gr M is annihilated by a power of the ideal Pn m0 I := i=1 OX [ξ1, . . . , ξn]ξi. Here, we note by ξi the principal symbol of ∂xi . Say I gr M = 0. In particular, n P this means that, if α ∈ Z≥0 satisfies αi = m0 we have:

∂αF ≤jM ⊆ F ≤j+m0−1M for every j ≥ 0. On the other hand, we have

≤m +j ≤m ≤j F 0 M = (F 0 DX )(F M) α ≤j = P n O ∂ F M α∈Z≥0,|α|≤m0 X ⊆ F ≤m0+j−1M for j 0. Thus, the ﬁltration F is eventually constant, and equal to M. Thus, M is OX -coherent.

2.4.1. The analytic setting. Let us remark that, in the analytic setting, it is not the case that a coherent DX -module admits a global good filtration. In general, it only admits a good filtration locally. We can still ∗ −1 define the singular support of a D-module M as the unique subvariety of T X satisfying SS(M) ∩ πX (U) = ∗ Supp(M|]U ) for any open subset U ⊆ X such that M|U admits a good filtration, and πX : T X → X is the natural projection. Theorem 2.7 is still valid in the analytic setting, see for example [K, Section 7.6].

2.5. Holonomic D-modules. Let M be a nonzero coherent DX -module. We know that SS(M) is coisotropic, in particular, dim SS(M) ≥ n.

Deﬁnition 2.9. We say that a nonzero DX -module M is holonomic if SS(M) is lagrangian or, equivalently, if dim SS(M) = dX .

We denote by DX -modhol the category of holonomic DX -modules. By convenience, we will also say that the DX -module 0 is holonomic. We remark, first, that DX -modhol is a Serre subcategory of DX -mod, this 00 0 0 00 follows since, for every M ∈ ExtDX (M , M ) we have SS(M) = SS(M ) ∪ SS(M ), an easy exercise. It is also true that every holonomic DX -module has finite length - this follows from considering a finer invariant, the characteristic cycle, which counts multiplicities of the components of SS(M). Note, Proposition 2.8, that an OX -coherent DX -module is holonomic. Let us see another example of a holonomic DX -module. −1 Example 2.10. Let X = C with coordinate x, and consider the DX -module “generated by x ”. In other words, M = OC× with standard action of differential operators. In particular, M is not OX -coherent. We ≤i −i can take the filtration on M given by F M = x OX . We leave it as an exercise to check, using this 2 filtration, that SS(M) ⊆ C = Spec(C[x, ξ]) is given by the equation xξ = 0. In particular, M is holonomic. Note that M|C× is a coherent OC× -module. Exercise 2.11. Find the composition series of the module M from the previous example. In the previous example, we saw that the holonomic module M becomes a vector bundle with a flat connection when restricting to a dense, open subset U ⊆ X. The next result tells us that, indeed, “every holonomic module is generically a vector bundle with a flat connection.”

Theorem 2.12. Let M be a holonomic DX -module. Then, there exists an open, dense subset U ⊆ X such that M|U is coherent over OU . × ∗ ∗ Proof. Consider SS(M). This is a lagrangian, C -stable subset of T X, and denote by S := SS(M) \ TX X. ∗ If S = ∅, then SS(M) ⊆ TX X and M is OX -coherent, Proposition 2.8. So we assume that S 6= ∅. Now ∗ × consider the projection π : T X → X. Since SS(M) is C -stable, the dimension of each ﬁber of π|S is nonzero. Thus, dim(π(S)) < dim(S) ≤ dim(SS(M)) = dim X. So we can ﬁnd a nonempty, open subset U of X such that U ⊆ X \ π(S). It is immediate, by Proposition 2.8, that M|U is OU -coherent. D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 5

3. Functors on D-modules Let f : X → Y be an algebraic map. In this section, we construct several functors between categories of DX and DY -modules. In particular, we are interested in preservation of coherent or holonomic modules. We will see that a direct image functor is more easily deﬁned for right D-modules. Thus, our ﬁrst subsection will explore the connections between left and right DX -modules.

3.1. Left vs. right D-modules. Since the sheaf DX is noncommutative, we need to be careful to distin- guish left from right DX -modules. The following is an example of a right DX -module. Consider the sheaf of n n-forms ΩX := ΩX . Then, ΩX is a right DX -module, where OX ⊆ DX acts by multiplication, while VectX acts by the negative of the Lie derivative. That this is actually deﬁnes a right DX -module structure on ΩX follows immediately from Cartan’s formula: if ξ ∈ VectX and ω is a k-form, Lieξ(ω) = d(ιξ(ω)) − ιξ(dω), where ι denotes the contraction operator.

In fact, more is true. Assume that M is a left DX -module. Then, M⊗OX ΩX is a right DX -module, where the action of vector ﬁelds on the latter sheaf is given by the Leibniz rule: (m ⊗ ω)ξ = ξm ⊗ ω − m ⊗ Lieξ ω. On the other hand, if N is a right DX -module, then we have a structure of a left DX -module on

HomOX (ΩX , N ), the action of vector ﬁelds is given by ξψ(ω) = −ψ(ω)ξ + ψ(Lieξ ω). Thus, we have functors:

opp (5) • ⊗OX ΩX : DX -Mod DX -Mod : HomOX (ΩX , •) and it is easy to check that these deﬁne a category equivalence. Moreover, this induces an equivalence between opp the categories DX -mod and DX -mod of coherent D-modules, and between the categories DX -modhol, opp DX -modhol of holonomic modules.

Remark 3.1. We remark that the equivalence (5) comes from an antiautomorphism of DX if and only if X is a Calabi-Yau manifold, that is, ΩX is the trivial bundle. 3.2. Inverse image. Let f : X → Y be a map of algebraic varieties. ∗ −1 We define a functor f : DY -Mod → DX -Mod. To do this, first we define a DX -f DY -bimodule DX→Y . −1 −1 −1 As an OX -module, this is simply OX ⊗f OY f DY , which is manifestly an OX -f DY -bimodule. We would like to extend the left action of OX to an action of DX , so we need the action of vector fields. To do −1 P this, recall that we have a natural morphism VectX → OX ⊗f −1O f VectY , say ξ 7→ ai ⊗ ηi. Then, P Y we may define ξ(a ⊗ P ) = ξ(a) ⊗ P + i aai ⊗ ηiP . It is easy to see that this is well-defined and actually defines a left DX -module structure on DX→Y .

−1 −1 Remark 3.2. Let us be more explicit about the map VectX → OX ⊗f OY f VectY . Assume we have local coordinates x1, . . . , xk, ∂x1 , . . . , ∂xk on X and y1, . . . , yn, ∂y1 , . . . , ∂yn on Y . Then, the map VectX → OX is given by

X ∂(yj ◦ f) ∂ 7→ ⊗ ∂ xi ∂ yj j xi

∗ −1 −1 Now we deﬁne f M := DX→Y ⊗f DY f M. Note that as an OX -module this coincides with the pullback ∗ −1 OX ⊗f OY M. In particular, f : DY -Mod → DX -Mod is a right exact functor. It is clear that we have ∗ an induced functor f : DY -Modqc → DX -Modqc. However, it may not preserve coherence.

Example 3.3. Assume f : X,→ Y is a closed embedding. We may take local coordinates y1, . . . , yn on Y such that X is given by the equations yk+1 = ··· = yn = 0. So we have local coordinates xi = yi ◦ f of X, −1 −1 i = 1, . . . , k, and the morphism VectX → OX ⊗f OY f VectY is given by ∂xi 7→ ∂yi . It can easily be seen ∗ that f DY = DX→Y is DX ⊗CX C[∂yk+1 , . . . , ∂yn ], which is not a coherent DX -module. We will see, however, that under some conditions f ∗ indeed preserves coherent modules. As we will see in Proposition 3.5 this is the case, for example, if f is smooth. 6 JOSE´ SIMENTAL

3.3. Direct image. Before proceeding to define the direct image functor, let us give an intuitive reason on why this is more naturally defined for right D-modules. If we have a map f : X → Y , we can pullback functions on Y to obtain functions on X: the structure sheaf is naturally a left D-module and so the inverse image is naturally defined, as we have already seen, for left D-modules. However, there is no natural way to pushforward functions. There is, however, a way to pushforward forms. In the setting of differential geometry, if we assume that f : X → Y is a fiber bundle with compact oriented fibers, then for a top form α ∈ ΩX we may define a top form f∗α ∈ ΩY by integrating along fibers, namely for y ∈ Y , y1, . . . , yk ∈ TyY : Z (π α) (y , . . . , y ) = α ∗ y 1 k ey1,...,yk f −1(y) where α is the induced top form on f −1(y). As we have seen, top forms are naturally a right D-module. ey1,...,yk So we can expect that pushforward is more naturally defined for right modules. 3.3.1. First try. Again, let f : X → Y be a map of algebraic varieties. Now we define the direct image functor. As we have mentioned, this is more naturally defined for right DX -modules. Indeed, recall that we −1 opp −1 have the DX -f DY -bimodule DX→Y . So, for M ∈ DX -Mod, M ⊗DX DX→Y is a right f DY -bimodule. −1 −1 Thus, f∗(M⊗DX DX→Y ) is a right f∗f DY -bimodule, which under the adjunction DY → f∗f DY becomes a right DY -module. This is the (first) definition of the direct image functor. Since most of the time we work with left modules, let us translate this definition using the results from Subsection 3.1. Thus, the direct image functor is given by the following commutative diagram

DX -Mod / DY -Mod

ΩX ⊗OX • ΩY ⊗OY • M7→f (M⊗ D ) opp ∗ DX X→Y opp DX -Mod / DY -Mod −1 −1 ⊗−1 −1 It is easy to see that if we define a f DY -DX -bimodule by DY ←X := ΩX ⊗OX DX→Y ⊗f OY f (ΩY ) then we have that the direct image functor on the level of left modules is given by M 7→ f∗(DY ←X ⊗DX M). This is, however, not the best definition. The reason is that the functor f∗ is, in general, only left exact, while the functor DY ←X ⊗DX • is only right exact. To define a direct image functor, we have to pass to derived categories.

3.3.2. Derived categories. Here we set notations for our work with derived categories. We deﬁne D(DX ) to be the derived category of DX -Mod. We denote by Dqc(DX ) the full subcategory of complexes whose cohomology is OX quasi-coherent. By Dc(DX ) we mean the full subcategory of complexes with coherent cohomology, and Dhol(DX ) is the full subcategory of complexes with holonomic cohomology. The superscripts b, +, − have the usual meanings. The following result is due to Beilinson-Bernstein. Theorem 3.4. The natural functors

b b D (DX -Modqc) −→ Dqc(DX ) b b D (DX -mod) −→ Dc(DX ) b b D (DX -modhol) −→ Dhol(DX ) are equivalences of categories. b We remark that every object in Dqc(DX ) can be represented by a finite complex of locally projective DX -modules. Moreover, DX has finite homological dimension, this follows because gr DX already has finite ∗ homological dimension. So we can, for example, define the left derived functor to the inverse image, Lf : b b · L −1 · D (DY ) → D (DX ) namely M 7→ DX→Y ⊗ −1 f M . To use later, we define the shifted inverse qc qc f DY † b b † ∗ image, f : Dqc(DY ) → Dqc(DX ) by f := Lf [dim X − dim Y ]. For example, if f : U,→ X is a smooth, † ∗ open embedding, then f = Lf , and this is just the restriction to U. Proposition 3.5. Let f : X → Y be a smooth morphism of smooth algebraic varieties. i ∗ (1) For M ∈ DY -Modqc, H (Lf M) = 0 for i > 0. ∗ (2) For M ∈ DY -mod, f (M) ∈ DX -mod. D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 7

∗ −1 −1 −1 Proof. For (1), recall that f M = OX ⊗f OY f M. The result now follows since OX is flat over f OY . −1 −1 For (2), it is enough to show that the map DX → DX→Y = OX ⊗f OY f DY , P 7→ P (1 ⊗ 1) is surjective. This is an easy exercise. b 3.3.3. Second try. Now we define a direct image functor on the level of derived categories, D (DX ) → b · b D (DY ). For M ∈ D (DX ) we define the direct image by: Z M· := f (D ⊗L M·) R ∗ Y ←X DX f R b b It is a priori not clear that f should send Dqc(DX ) to Dqc(DY ). However, this is indeed the case, see e.g. R b [HTT, Section 1.5]. Moreover, is f is proper, then f preserves the category Dc - this is a consequence of Kashiwara’s theorem, which we review next.

3.4. Kashiwara’s theorem. In this subsection, we prove one of the most fundamental theorems in the theory of D-modules. To state it, let ι : X → Y be a closed embedding, and consider the category X DY -Modqc of OY -quasicoherent DY -modules whose set-theoretic support is contained in X. Intuitively, the 0 R X functor H ι should send DX -Modqc to DY -Modqc. In this section, we will see that this is indeed the case and that, moreover, this is an equivalence of categories. 0 R X Theorem 3.6. The functor ι0 := H ι : DX -Modqc → DY -Modqc is an equivalence of categories. Before proceeding to the proof, we make a few comments. • We remark that the side-changing functors of Subsection 3.1 preserve the subcategories of modules set-theoretically supported at X. Thus, we will prove Theorem 3.6 for right modules. An advantage in this case is that we already have a description of DX→Y , see Example 3.3. • In the first part of the proof, we will assume that Y and X are affine. Moreover we will assume that the codimension of X on Y is 1. Afterwards, we will see why this implies the general case. Proof of Theorem 3.6. We proceed in several steps. opp opp X Step 1. First, we will show that ι0(M) = M ⊗DX DX→Y sends DX -Modqc to DY -Modqc. To do this, let J ⊆ OY be the ideal defining X. Thanks to our computations in Example 3.3, DX→Y = DY /JDY . So n it suffices to see that for every P ∈ DY , we have an integer n 0 such that P J ⊆ J DY . We leave this as an exercise. • opp opp opp Step 2. Let us construct a functor ι : DY -Modqc → DX -Modqc. For M ∈ DY -Modqc, let

• ι (M) := HomDY (DX→Y , M) = {m ∈ M : mJ = 0} Since DX→Y is a DX -DY -bimodule, this is a right DX -module. By the usual tensor-hom adjunction, ι0 is left adjoint to ι•. • • Step 3. We have adjunction morphisms ι0ι N → N and M → ι ι0M. We will show that these are opp X isomorphisms when we take N ∈ DY -Modqc. We assume that we have a local system of coordinates opp y1, . . . , yn, ∂y1 , . . . , ∂yn on Y such that X = {yn = 0}. First, take M ∈ DX -Modqc. Thanks to our L k • computations in Example 3.3, we have ι0M = k M∂yn . Now, ι ι0M = ker(m 7→ myn : ι0M → ι0M). If we remember how to differentiate polynomials, it is easy to see that this kernel is precisely M. opp X • Step 4. Now take N ∈ DY -Modqc, in particular, yn acts locally nilpotently on N . Let N0 := ι N , and P j j consider Ne := j≥0 N0∂yn . By induction on j, we can see that N0∂yn is the j-eigenspace for the action L j • of yn∂yn on Ne. In particular, Ne = j N0∂yn . It remains to show that Ne = ι0ι N = N . To do this, set L := N /Ne. We remark that yn acts locally nilpotently on L. So it will suffice to check that yn has zero kernel on L. Let ` ∈ N be such that `yn ∈ Ne. Note that yn : Ne → Ne is surjective, cf. Step 3. So there exists è∈ Ne with (` − è)yn = 0, that is, ` − è∈ N0 ⊆ Ne. So ` ∈ Ne. We are done with the case where X, Y are affine and the codimension of X on Y is 1. Step 5. Globalizing. Let us remark that, in the general case, we may define the functor

• −1 −1 ι (M) = Homf DY (DX→Y , f M) 8 JOSE´ SIMENTAL

• and the proof of Kashiwara’s theorem reduces to check that ι0, ι are quasi-inverses when restricting to opp X DY -Modqc. This is a local statement. So, by induction on codimY X, the theorem follows by what we have done in Steps 1-4. Remark 3.7. We remark that, from the proof of Kashiwara’s theorem, we see that we also have an equiva- X lence DX -mod → DY -mod . R b b Corollary 3.8. Let f : X → Y be a proper map. Then, f : Dqc(DX ) → Dqc(DY ) preserves the categories b Dc of coherent modules. Proof. In this proof, we work again with right D-modules. Since f is proper and X, Y are quasi-projective, n we may decompose our map f into a composition of a closed embedding and a projection, X,→ Y ×P → Y . The case of a closed embedding is an easy consequence of Kashiwara’s theorem. Let us proceed to the case n of a projection, so we assume that X = Y ×P and f is just projection to Y . Note that we only need to show R b that f DX ∈ Dc(DY ) - here we are considering DX as a right DX -module. Now, we have DX→Y = OPn DY . Thus, Z n n n n DY ×P = Rf∗(DX→Y ) = Rf∗(DY OP ) = DY ⊗CY RΓ(P , OP ) f R b so that f DY ×Pn = DY ∈ Dc(DY ). This ﬁnishes the proof. If ι : X → Y is a closed embedding, we will use the notation:

X BX|Y := ι0OX ∈ DY -mod n In particular, if {yi, ∂yi }i=1 is a system of coordinates such that X is given by the equations ym+1 = ··· = yn = 0, then B = DY Pn Pn X|Y i=1 DY ∂yi + j=m+1 DY yj

In particular, if X = {p}, we denote δp := BX|Y , the DY -module of delta-functions at p. By Kashiwara’s theorem, every DY -module supported at {p} is the direct sum of copies of δp. Let us see how to use Kashiwara’s theorem to define D-modules on singular varieties. Note that we still have an algebra DX define when X is singular - this is basically defined by the order filtration. However, the algebra DX does not enjoy nice properties, for example, it may not be noetherian. Definition 3.9. Let X be a singular variety. Fix an embedding X,→ Y where Y is smooth. Then, we define X the category of quasi-coherent DX -modules by DX -Modqc := DY -Modqc. Similarly, we define the category X of coherent DX -modules by DX -mod := DY -mod . Of course, the previous definition is not very good. It explicitly depends on an embedding X,→ Y . X One can indeed show that for two distinct embeddings X,→ Y1,X,→ Y2 the categories DY1 -Modqc and X DY2 -Modqc are canonically equivalent. For details, see [BCEY, Section 2.7.2].

3.5. Preservation of holonomicity. As we have seen, for a general morphism f : X → Y it is not the case R b b † b b that f : Dc(DX ) → Dc(DY ), f : Dc(DY ) → Dc(DX ). The situation is diﬀerent, however, for holonomic modules. Theorem 3.10. Let f : X → Y be a morphism of algebraic varieties. Then, we have † b b (1) f : Dhol(DY ) → Dhol(DX ). R b b (2) f : Dhol(DX ) → Dhol(DY ). We will not give a proof of this theorem. For a proof, see e.g. [BCEY, Chapter 3]. The result is proved by decomposing a map as the composition of a closed embedding and a projection. Let us give the case of the direct image under a closed embedding as an exercise. D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 9

∗ Exercise 3.11. Let ι : X,→ Y be a closed embedding, and consider the fibered product X ×Y T Y , which ∗ ∗ ∗ ∗ comes with maps ρY : X ×Y T Y → T Y and ρX : X ×Y T Y → T X. Show that, for M ∈ DX -mod, Z −1 SS M = ρY ρX SS(M). ι R Deduce that M is holonomic if and only if ι M is. The case of a projection is harder. We remark, however, that this result is not valid in the analytic case. In fact, in the analytic case neither the direct nor the inverse image of a holonomic module even need to be coherent! We will see an example of this for the direct image in Subsection 4.1. For the inverse image, consider X = Y = C and f : X → Y the exponential map. Let y ∈ Y be nonzero, and consider δy, the ∗ L DY -module of δ-functions on Y . This is clearly holonomic. Then, Lf δy = x∈f −1(y) δx, which is an infinite sum and is therefore not coherent. 3.6. Simple holonomic modules. We state, for future use when defining regular holonomic D-modules (cf. Section 5) the following classification of simple holonomic D-modules. This can be proved using the theory of minimal extensions. For details, we refer the reader to [HTT, Chapter 3]. Theorem 3.12. Let X be a smooth algebraic variety. Let Y be a smooth, locally closed subvariety of X such that the inclusion ι : X → Y is affine, and let M be a simple, holonomic DY -module. Then: R R (1) The cohomology of ι M is concentrated in degree 0. Moreover, the module ι M has a unique simple submodule, which is denoted by L(Y, M). (2) Any simple, holonomic DX -module is of the form L(Y, M), where Y is as in the statement of this theorem and M is a simple, OY -coherent DY -module. b 3.7. Duality. Let us finish this section by introducing a duality functor in the category Dqc(DX ). First b opp b of all, we have the functor RHomDX (•, DX )[dim X]: Dqc(DX ) → Dqc(DX ). Using the side-changing b opp b opp functors, we obtain a functor D : Dqc(DX ) → D (DX ) . The first part of the following result is standard. The second part, a little less so.

Proposition 3.13. (1) The functor D is an equivalence. Moreover, D ◦ D = id. b · b · · (2) The functor D induces a duality in Dc(DX ). For M ∈ D (DX ), we have SS(M ) = SS(D(M )), · i · b where SS(M ) = ∪i SS(H (M )). In particular, we also get a duality in Dhol(DX ).

We remark that, moreover, the functor D induces a duality in the category DX -modhol. This is, basically, i a consequence of the fact that for an appropriate ﬁltration we have an epimorphism Ext (gr M, O ∗ ) OT ∗X T X i i gr Ext (M, D ) and a theorem of commutative algebra that tells us that Ext (gr M, O ∗ ) = 0 for DX X OT ∗X T X i 6= dim X.

4. The De Rham and Solutions complexes 4.1. Analytiﬁcation. Let X be an algebraic variety. We will denote by Xan the corresponding analytic n space. More precisely, we can embed X as a locally closed subset of some projective space P , which has a natural complex structure and Xan is X with the induced structure. In particular, if X is smooth, then Xan is a complex manifold. We have a morphism ι : Xan → X of topological spaces, and a canonical morphism −1 −1 an an an −1 ι DX ,→ DX . Moreover, DX = OX ⊗ι OX ι DX . Thus, we obtain a functor

an • : DX -Mod −→ DXan -Mod (6) −1 an −1 M 7→ OX ⊗ι OX ι M an an We remark that the functor is exact, and it induces functors • : DX -mod −→ DXan -mod and • : b b Dc(DX ) −→ Dc(DXan ). We would like to mention how the analytiﬁcation functor interacts with the functors introduced in Section 3. First, it is easy to see that duality commutes with analytiﬁcation, that is · an · an an (DX M ) = DXan (M ) . Now, for a map f : X → Y of smooth algebraic varieties, denote by f the map considered as a map between complex manifolds. · b · b Proposition 4.1. Let f : X → Y be a map of algebraic varieties, M ∈ Dc(DX ) and N ∈ Dc(DY ). 10 JOSE´ SIMENTAL

(1) We have (f †N ·)an =∼ (f an)†(N ·)an. R · an R · an (2) There is a canonical morphism ( f M ) → f an ((M ) ). This map is an isomorphism if f is proper. For a proof of Proposition 4.1, see e.g. [HTT, Proposition 4.7.2]. Here, we will provide an example × showing that the map in Proposition 4.1(2) may fail to be an isomorphism if f is not proper. Take X = C , · Y = C (with coordinate y) and ι : X → Y the inclusion. For M we take OX concentrated in degree 0. 0 R −1 Now, H ( ι OX ) = ι0OX = OY [y ]. As we have seen in Example 2.10, this is a holonomic DY -module. −1 an −1 Moreover, (OY [y ]) = OY an [y ], which can be seen to be a holonomic DY an -module. an On the other hand, take ι0 OXan . This contains all meromorphic functions at 0, but also functions with −1 essential singularities at 0, e.g. exp(y ). In fact, this is not even a coherent DY an -module. So the modules an an (ι∗OX ) and ι∗ OXan cannot be isomorphic. 4.2. The De Rham functor.

4.2.1. The analytic case. For this subsection, we let X be a complex manifold. Recall that ΩX is a right

DX -module. So, for any left DX -module M, we can take the space ΩX ⊗DX M. We remark that this space does not admit any natural structure of a (left or right) DX -module. It is only a sheaf of C-vector spaces.

Deﬁnition 4.2. For a coherent DX -module M, we deﬁne the de Rham complex of M by DR(M) := Ω ⊗L M. We note that this extends to a functor: X DX

DR := Ω ⊗L • : Db(D ) → Db( ) X DX c X CX opp Let us give a locally free resolution of the DX -module ΩX . Thanks to the side-changing operations, cf. Subsection 3.1, it is enough to give a locally free resolution of the DX -module OX . Consider the complex

n n−1 ^ ^ (7) 0 → DX ⊗OX VectX → DX ⊗OX VectX → · · · → DX ⊗OX VectX → DX → OX → 0 Vk where the rightmost part of the complex is as in (2) and the diﬀerential d : DX ⊗OX VectX → D N Vk−1 Vect is given by X OX X

P i+1 d(P ⊗ ξ1 ∧ · · · ∧ ξk) = i(−1) P ξi ⊗ ξ1 ∧ · · · ∧ ξbi ∧ · · · ∧ ξk (8) P i+j + i

This is known as the Spencer resolution of OX . The fact that the complex is indeed acyclic follows by relating it to the Koszul resolution of the Sym Vect -module O . Indeed, we have a natural filtration on OX X X the complex (7) induced from the filtration on DX by the order of a differential operator, and its associated graded is the Koszul resolution. By applying side-changing operations, we get a resolution of ΩX :

1 n−1 0 → DX → Ω ⊗O DX → · · · → Ω ⊗O DX → ΩX ⊗O DX → ΩX → 0 (9) X X PX X d(ω ⊗ P ) = dω ⊗ P + i dxi ∧ ω ⊗ ∂xi P where {xi, ∂xi } are local coordinates of X. In particular, this gives a complex for DR(M) when M ∈ DX -mod, that is given by very similar formulae. We remark that the leftmost term in the complex DR(M) 1 is given precisely by the connection ∇ : M → ΩX ⊗OX M determined by the DX -module structure on M. In particular

−dX ∇ (10) H DR(M) = M := {m ∈ M : VectX m = 0}

In particular, in the case where M is OX -coherent, we get a local system on X that is, a locally free CX -module of finite rank - this is a consequence of the usual existence and uniqueness theorems for solutions of first order ODE’s with initial conditions. On the other hand, from a locally free CX -module of finite rank L we can get a vector bundle with a flat connection if we define ML := OX ⊗CX L and the connection 1 ∇ ∇ : OX ⊗CX L → ΩX ⊗CX L is given by d ⊗ idL. It is clear that the assignments M → M ,L 7→ ML are i mutually inverse. Moreover, it is easy to see that H (DR(ML)) = 0 for i 6= −n. Thus, we get. D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 11

Theorem 4.3. Let Conn(X) denote the category of OX -coherent DX -modules, and Loc(X) the category of local systems on X. We have an equivalence of categories:

H−dX (DR(•)) : Conn(X) → Loc(X) The following theorem is sometimes useful for computations. · b Theorem 4.4. Let f : X → Y be a morphism of complex manifolds, and let M ∈ D (DX ). Then, in b D (CY ) we have: Z · ∼ · Rf∗ DRX M = DRY M f

∼ −1 L Proof. This follows easily from the isomorphism ΩX = f ΩY ⊗ −1 DY ←X , which is an exercise. f DY 4.2.2. The algebraic case. Now assume that X is an algebraic variety. We would like to have an analog of b b the de Rham functor DR : Dc(DX ) → D (CX ). We have a resolution of ΩX given by the same formulas as in (9). Hence, we could be tempted to define DR(•) = Ω ⊗L •. This is, however, not a good definition. X DX Grosso modo, the reason is that solutions to differential equations with polynomial coefficients need not be polynomial.

Example 4.5. Let X = C, λ ∈ C \ Z and consider the DX -module M = DX /DX (∂x − λ). We can think of M as the DX -module “generated by the function exp(λx).”In particular, it is easy to see that M is OX - coherent. However, we have that M∇ = 0. In fact, it is not hard to see that Ω ⊗L M is an acyclic X DX complex. Thus, in order to obtain a nice functor, we need to relate to the analytic case. b Deﬁnition 4.6. Let X be a smooth algebraic variety. We deﬁne the de Rham functor, DR : Dc(DX ) → b D (CXan ) by

· · an DR(M ) := DRXan ((M ) ) Remark 4.7. We remark that the functor DR commutes with taking restriction to an open set, and with pullback/pushforward under an ´etalecovering. This is a consequence of Proposition 4.1, and it will be important for our proof of the Kashiwara constructibility theorem. We remark that an analog of Theorem 4.3 is not valid in the algebraic situation. The reason now is that two connections may be not algebraically isomorphic, and yet still be analitically isomorphic. For example, let X = C and M = (OX , ∇), N = (OX , ∇e ), where the connections are

∇(f) = f 0dx ∇e (f) = (f 0 − f)dx The connection ∇ has the constant function 1 as a flat section. In the second case, a flat section must be ∇ ∇e an ∼ an a multiple of exp(x), which is not algebraic. So M = C while N = {0}. However, M = N , an isomorphism is given by f 7→ f exp(x). A solution to this problem is to restrict our attention to certain connections with “regular”singularities. We will make this precise in Section 5. 4.3. The solutions functor. 4.3.1. The analytic case. Let X be a complex manifold. We would like to mention another functor that is closely related to the de Rham functor. To motivate it, take a differential operator P ∈ DX , and consider the cyclic DX -module MP := DX /DX P . Note that for any N ∈ DX -mod, HomDX (M, N ) = {n ∈ N : P n =

0}. So, in particular, we think of HomDX (M, OX ) as the space of holomorphic solutions of the equation P f = 0. · b · · Definition 4.8. Let M ∈ Dc(DX ). We define the solution complex Sol(M ) := RHomDX (M , OX ). In particular, this defines a functor: b b opp Sol = R HomDX (•, OX ): Dc(DX ) → D (CX ) 12 JOSE´ SIMENTAL

Let us remark that this functor is very closely related to the de Rham functor. Indeed, we have:

· · Sol(DM ) = RHomDX (DM , OX ) =∼ Hom ( M·, D ) ⊗L O R DX D X DX X =∼ Ω ⊗ M· ⊗L O [−d ] X OX DX X X =∼ M· ⊗L Ω [−d ] DX X X · = DR(M )[−dX ]

4.3.2. The algebraic case. Now let X be a smooth algebraic variety. By the relation between Sol and DR in · · the analytic case, we see that it is not a good idea to simply deﬁne Sol(M ) = R HomDX (M , OX ). Rather, · b we deﬁne for M ∈ Dc(DX ):

· · an Sol(M ) = SolXan ((M ) ).

4.4. Kashiwara’s constructibility theorem. Now we take X to be a smooth algebraic variety. In this section we prove, following Bernstein, a remarkable result of Kashiwara that relates holonomic DX -modules to constructible sheaves on X. We remark that this result is valid in the analytic situation - in fact, by the definition of DR, the analytic setting effectively generalizes the algebraic one. However, the analytic setting b is harder, see e.g. [HTT, Section 4.6]. Denote by Dc(CX ) the subcategory of the bounded derived category of sheaves of vector spaces on X consisting of complexes of CX -modules with constructible cohomology. · b Theorem 4.9 (Kashiwara’s constructibility theorem). Let X be an algebraic variety, and M ∈ Dhol(DX ). b Then, DR(M) ∈ Dc(CX ). · · First of all, note that we may assume that M is concentrated in degree 0, that is, M = M ∈ DX -modhol. Thanks to Theorem 2.12, we may find a dense, open set U ⊆ X such that M|U is OU -coherent. Thus, b DRU (M|U ) ∈ Dc(CU ), cf. Theorem 4.3. Then, to prove Kashiwara’s constructibility theorem, we need to show the following result.

Lemma 4.10. Let M ∈ DX -modhol, and assume that for a dense, open subset U ⊆ X, DRU (M|U ) ∈ b b Dc(CU ). Then, there exists an open, dense subset Y of X \ U such that DRU∪Y (M|U∪Y ) ∈ Dc(CU∪Y ). Proof. Step 1. First, let Z ⊆ X \ U be irreducible. By considering local coordinates around a point p ∈ Z, 0 n 0 we see that we may find open subsets V ⊆ X, V ⊆ C (n = dX ) and an étalemap f : V V such that : • V ∩ (X \ U) ⊆ Z is open and dense. 0 n−k n−k • V ∩ C ⊆ C is open and dense. −1 0 n−k • f (V ∩ C ) = V ∩ (X \ U). b Since f is étale,we have that DRV (M|V ) ∈ D (CV ) is constructible if and only if f∗ DRV (M|V ) ∈ b R n D (CV 0 ) is constructible. Moreofer, f∗ DRV (M|V ) = DRV 0 ( f M|V ). Thus, we may assume that X ⊆ C , n−k n−k n−k X \ U ⊆ C and T := X ∩ C ⊆ C is dense. By shrinking X, we may assume that X is open in k C × T . k k k · R Step 2. Now consider C as a subset of P . In particular, we have ιX : X,→ P ×T . Consider N := ι M ∈ b k · ∼ R · Dhol(DPk×T ). Since the projection π : P × T → T is proper, we have that π∗ DRPk×T (N ) = DRT ( π N ). R · b · · Now, π N ∈ Dhol(DT ). So, denoting K := DRPk×T (N ) and replacing T by an open subset Y , we may · assume that π∗K is locally constant. k k Step 3. Now let S := (P × Y ) \ X. We have P × Y = Y t S t U, and U is open. Let ιX , ιU , ιY , ιT be the corresponding inclusions. Then, we have a distinguished triangle

−1 · · −1 · −1 · +1 (11) ιU!ιU K −→ K −→ ιS!ιS K ⊕ ιY !ιY K −→ b By applying π∗ (= π!) to the distinguished triangle (11) we obtain an exact triangle in D (CY ):

−1 · · −1 · −1 · +1 (12) (π ◦ ιU )!ιU K −→ π∗K −→ (π ◦ ιS)1ιS K ⊕ (π ◦ ιY )!ιY K −→ D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 13

b b By our choice of U, the ﬁrst term in this triangle is in Dc(CY ). The second term is in Dc(CY ) by Step 2. b −1 · b Thus, the last term is also in Dc(CY ). But π ◦ ιY = idY . Thus, ιY K ∈ Dc(CY ). This ﬁnishes the proof of the lemma, and also of Kashiwara’s constructibility theorem. · b The following is a more precise version of Theorem 4.9 in the case where M ∈ Dhol(DX ) is concentrated in degree 0.

Theorem 4.11. Let M ∈ DX -modhol. Then, DR(M) is a perverse sheaf on X. For a proof of Theorem 4.11 see e.g. [HTT, Theorem 4.6.6]. Let us give an example. If ι : X → Y is a closed embedding, then BX|Y = ι0OX is a holonomic DY -module. We have:

DRY (BX|Y ) = Rι∗ DRX (OX ) = CX [dim X] which is a perverse sheaf on Y .

5. Regular Holonomic D-modules b b an So far, we know that we have a functor DR : Dhol(X) → Dc(X ). This functor, however, is not an equivalence - for example, we have seen that it may carry nonisomorphic objects to isomorphic ones. Our goal in this section is to ﬁx this, by restricting to a special class of holonomic D-modules. Roughly speaking, we will restrict to a class of holonomic modules coming from connections on vector bundles whose singularities are at worst simple poles. We will start by deﬁning exactly what this means in the 1-dimensional case, to later generalize it to arbitrary dimensions. At the end we mention, without proof, the Riemann-Hilbert correspondence.

5.1. The 1-dimensional case.

5.1.1. The analytic setting. Let us ﬁx the notation D := {z ∈ C : |z| < 1}, the open complex disc, and × D := D \{0}, the punctured disc. We will consider these as complex manifolds, so OD is the sheaf −1 of holomorphic functions on D. Thus, OD[z ] is the sheaf of meromorphic functions on D which are × −1 holomorphic on D . In particular, OD[z ] ⊆ OD× . By ΩD we mean the sheaf of 1-forms on D, and −1 × ΩD[z ] are the meromorphic 1-forms on D which are holomorphic on D . We have a distinguished subsheaf −1 ΩD,log ⊆ ΩD[z ]. This is the sheaf consisting of those 1-forms that have a pole of order at most 1 at 0. Deﬁnition 5.1. A meromorphic connection on D, holomorphic on D× consists of a vector bundle M on D together with a morphism

1 −1 (13) ∇ : M → ΩD[z ] ⊗OD M 0 0 that satisﬁes the Leibniz rule, that is ∇(fm) = df⊗m+f∇(m), f ∈ OD.A morphism ϕ :(M, ∇) → (M , ∇ ) × is a morphism of vector bundles on C , which is meromorphic at 0 and intertwines the connections.

Of course, a vector bundle M on D is trivial. Choose a trivialization M = OD ⊗C E, where E is a finite dimensional vector space. Using the Leibniz rule, it is easy to see that a connection has to be given by −1 ∇ = d + A(z), where A(z) ∈ Matn×n(ΩD[z ]). Now assume that we choose two trivializations of the same connection. These trivializations on D× differ by a meromorphic map D → GL(E) holomorphic on D×. If ∇ = d + A(z), ∇0 are the connections with respect to the two trivializations, we must have ∇0 = (1⊗g)∇g−1 = (1⊗g)(d+A)g−1 = gAg−1 +gd(g−1)+d. This motivates the following definition.

−1 −1 Deﬁnition 5.2. The gauge action of GLn(OD[z ]) on Matn×n(ΩD[z ]) is given by

g ◦ A := gAg−1 + gd(g−1)

0 −1 Thus, two matrices A, A ∈ Matn×n(ΩD(z )) deﬁne isomorphic connections on a rank n vector bundle if and only if A and A0 are gauge equivalent. 14 JOSE´ SIMENTAL

Deﬁnition 5.3. Let (M, ∇) be a meromorphic connection on D, holomorphic on D×. We say that this connection has regular singularities if there exists an isomorphism (M, ∇) =∼ (M0, ∇0) where

0 0 0 ∇ : M → ΩD,log ⊗OD M equivalently, if there exists a trivialization of the connection such that A ∈ Matn×n(ΩD,log). Let us remark that a trivial vector bundle with meromorphic connection d + A may have regular singularities even if A 6∈ Matn×n(ΩD,log). This is why we had to introduce the gauge action. Indeed, let us consider −1 the case where the ﬁbers of the bundle have dimension 2. Take any meromorphic function b(z) ∈ OD[z ] and consider

0 b(z)dz A(z) = ∈ Mat (Ω [z−1]) 0 0 2×2 D −1 −1 Now let eb(z) = b(z) − c−1(b)z be the function obtained by deleting the z term on the Laurent expansion R 0 × around 0 of b, and let u := eb(z)dz. Thus, b − u is meromorphic on D, holomorphic on D , with a simple pole at 0 and, if

1 u g = ∈ GL (O [z−1]) 0 1 2 D then it is easy to see that

0 (b − u0)dz g ◦ A = ∈ Mat (Ω ) 0 0 2×2 D,log So the connection defined by g ◦ A, and therefore also that defined by A, has regular singularities. Let us now rephrase the definition of regularity in terms of the theory of D-modules. A meromorphic flat × −1 connection, holomorphic on D as in (13) allows us to define a DD-module structure on M[z ]. Assume that this bundle has regular singularities and that the connection ∇ = d + A(z) is already trivialized so that −1 A(z) ∈ Matn×n(ΩD,log). Then, we have that ∂zM ⊆ z M so that, in particular, z∂zM ⊆ M. Thus, we have the following result. Proposition 5.4. Let (M, ∇) be a meromorphic connection on D, holomorphic on D×. Consider the −1 induced DD-module structure on M[z ]. Then, the connection (M, ∇) has regular singularities if and only −1 −1 if there exists a OD-coherent submodule N ⊆ M[z ] with N |D× = M[z ] that is stable under the action of z∂z. Before moving on to the algebraic case, let us first mention that the categories of meromorphic connections on D, holomorphic on D× with regular singularities is closed under taking tensor products, duals and internal homs. Moreover we have the following proposition, which can be proved by analizing the growth of the flat sections of a regular connection on D. Theorem 5.5. Restriction to D× is an equivalence of categories from the category of regular meromorphic flat connections on D holomorphic on D× with regular singularities to the category of holomorphic flat connections on D×. For a proof of this Theorem, see [HTT, Section 5]. 5.1.2. The algebraic setting. For us, Proposition 5.4 is a motivation for the definition of regular singularities × in the algebraic setting. Let X := C and U := C . Let us denote by ι : U → X the open embedding. Of −1 −1 course, we have that OX = C[z], OU = C[z, z ], DX = Chz, ∂zi and DU = Chz, z , ∂zi. Finally, let s := z∂z 0 0 be the Euler vector field, and DX ⊆ DX is DX := Chz, si. Note that, for any OU -coherent DU -module M, 0 R the pushforward ι0M = H ι M is holonomic.

Deﬁnition 5.6. We say that a OU -coherent DU -module M is regular at 0 if the pushforward ι0M is a 0 union of OX -ﬁnitely generated DX -submodules. Let us give examples of regular and non-regular modules. D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE 15

Example 5.7. Let M = OU with trivial connection. This is a regular DU -module. This follows easily from m m z∂z(z ) = mz for any m. −1 −1 Example 5.8. Now consider M = OU exp(z ). This is the DU -module “generated by exp(z )”, the −1 −2 −1 action of ∂z is given, of course, by ∂z exp(z ) = −z exp(z ). We claim that this is not regular. Indeed, −1 −1 −1 −1 −1 −1 ι0M = OX [z ] exp(z ). Since z∂z exp(z ) = −z exp(z ), we can see that Chz, si exp(z ) = ι0M. −i Thus, M is not regular. More generally, M = OU exp(z ) is not regular for i > 0.

Example 5.9. Let M = OU log(z). We leave it as an exercise to check that this is regular. 5.1.3. Regular singularities on curves. Now we define regular singularities on a curve C. So from now on we fix a smooth curve C and a projective, smooth curve Cb with an embedding ι : C → Cb where C is open, dense in Cb. Let Z := Cb \ C, and let DZ be the subsheaf of D locally generated by O and vector fields Cb Cb Cb vanishing at Z.

Definition 5.10. We say that a OC -coherent DC -module M has regular singularities if ι0M is a union of O -finitely generated DZ -submodules. Cb Cb We remark that, a priori, the definition depends on the embedding ι : C → Cb. It is a theorem of Deligne that the definition is actually independent of this. From Proposition 5.4 we get the following.

Proposition 5.11. Let M be a OC -coherent DC -module with regular singularities. Take a vector bundle Z extension Mc to Cb that has an action of DX - this exists by definition. Take a point s ∈ Z, and an analytic an disc neighborhood of s, say D. Then, McD is a flat meromorphic connection, holomorphic on D \{s}, with regular singularities. The following result is now, basically, a corollary of Theorem 5.5 and Serre’s GAGA. Theorem 5.12. Let C be a smooth algebraic curve. Then, the analytification functor defines an equivalence between the following categories:

• The category of OC -coherent DC -modules with regular singularities. • The category of holomorphic vector bundles on C with ﬂat connection.

We remark that another consequence of Serre’s GAGA is that the category of OC -coherent DC -modules with regular singularities is closed under extensions in the category of DC -modules. We leave this as an exercise. Recall that a holonomic D-module is generically O-coherent. Then, we have the following deﬁnition.

Deﬁnition 5.13. Let X be a smooth algebraic curve, and let M be a holonomic DX -module. We say that M has regular singularities (or that M is regular holonomic) if there exists an open, dense subset U ⊆ X such that M|U is a OU -coherent DU -module with regular singularities.

For an algebraic curve C, we denote by DX -modrh the category of regular holonomic DC -modules, and b by Drh(DC ) the category of bounded complexes of DC -modules whose cohomology is regular holonomic. 5.2. General algebraic varieties. For a general smooth algebraic variety X, the deﬁnition of regular holonomic D-modules is more complicated. Let us start by deﬁning what does it mean for a OX -coherent DX -module to have regular singularities.

Deﬁnition 5.14. Let X be a smooth algebraic variety. Then, a OX -coherent DX -module M is regular holonomic if, for any smooth curve C ⊆ X, the restriction M|C is regular holonomic.

Recalling the classiﬁcation of simple holonomic DX -modules, Subsection 3.6, we can now give a deﬁnition of a simple regular holonomic DX -module.

Definition 5.15. Let X be a smooth algebraic variety, and let L(Y, M) be a simple holonomic DX -module, where Y is a locally closed subvariety of X such that the inclusion is affine and M is a simple OY -coherent DY -module. Then, we say that L(Y, M) is regular holonomic if M is regular holonomic in the sense of Definition 5.14.

And now we can deﬁne a regular holonomic DX -module in general. 16 JOSE´ SIMENTAL

Definition 5.16. Let M be a holonomic DX -module. Then, we say that M is regular holonomic if every simple subquotient of M is regular holonomic in the sense of Definition 5.15. Of course, the chain of definitions 5.14, 5.15, 5.16 is annoying. It has a slight advantage: if we denote by DX -modrh the category of regular holonomic DX -modules, it is obvious by definition that this category is b b closed under extensions. Let us denote by Drh(DX ) the triangulated subcategory of Dhol(DX ) consisting of complexes whose cohomology is regular holonomic. The following criterion is used in practice much more than the chain of definitions 5.14, 5.15, 5.16. For a proof, see [HTT, Chapter 6]. · b Theorem 5.17. Let X be a smooth algebraic variety. For M ∈ Dhol(DX ), the following are equivalent. · b (1) M ∈ Drh(DX ). † · b (2) ιC (M ) ∈ Drh(DC ) for any locally closed embedding of a smooth algebraic curve ιC : C,→ X.

So, while slightly imprecise, perhaps the best way to think of a regular holonomic DX -module is as a holonomic DX -module whose restriction to every curve is regular holonomic, i.e., it is generically a flat connection whose worst singularities are simple poles. 5.3. The Riemann-Hilbert correspondence. We finish these notes by stating, without proof, the (al- b b an gebraic) Riemann-Hilbert correspondence. Recall that we have a functor DR : Dh(DX ) → Dc(X ). This functor, of course, has no chance of being an equivalence: as we have seen, analytification may send nonisomorphic modules to isomorphic ones. This problem is fixed if we restrict our attention to the class of regular holonomic modules. A first hint of this is Theorem 5.12 for a curve. The following theorem is Deligne’s version of the Riemann-Hilbert correspondence.

Theorem 5.18. Let X be a smooth algebraic variety. Then, the functor DRX induces equivalences. b b an (1) Between the categories Drh(DX ) and Dc(X ). an (2) Between the categories DX -modrh and that of perverse sheaves on X . For a proof, the reader may consult [B], [HTT, Chapter 7] or [AN, Chapter 6].

References [AN] S. Arkhipov, N. Nikolaev. D-modules. Lecture notes. Available at http://www.math.toronto.edu/nikolaev/files/ 130726232840.pdf [B] J. Bernstein, Algebraic theory of D-modules. Available at http://www.math.uchicago.edu/~mitya/langlands/Bernstein/ Bernstein-dmod.ps [BCEY] A. Braverman, T. Chmutova, Lectures on algebraic D-modules. Edited and expanded by P. Etingof and D. Yang. Available at http://www-math.mit.edu/~etingof/dmlec1.pdf [HTT] R. Hotta, K. Takeushi, T. Tanisaki, D-modules, Perverse Sheaves, and Representation Theory. Progress in Mathematics 236. Birkh¨auser,Boston, MA. 2008. [K] M. Kashiwara, D-modules and Microlocal Calculus. Translations of Mathematical Monographs, 217. American Mathemat- ical Society, Providence, RI, 2003. [PS] C. Peters, J. Steenbrink, Mixed Hodge structures. Results in Mathematics and Related Areas. 3rd Series, 52. Springer- Verlag, Berlin, 2008.

Department of Mathematics, Northeastern University. Boston, MA 02115. USA. E-mail address: [email protected]