Algebraic Theory of D-modules

J. Bernstein

Lecture 1. D-modules and functors.

§0. Introduction.

1. In my lecture I will discuss the theory of modules over rings of differential operators (for short Dmodules). This theory started about 15 years ago and now it is clear that it has very valuable applications in many fields of mathematics. Names: Sato, Kashiwara, Kawai, Bernstein, Roos, Björk, Malgrange, Beilinson.

2. I will speak on an interpretation of the theory, given by Beilinson and myself. We restrict ourselves to purely algebraic theory of D-modules over any algebraically closed field k of characteristic 0. Sato and Kashiwara worked for analytic varieties over ℂ, so as usual, our theories are interlapped like this . I should mention from the very beginning, that some of the most important technical notions and results are due to Kashiwara.

§1. 1. O-modules.

So we fix an algebraically closed field k of char 0. One can assume k = ℂ.

Let X be an algebraic variety (over k), O_{X} the structure sheaf. Let F be a sheaf of O_{X}-modules. I call F quasi-coherent sheaf of O_{X}-modules (or O_{X}-module) if it satisfies the condition: (*) If U ⊂ X is open affine subset, f ⊂ O(U), U_{f} = {u ∈ U|f(u) ≠ 0}, then F (U_{f}) = F (U)_{f} =^{def} O(U_{f})⨂_{o(U)}F.

By Serre’s theorem this condition is local.

Let μ(O_{X}) be the category of O-modules. Locally, i.e., on an open affine subspace, U ⊂ X, I will identify μ(O_{U}) with the category of C-modules, where C = O(U).

2. Differential operators and D-modules.

By definition, a differential operator of order ≤ k on U is a k-linear morphism d : C → C, such that [f̂_{k} … [f̂_{1}[f̂_{o}, d]]] = 0 for any f_{o}, …, f_{k} ∈ C, where f̂: C → C is an operator of multiplication by f.

The ring of different operators on U I denote by D(U), O(U) ⊂ D(U).

Proposition. D(U_{f}) = O(U_{f}) ⨂_{o(U)} D(U) = D(U) ⨂_{o(U)} O(U_{f}). Hence U → D(U) is a quasicoherent sheaf of O_{X}-modules. I denote it by D_{X} and call the sheaf of differential operators on X.

D-module is by definition a sheaf F of left D_{X}-modules which is quasi-coherent as O_{X}-module. Category of D_{X}-modules I will denote by μ(D_{X}). Locally, on affine open set U, μ(D_{U}) ≅ D(U)-mod.

If X is singular, D_{X} can be bad (for instance, it can be not locally noetherian). So from now on I assume X regular, if I don’t say otherwise. Lemma. 1. For each x ∈ X there exist an affine neighbourhood U ⊃ x functions x_{1}, …, x_{n} on U and vector fields ∂_{1}, …, ∂_{n} on U such that ∂_{i}(x_{j}) = δ_{ij}, ∂_{i} generate tangent bundle of X.

2. D(U) = O(U) ⨂_{k}k[∂_{1}, …, ∂_{n}].

The system (x_{i}, ∂_{i}) I will call the coordinate system in D_{X}.

3. So I introduced main characters of my story and can begin the play. It is very useful, though formally not necessary, to have in mind some analytic picture, corresponding to D-modules. Let me describe it. Analytic picture. Suppose we have a system S of p linear differential equations on q functions f_{1} … f_{q}, S = {∑_{j=1}^{q}d_{ij}f_{j} = 0, i = 1, …, p}. Then we can assign to S a D-module M given by q generators e_{1}, …, e_{q} and p-relations M = ⊕D · e_{j}/(+D(∑ d_{ij}e_{j})). In this language, a solution s of the system S in some space of functions F is nothing else than a morphism of D-modules α_{S} : M → F.

Having in mind this picture we can start investigation of D-modules.

4. Left and right D-modules.

Let us denote by μ^{R}(D_{X}) the category of right D-modules. How is it connected with μ(D_{X})? Motivation. In analytic picture, the space of functions F is a left D-module. But if we consider the space of distributions F^{∗}, it has a natural structure of a right D-module. Hence systems of differential equations for distributions correspond to right D-modules.

But if we fix a differential form w of highest degree, we can identify F and F^{∗} by ϕ ∈ F ⟼ ϕw ∈ F^{∗}. Proposition-Definition. Let Ω = Ω_{X} be the O_{X}-module of differential forms of highest degree on X. For any D_{X}-module F denote by Ω(F) the right D_{X}-module, given by Ω(F) = Ω ⨂_{o_{X}} F

f(w ⊗ u) = fw ⊗ u, ξ(w ⊗ u) = −Lie_{ξ}(w) ⊗ u − w ⊗ u.

Functor Ω : μ(D_{X}) → μ^{R}(D_{X}) is an equivalence of categories.

I prefer to use a slightly different description of Ω. Consider the module ^{Ω}D_{X} = Ω(D_{X}) = Ω ⨂_{o_{X}} D_{X}. It has two different structures of a right D_{X}-module – one as Ω( ), and another from the endomorphism of left D_{X}-module D_{X}, which are given by right multiplications. It is easy to check that there exists a unique involution v of ^{Ω}D_{X}, which interchanges these two structures of right D_{X}-module and is identical on Ω ⊂ ^{Ω}D_{X}. By definition,

Ω(F) = ^{Ω}D_{X} ⨂_{D_{X}} F.

The inverse function Ω^{−1} : μ^{R}(D_{X}) → μ(D_{X}) is given by multiplication on the module D_{X}^{Ω} = D_{X} ⨂_{o_{X}} Ω^{−1} = Hom_{D_{X}} (^{Ω}d_{X}, D_{X}), which has two structures of left D-module.

We will work with left D-modules but remember that we can go freely to right D-modules and back.

5. Inverse image of D-modules.

Let π : Y → X be a morphism of algebraic varieties.

Motivation. We can lift a function from X to Y. If they satisfy some system of equations S, then their images also would satisfy some system of equations S′. Is it possible to describe this system?

It turns out that we can do some algebraic version of this. Namely, I will describe a functor

π^{Δ} : μ(D_{X}) ⟶ μ(D_{Y}).

First do it locally, i.e., suppose X and Y are affine, and D_{X}-module is given by a D_{X} = D(X)-module M. Then put

π^{Δ}(M) = O_{y} ⨂_{o_{X}} M

and define the action of D_{Y} on π^{Δ}(M) by

(*)  f′(f ⊗ m) = f′f ⊗ m, ξ(f ⊗ m) = ξf ⊗ m + f(∑_{i}ξ(x_{i}) ⊗ ∂_{i}m),

where (x_{i}, ∂_{i}) is a coordinate system in D_{X}. It is easy to check that this definition is correct. Intuitively, it is a version of the chain rule.

Now we can write the general definition

π^{Δ}(F) = O_{Y} ⨂_{π·(o_{X})} π^{·}(F),

where π^{·} is an inverse image in the category of sheaves and the action of D_{Y} is given by (*). Again, it is convenient to rewrite this definition slightly. Put

D_{Y →X} = π^{Δ}(D_{X}).

D_{Y →X} is a sheaf on Y, which is D_{Y} − π^{·}(D_{X})-bimodule. By definition

π^{Δ}(F) = D_{Y →X} ⨂_{π·(D_{X})} π^{·}(F).

Note that as an O_{Y} -module π^{Δ}(F) coincides with an inverse image π^{∗}(F), but I would like to save notation π^{∗} for other case.

Lemma. If τ : Y → Z is a morphism we have (τπ)^{Δ} = π^{Δ}τ^{Δ}.

6. Direct image of D-modules.

Motivation. We can integrate functions on Y (say with compact support) to get functions on X. How does this affect systems of equations they satisfy?

First of all, we should realize that there is no natural way of integrating functions, but there is a natural way of integrating distributions (namely ⟨∫ E, ϕ⟩ = ⟨E, π^{∗}(ϕ)⟩). Hence we should try to construct a functor π_{+} : μ^{R}(D_{Y}) ⟶ μ^{R}(D_{X}).

First consider a local case. Then we can put π_{+}(N) = N_{D_{Y}} ⊗ D_{Y →X}, where N is a right D_{Y} -module. Or, in terms of sheaves,

π_{+}(H) = π_{·}(H_{D_{Y}} ⊗ D_{Y →X}),

where π_{·} is the direct image in the category of sheaves.

Since we can freely go from left D-modules to right D-modules and back, we can rewrite this functor for left D-modules. Since I prefer to work with left D-modules, let us do it.

Put D_{X→Y} = Ω(π^{Δ}(D_{X}^{Ω})) = Ω_{Y π·(o_{X})} ⊗ Ω_{X}^{−1}). This is a sheaf on Y, which is π^{·}(D_{X}) − D_{Y} -bimodule. Now we define the functor π_{+} : μ(D_{Y}) ⟶ μ(D_{X}) by (*)

(*)  π_{+}(H) = π_{·}(D_{X←Y} ⨂_{D_{Y}} H).

Now let us try to handle the general case (X and Y are not affine). Then we immediately run into trouble. The matter is, that formula (*) describes π_{+} as a composition of left exact functor π_{·} and right exact functor D_{X←Y} ⊗, and this composition apparently does not make much sense (formally, it affects the composition rule, (τπ)_{+} ≠ τ_{+}π_{+}).

Definition (*) makes some sense for affine morphism π [when π^{−1} (affine open subset) is affine], since in this case π_{·} is an exact functor. But in order to study the general case, we should work in derived categories.

7. D-complexes and functors.

Henceforth I assume all algebraic varieties to be quasiprojective.

Proposition. Category μ(D_{X}) has enough injective and locally projective objects. It has a finite homological dimensin (we will see that it is ≤ 2dimX).

Definition. D_{X}-complex is a bounded complex of D_{X}-modules. Corresponding derived category, which consists of D_{X}-complexes up to quasi-isomorphisms we denote by D(D_{X}).

D_{X}-complexes I will often denote by Ḟ , Ḣ, …. We define functors

Ω : D(D_{X}) ⟶^{∼} D^{R}(D_{X})

Lπ^{Δ} : D(D_{X}) ⟶ D(D_{Y}), for π : Y → X, by

Lπ^{Δ}(Ḟ) = D_{Y →X} ⨂_{π·(D_{X})}^{L}π^{·}(Ḟ).

π_{∗} : D(D_{Y}) → D(D_{X}) by

π_{∗}(Ḣ) = Rπ_{·}(D_{X←Y} ⨂_{D_{Y}}^{L}Ḣ).

Proposition. L(τ ∘ π)^{Δ} = Lπ^{Δ} ∘ Lτ^{Δ}

(τπ)_{∗} = τ_{∗}π_{∗}.

Usually we will decompose π as a product of a locally closed imbedding and a smooth morphism. So let us consider these cases in more detail.

8. Case of a closed imbedding i : Y → X.

Let us define functors

i_{+} : μ(D_{Y}) ⟶ μ(D_{X}) by i_{+} = i_{·}(D_{X→Y} ⊗_{D_{Y}} H)

i^{+} : μ(D_{X}) ⟶ μ(D_{Y}) by i^{+}(F) = Hom_{i^{·}D_{X}} (D_{X←Y′}i^{·}(F)).

Lemma. i_{+} is left adjoint to i^{+}; i_{+} is exact and i^{+} is left exact.

\begin{aligned}Ri_{+} & = i_{∗′} \\ Ri^{+} & = Li^{Δ}[dim Y − dimX]. \end{aligned}

It turns out that it is quite convenient to use shifted functor Lπ^{Δ}[dim Y − dim X] which in the case of imbedding coincides with Ri^{+}. So I put

π^{!} = Lπ^{Δ}[dim Y − dim X] : D(D_{X}) ⟶ D[D_{Y}].

For any closed subset Z ⊂ X I denote by μ_{Z}(X) the full subcategory of μ(X), consisting of D_{X}-modules F such that supp F ⊂ Z.

Theorem (Kashiwara). Let i : Y → X be a closed imbedding. Then functors i_{+} : μ(D_{Y}) → μ_{Y} (D_{X}) and i^{+} : μ_{Y} (D_{X}) → μ(D_{Y}) are inverse and define an equivalence of categories.

This simple technical statement is very important and very useful.

9. D-modules on singular varieites.

Let Z be a singular variety. Then the algebra D_{Z} can be very bad, so it does not make sense to study modules over D_{Z}. But using Kashiwara’s theorem we can define category of D-modules on Z (which we denote by μ(D_{Z}) though it is not category of D_{Z}-modules) in the following way.

Let us realize Z as a closed subvariety of a nonsingular variety X and put by definition

μ(D_{Z}) = μ_{Z}(D_{X}).

Even if we cannot realize Z as a subvariety, we can do it locally. Now, Kashiwara’s theorem implies, that at least locally, μ(D_{Z}) is correctly defined. Glueing pieces together we can construct μ(D_{Z}) globally.

We define D(D_{Z}) as derived category of μ(D_{Z}). If Z is a closed subset of X, one can show that

D(D_{Z}) = D_{Z}(D_{X}) = {Ḟ ∈ D(D_{X})|supp Ḟ ⊂ Z, i.e., Ḟ |_{X\Z} = 0}.

Later I will discuss only nonsingular varieties, but all results can be transferred to the singular case.

10. Proof of Kashiwara’s theorem.

We should prove that natural morphisms of functors

Id_{μ(D_{Y} )} ⟶ i^{+}i_{+}, i_{+}i^{+} ⟶ Id_{μ_{Y} (D_{X})}

are isomorphisms. It is sufficient to check locally, so I can assume that X is affine, Y is given by equations x_{1}, …, x_{ℓ}. Using induction by ℓ I can assume that Y is given by one equation x. Locally I can choose a vector field ∂ such that ∂(x) = 1, i.e., [∂, x] = 1.

If F ∈ μ_{Y} (D_{X}), then supp F ⊂ Y and since F is quasicoherent, any section ξ ∈ F is annihilated by large powers of x.

Consider the operator I = x∂ and put F^{i} = {ξ|Iξ = iξ}. Then it is clear that x : F^{i} → F^{i+1}, ∂ : F^{i} → F^{i−1}, x∂ : F^{i} → F^{i} is an isomorphism for i < 0, ∂x = x∂ + 1 is an isomorphism for i < −1. Hence x : F^{i} → F^{i+1} and ∂ : F^{i+1} → F^{i} are isomorphisms for i < −1. If ξ ∈ F and xξ = 0, then x∂ξ = ∂xξ − ξ = −ξ, i.e., ξ ∈ F^{−1}. By induction on k it is easy to prove, that if x^{k}ξ = 0, then ξ ∈ F^{−1} ⊕ … ⊕ F^{−k}. Hence F = ⨁_{i=1}^{∞}F^{−1} = k[∂]⨂_{k}F^{−1} and Ker (x, F) = F^{−1}. This is the statement of Kashiwara’s theorem.

Lecture 2.

1. Some applications of Kashiwara’s theorem.

a) Structure of O-coherent D_{X}-modules.

We say that D_{X}-module F is O-coherent if locally it is a finitely generated O_{X}-module. Proposition. O-coherent D_{X}-module F is locally free as O_{X}-module. Proof. Let x ∈ X, m_{X} corresponding maximal ideal of O_{X}. The space F_{x} = F/m_{x}F is called the fiber of F at x. Since F is coherent as O_{X}-module, it is sufficient to check that dim F_{x} is a locally constant function on X. This we can check for restriction of F on any nonsingular curve C ⊂ X. Hence we can replace X by C and F by i_{C→X}^{Δ}(F), and assume that X is a curve.

If F has a torsion at a point x, then F contains a nonzero subsheaf (i_{x})_{+}i_{x}^{+}(F), which is not O-coherent. Hence F has no torsion, and, since X is a curve, F is locally free. Q.E.D.

Recall that locally free O_{X}-modules F naturally correspond to the algebraic vector bundles E on X (F is a sheaf of sections of E). Action of D_{X} on F defines a connection on E, by ∇_{ξ}(ϕ) = ξϕ. Since [∇_{ξ}, ∇_{η}] = ∇_{[ξ,η]} this connection is flat.

This gives an equivalence of categories

\{0 − coherent D_{X} − modules\} = \{ \begin{matrix}algebraic vector bundles on X \\ with flat connection \end{matrix} .

b) D-modules on projective space.

Let V = k^{n} be an affine space over k, V^{∗} = V \ {0}, X = ℙ(V)-corresponding projective space, pr : V^{∗} → ℙ(V) the natural projection. Theorem. Functor of global sections Γ : μ(D_{X}) → Vect, F → Γ(X, F) is exact, and each D_{X}-module F is generated by its global sections (i.e., D_{X} ⊗ Γ(F) → F is an epimorphism). Remark. Note that Γ(F) = Hom_{μ(D_{X})}(D_{X}, F). Hence theorem simply means that D_{X} is a projective module and is a generator of category μ(D_{X}). Proof. For any D_{X}-module F put F^{Δ} = pr^{Δ}(F) ∈ μ(D_{V^{∗}}). This sheaf carries a natural action of the homotety group k^{∗} and hence the space of sections Γ(F^{Δ}) is a graded space ⨁_{n=−∞}^{∞}Γ(F^{Δ})^{n}. It is clear that Γ(F) = Γ(F^{Δ})^{0} – zero component. If we denote by I ∈ D_{V} the Euler operator ∑ x_{i}∂_{i}, which is an infinitesimal generator of the group k^{∗}, then it defines a grading on Γ(F^{Δ}), i.e., its action on Γ(F^{Δ})^{n} is multiplication by n.

Functor F → F^{Δ} is exact, hence all nonexactness can come only from the functor Γ_{V^{∗}}. Let us decompose it as Γ_{V^{∗}} = Γ_{V} ∘ j_{+} : μ(D_{V^{∗}}) → μ(D_{V}) → Vect, where j : V^{∗} ↪ V. Since V is affine, functor Γ_{V} is exact.

Let 0 → F_{1} → F_{2} → F_{3} → 0 be an exact sequence of D_{X}-modules. Then the sequence 0 → j_{+}(F_{1}^{Δ}) → j_{+}(F_{2}^{Δ}) → j_{+}(F_{3}^{Δ}) → 0 is exact when restricted to V^{∗}, hence its cohomologies are sheaves on V, concentrated at 0.

By Kashiwara’s theorem each sheaf concentrated at 0 is a direct sum of many copies of a standard D_{V} -module Δ = P ∘ 1[∂_{1}, …, ∂_{k}]δ, where x_{i}δ = 0. This implies that eigenvalues of I on Γ(Δ) are equal −n, −n − 1, −n − 2, … −. Hence the sequence

0 ⟶ Γ(F_{1}^{Δ})^{o} ⟶ Γ(F_{2}^{Δ})^{o} ⟶ Γ(F_{3}^{Δ})^{o} ⟶ 0

is exact, since Γ = Γ_{V} is an exact functor and sheaves, concentrated at 0, do not affect 0-graded part.

The statement, that any D_{X}-module is generated by its global sections can be reduced, using exactness of Γ, to the statement F ≠ 0 ⟹ Γ(F) ≠ 0. This is proved in the same way as exactness of Γ.

2. Case of an open imbedding.

Let j : V → X be an open imbedding. Then j^{Δ} is an exact functor of restriction, i.e., j^{!} = j^{Δ}, and j_{+} is the usual functor of direct image in category of sheaves. Its derived functor Rj_{+} equals j_{∗}. In particular case when j is an affine imbedding the functor j_{+} is exact, i.e., j_{∗} = j_{+}.

Functor j^{Δ} is left adjoint to j_{+} and j^{Δ}j_{+} = Id_{U}. For arbitrary D_{X}-module F the kernel and cokernel of the morphism F ⟶^{α} j_{+}j^{Δ}F are supported on the closed subset Z = X \ U.

Let us consider the functor Γ_{Z} : μ(D_{X}) ⟶ μ(D_{X}) given by Γ_{Z}(F) = {ξ ∈ F |supp ξ ⊂ Z}. Then we have an exact sequence

0 ⟶ Γ_{Z}(F) ⟶ F ⟶^{α} j_{+}j^{Δ}F.

If F is an injective D_{X}-module, α is onto. Hence in derived category we always have an exact triangle

(*)  RΓ_{Z}(Ḟ) ⟶ Ḟ ⟶ j_{∗}j^{!}Ḟ .

We will call this triangle a decomposition of Ḟ with respect to (U, Z).

Denote by D_{Z}(D_{X}) the full subcategory of D(D_{X}), consisting of D_{X}-complexes Ḟ such that Ḟ |_{U} = 0. Then (*) implies that the natural inclusion D(μ_{Z}(D_{X})) → D_{Z}(D_{X}) is an equivalence of categories.

3. Base change.

Theorem. Consider Cartesian square

Z ⟶^{α̌} Y
↓π̌ ↓π
S ⟶^{α} X

i.e., Z = Y ×_{X} S. Then functors α^{!}π_{∗} and π̌_{∗}α̌^{!} : D(D_{Y}) ⟶ D(D_{S}) are naturally isomorphic.

Corollary. If Z = ∅, i.e., α(S) ∩ π(Y) = ∅, then α^{!}π_{∗} = 0.

Sketch of the proof. It is sufficient to consider 2 cases

i) α is a projection T × X → X ii) α is a closed imbedding.

The case (i) is straightforward. In (ii) let U be a complement of S,

V = π^{−1}(U) = Y \ Z,  j : U → X,  ǰ : V → Y.

We have natural exact triangles

α̌_{∗}α̌^{!}Ḣ ⟶ Ḣ ⟶ j_{∗}ǰ ^{ ˇ !}Ḣ
α_{∗}α^{!}Ḟ ⟶ Ḟ ⟶ j_{∗}j^{!}Ḟ .

Put Ḟ = π_{∗}Ḣ. Then since we clearly have a base change for an open subset U, we have π_{∗}(j_{∗}ǰ ^{ ˇ !}H) ≅ j_{∗}j^{!}Ḟ. Hence, since π_{∗} is an exact functor in derived categories, we have

π_{∗}(α̌_{∗}α̌^{!}Ḣ) ≅ α_{∗}α^{!}Ḟ .

But π_{∗}α̌_{∗} ≅ α_{∗}π̌_{∗}, i.e., α_{∗}(π̌_{∗}α̌^{!}H) ≅ α_{∗}(α^{!}π_{∗}H). By Kashiwara’s theorem we can remove α_{∗}, which gives us the base change.

4. Let S = {X = ⋃_{i=0}^{n}X_{i}} be a smooth stratification of X, i.e., each X_{i} is a locally closed nonsingular subvariety, and X_{o}∪X_{i}∪…∪X_{j} is closed for each j. For each i consider the functor S_{i} : D(D_{X}) → D(D_{X}), where S_{i} = r_{I^{∗}}r_{i}^{!}, r_{i} : X_{i} → X. Then each D_{X}-complex Ḟ is glued from S_{i}(Ḟ), i.e., we have D_{X}-complexes Ḟ_{i} and exact triangles

Ḟ_{i−1} ⟶ Ḟ_{i} ⟶ S_{i}(Ḟ) such that

Ḟ_{−1} = 0, Ḟ_{n} = Ḟ.

We will call {S_{i}(Ḟ)} the stratification of Ḟ with accordance to S, and D_{X_{i}} complexes r_{i}^{!}(Ḟ) components of the stratification.

5. Case of smooth (submersive) morphism π : Y → X.

For any smooth variety Y let us denote by DR_{Y} the de Rham complex Ω_{Y}^{0} → Ω_{Y}^{1} → … → Ω_{Y}^{k} of sheaves on Y. More generally, if H is a D_{Y} -module, we can by the same formulae define de Rham complex DR_{Y} (H) with components DR_{Y} (H)^{i} = Ω_{Y}^{i} ⊗_{o_{Y}} H.

It is clear that DR_{Y} (D_{Y}) is a complex of right D_{Y} -modules. Now, let π : Y → X be a smooth morphism. Denote by Ω_{Y/X}^{i} sheaves of relative i-forms on Y. In the same way as earlier we can define the relative de Rham complex DR_{Y/X}(H) for any D_{Y} -module H.

Lemma. DR_{Y/X}(D_{Y})[k] = D_{X←Y} as a complex of right D_{Y} -modules.

Hence we can calculate the direct image functor π_{∗} using this complex:

π_{∗}(H) = Rπ_{·}(D_{X←Y} ⨂_{D_{Y}}^{L}H) = Rπ_{·}(DR_{Y/X}(D_{Y})⨂_{D_{Y}}H)[k] = Rπ_{·}(DR_{Y/X}(H)[k].

The only trouble here is that this formula defines π_{∗}(H) only as a complex of O-modules. Action of vector fields in general is described by quite unpleasant formulae. In the case when π is a projection τ : Y = T × X → X, action of vector fields is given by their action on H.

6. Coherent D_{X}-modules and D_{X}-complexes.

D_{X}-module F is called coherent if locally it is finitely generated. We’ll see that locally D_{X} is a noetherian ring, hence any submodule of a coherent D_{X}-module F is coherent.

Any D_{X}-module F is a union of coherent O_{X}-submodules L_{α}. If we put F_{α} = D_{X}L_{α} we see that F is a union of coherent D_{X}-submodules F_{α}. It implies:

(i) Any coherent D_{X}-module F is generated by a coherent O_{X}-submodule F^{o}. (ii) Extension principle. If H is a D_{X}-module, U ⊂ X an open subset, F ⊂ H|_{U} – a coherent D_{U}-

submodule, then then exists a coherent D_{X}-submodule H′ ⊂ H such that H′|_{U} = F. Category of coherent D_{X}-modules I denote by μ_{coh}(D_{X}). D_{X}-complex Ḟ is called coherent if all its homology sheaves H^{i}(Ḟ) are coherent D_{X}-modules. The

full subcategory of D(D_{X}) consisting of coherent D_{X}-complexes I will denote by D_{coh}(D_{X}).

Properties of coherent D_{X}-modules imply Lemma. The natural morphism D(μ_{coh}(D_{X})) ⟶ D_{coh}(D_{X}) is an equivalence of categories.

7. Direct image of proper morphism. Proposition. Let π : Y → X be a proper morphism. Then π_{∗}D_{coh}(D_{Y}) ⊂ D_{coh}(D_{X}). Proof. If π is a closed imbedding, proposition follows from Kashiwara’s theorem. So consider the case when π : Y = ℙ × X ⟶ X is a projection, where ℙ is a projective space.

We can assume X to be affine. Then by 1(b) D_{Y} is a generator in μ_{coh}(D_{Y}) and hence it is sufficient to prove that π_{∗}(D_{Y}) ⊂ D_{coh}(D_{X}). But

π_{∗}(D_{Y}) = Rπ_{·}(D_{X←Y} ⨂_{D_{Y}}D_{Y}) = Rπ_{·}(D_{X} ⨂_{k}Ω_{ℙ})

= D_{X} ⨂_{k}Rπ_{·}(Ω_{ℙ}) = D_{X}[−dim ℙ] ∈ D_{coh}(D_{X}).

8. Good filtration and singular support of a D-module.

Consider the filtration D_{X}^{0} ⊂ D_{X}^{1} ⊂ … of D_{X} by order of an operator. Each D_{X}^{i} is a coherent O-module, D_{X}^{0} = O_{X} and D^{i} · D^{j} ⊂ D^{i+j}.

Let Σ = ⨁_{i=0}^{∞}Σ^{i}, Σ^{i} = D^{i}/D^{i−1} be the associated graded sheaf of algebras. Then Σ is commutative and naturally isomorphic to the algebra of regular functions on the cotangent bundle T^{∗}(X).

Let F be a D_{X}-module. A filtration on F is a filtration Φ = {F^{o} ⊂ … F^{k} ⊂ … } of F by Osubmodules such that F = ∪F^{j}, D^{i}F^{j} ⊂ F^{i+j}. The associated graded module F_{Σ} = ⊕F^{i}/F^{i−1} has a natural structure of Σ-module.

We say that filtration Φ is good if F_{Σ} is a coherent Σ-module. An equivalent condition is

(*)  Each F^{j} is a coherent O_{X} − module and D^{1}F^{j} = F^{j+1} for large j.

It is clear that D_{X}-module F with a good filtration is coherent. Conversely, if F is a coherent D_{X}- module, then it is generated by a coherent O_{X}-module F^{o} and we can define a good filtration Φ on F by F^{j} = D^{j}F^{o}.

Let F be a coherent D_{X}-module. Choose a good filtration Φ on F and denote by F_{Σ} the corresponding Σ-module. As a coherent Σ-module F_{Σ} has a support supp(F_{Σ}) ⊂ T^{∗}(X) (this support is a closed subvariety which is defined by the ideal J_{F} ⊂ Σ, equal to the annulator of F_{σ} in Σ).

Proposition. Supp(F_{Σ}) depends only on F and not on a particular choice of a filtration Φ.

We will denote this supp(F_{Σ}) as S.S.(F) ⊂ T^{∗}X and call it the singular support or the characteristic variety of F.

Proof. Let Φ, Ψ be two good filtrations of F. We say that Φ and Ψ are neighbour if F_{Φ}^{i+1} ⊃ F_{Ψ}^{i} ⊃ F_{Φ}^{i} for all i. For neighbour filtrations consider the natural morphism F_{ΣΦ} ⟶ F_{ΣΨ} and include it in the exact sequence

0 ⟶ K ⟶ F_{ΣΦ} ⟶ F_{ΣΨ} ⟶ C ⟶ 0.

It is easy to check that Σ-modules K and C are isomorphic (only the grading is shifted by 1). This proves the proposition for neighbour filtrations.

If Φ and Ψ are arbitrary good filtrations, we define the sequence of filtrations Φ_{k} by F_{Φ_{k}}^{i} = F_{Φ}^{i} + F_{Ψ}^{i+k}. It is clear that Φ_{k} and Φ_{k+1} are neighbour, Φ_{k} = Φ for k ≪ 0 and Φ_{k} = Ψ shifted on k for k ≫ 0. This proves the proposition.

Remarks. 1. Let F be a D_{K}-module with a good filtration Φ, H ⊂ F a D_{X}-submodule. Consider induced filtrations on H and F/H. Then we have an exact sequence 0 → H_{Σ} → F_{Σ} → F/H_{Σ} → 0. In particular, filtration on H is good, i.e., H is D_{X}-coherent. Also we have

S.S.F = S.S.H ∪ S.S.(F/H).

Moreover, let k = dim S.S.F. Then we can assign to each k-dimensional component W of S.S.F some multiplicity (the multiplicity of supp F_{Σ} at W; the proposition above really proves that this multiplicity is well defined). Put m_{k}(F) = sum of multiplicities of all k-dimensional components of S.S.F. Then

m_{k}(F) = m_{k}(H) + m_{k}(F/H).

2. It is easy to see that D_{X}-module F is O-coherent if and only if S.S.F ⊂ X ⊂ T^{∗}X.

9. Singular support and functors.

Usually it is very difficult to describe the effect of functors π_{+}, π^{Δ} on singular support. (For instance, these functors usually do not preserve D-coherency.) But there are 2 cases when it can be done.

a) Let i : Y → X be a closed imbedding, H ∈ μ(D_{Y}). Then i_{+}(H) is coherent if and only if H is coherent and

S.S.(i_{+}H) = {(x, ξ)|x ∈ Y, (x, P r_{T^{∗}(X)→T^{∗}(Y)}ξ) ∈ S.S.H}.

b) Let π : Y → X be a smooth (i.e., submersive) morphism, F ∈ μ(D_{X}). Then π^{Δ}(F) is coherent if and only if F is coherent and

S.S.(π^{Δ}F) = {(y, ξ)|ξ = d^{∗}π_{T^{∗}(πy)→T^{∗}(y)}η, (π(y), η) ∈ S.S.F }.

Let us note that in these two cases one important characteristic of S.S. is preserved. Namely, if we define the defect of F as def(F) = dim S.S.F − dim X, then the defect is preserved.

10. Theorem on defect. Theorem. Let F ≠ 0 be a coherent D_{X}-module. Then def(F) ≥ 0, i.e., dim S.S.(F) ≥ dim X. Proof. Suppose that dim S.S.(F) < n = dim X. Then F is supported on some proper closed subset Z ⊂ X. Restricting to an appropriate open subset we can assume that Z is not empty and nonsingular. By Kashiwara’s theorem F = i_{+}(H), where i : Z → X, H be a coherent D_{Z}-module. Then d(F) = d(H) < 0 and we have a contradiction by induction on dim X.

11. Holonomic D-modules.

Coherent D_{X}-module F is called holonomic if def(F) ≤ 0, i.e., dim S.S.(F) ≤ dim X, i.e., F has “minimal possible size”. Holonomic modules will play a central role in our discussion. Example. O-coherent D-modules are holonomic. The full subcategory of μ_{coh}(D_{X}), consisting of holonomic D_{X}-modules I will denote by Hol(D_{X}).

Proposition. a) Sub-category Hol is closed with respect to subquotients and extensions.

b) Each holonomic D_{X}-module has a finite length.

c) If F is a holonomic D_{X}-module, then there exists an open dense subset U ⊂ X such that F |U is O-coherent D_{U}-module.

Proof. a) and b) easily follow from Remark 1 in 8. Indeed if n = dim X, then m_{n}(H) is an additive characteristic on subquotients of F which is strictly positive by the theorem on defect. Hence F has a finite length. Another proof is based on the existence of a contravariant duality D : Hol → Hol, such that D^{2} = id_{Hol}, which will be proved next time. This duality implies that F satisfies together ascending and descending chain conditions, i.e., F has a finite length.

In the proof of c) put S = S.S.(F) \ X. Since F_{Σ} is a graded Σ-module, S is invariant with respect to homotety in fibers of T^{∗}X. It means that projection p : T^{∗}X → X has at least 1-dimensional fibers on S. Hence dim p(S) < dim S ≤ dim X. After replacing X by a suitable open subset U ⊂ X \ p(S) we can assume that S = ∅, i.e., S.S.F ⊂ X, i.e., F is O-coherent.

1. Main theorem A.

We call a D_{X}-complex Ḟ holonomic if all its cohomology sheaves H^{i}(Ḟ) are holonomic D_{X}-modules. The full subcategory of D(D_{X}) consisting of holonomic D_{X}-complexes we denote by D_{hol}(D_{X}).

Remark. I do not know whether the natural inclusion d(Hol(D_{X})) → D_{hol}(D_{X}) is an equivalence of categories. In a sense, I do not care.

Main theorem A. Let π : Y → X be a morphism of algebraic varieties. Then

π_{∗}D_{hol}(D_{Y}) ⊂ D_{hol}(D_{X}), π^{!}D_{hol}(D_{X}) ⊂ D_{hol}(D_{Y}).

The proof of the theorem is based on the following

Key lemma. Let i : Y → X be a locally closed imbedding, Ḣ ⊂ D_{hol}(D_{Y}). Then i_{∗}(Ḣ) ⊂ D_{hol}(D_{X}). We will prove the lemma in the subsection 8.

2. Proof of theorem A for π^{!}.

It is sufficient to check 2 cases

a) π is a smooth morphism (e.g., π is a projection π : Y = T × X → X). In this case π^{Δ} is exact and π^{Δ}(Hol) ⊂ Hol by 2.9, i.e., π^{!}T D_{hol}(D_{X}) ⊂ D_{hol}(D_{Y}).

b) i : Y → X is a closed imbedding. Let j : U = X \ Y → X be the imbedding of the complementary open set. For Ḟ ∈ D_{hol}(D_{X}) consider the exact triangle

i_{∗}(i^{!}Ḟ) ⟶ Ḟ ⟶ j_{∗}(Ḟ |U).

By the key lemma ki_{∗}(Ḟ |U) is a holonomic D_{X}-complex. Hence i_{∗}(i^{!}Ḟ) is also holonomic. Now since the functor i_{∗} is exact and preserves the defect of a module, we can conclude that i^{!}Ḟ is a holonomic D_{Y} -complex.

3. Criteria of holonomicity.

Criterion. Let Ḟ be a D_{X}-complex. Then Ḟ is holonomic iff Ḟ is coherent and for any point x ∈ X the fiber (i_{x}^{!}Ḟ) of F at x is finite dimensional.

Proof. Direction “only iff” follows from 2. To prove “if” direction we need some general

Lemma. Let F be a coherent D_{X}-module. Then there exists an open dense subset U ⊂ X such that F |U is locally free as O_{U}-module.

Proof. We assume X to be affine and irreducible. Consider a good filtration Φ on F and the associated Σ-module F_{Σ}. Since F_{Σ} is a finitely generated Σ-module and Σ is a finitely generated algebra over O_{X}, general results of commutative algebra imply that we can replace X by an open dense affine subset U ⊂ X such that F_{Σ}|U is free as O_{U}-module (see EGA IV, 6.9.2). Since F_{Σ} = ⊕F_{Σ}^{n} = ⊕(F^{n}/F^{n−1}), all modules F_{Σ}^{n} are projective as O_{U}-module. This proves the lemma.

Now let us prove that a coherent D_{X}-complex Ḟ with finite dimensional fibers is holonomic.

We use induction on dim S = supp Ḟ. Choose an open nonsingular subvariety Y ⊂ S such that dim(S \ Y) < dim S and put Ḣ = i_{Y}^{!} Ḟ ⊂ D(D_{Y}). Then Ḣ is coherent and hence, replacing Y by a suitable open dense subset, I can assume that all cohomology sheaves of Ḣ are locally free as O_{Y} -modules.

At each point y ∈ Y the fiber i_{Y}^{!} Ḣ = i_{Y}^{!} Ḟ is finite dimensional. Since i_{y}^{!} up to a shift is equal to Ri_{y}^{Δ}, and all cohomology sheaves of Ḣ are i_{y}^{Δ} acyclic (since they are O-free), it simply means that fibers of all these sheaves are finite dimensional, i.e., these sheaves are O-coherent. Hence Ḣ is holonomic and by the key lemma i_{∗}(Ḣ) is also holonomic.

Replacing Ḟ by Ḟ′ = cocone (Ḟ → i_{∗}(Ḣ)) we see that Ḟ′ is coherent, since Ḟ and i_{∗}(Ḣ) are, and all its fibres are finite dimensional (they are 0 outside of S \ Y and coincide with fibers of Ḟ on S \ Y, since by base change i_{x}^{!}i_{∗}(Ḣ) = 0 for X ∉ Y). Since dim supp Ḟ′ < dim S, we see by induction that Ḟ′ is holonomic and hence Ḟ is holonomic. Remark. The proof above proves also the following Criterion. A D_{X}-complex Ḟ is holonomic if and only if there exists a smooth stratification S = {X = UX_{i}} of X such that all components (see 2.4) H_{i} = r_{i}^{!}Ḟ ⊂ D(D_{X_{i}}) of the corresponding stratification of Ḟ are 0-coherent (i.e., all their cohomology sheaves are 0-coherent).

4. Proof of theorem A for π_{∗}.

Since the case of locally closed imbedding is contained in the key lemma, it is sufficient to consider morphism π : Y = T × X → X, where T is a complete variety.

Let Ḣ ∈ D_{hol}(D_{Y}), Ḟ = π_{∗}(Ḣ) ∈ D(D_{X}). In order to prove that Ḟ is holonomic we use criterion ¿from 3. Since π is proper, Ḟ is coherent by 2.7. For any point x ∈ X using base change we have

i_{x}^{!}Ḟ = (π_{x})_{∗}(i_{T_{x}}^{!} Ḣ), where
T_{x} = π^{−1}(x) ≃ T, i_{T_{x}} : T_{x} ⟶ Y and π_{x} : T_{x} ⟶ x

are natural inclusion and projection. By 2., i_{T_{x}}^{!} Ḣ is holonomic. Since π_{x} is proper, it maps this complex into a coherent complex, i.e., i_{x}^{!}Ḟ is coherent, which means finite dimensional. QED

5. Theoremof J.E. Roos.

In order to prove the key lemma and introduce a duality on holonomic modules we need the following important result, due to J.E. Roos, which gives a connection between S.S.F. and homological properties of F.

Consider the D_{X}-module D_{X}^{Ω}, described in 1.4, which has a second structure of a left D_{X}-module. For any coherent D_{X}-module F this structure defines the structure of D-module on all sheaves Ext_{D_{X}}^{i} (F, D_{X}^{Ω}). Note that if F is not coherent, these sheaves are not quasicoherent; we will not consider this case.

Theorem. 0. F has a finite resolution by locally projective D_{X}-modules.

1. codim S.S.(Ext_{D_{X}}^{i} (F, D_{X}^{Ω}) ≥ i.

2. If codim S.S.F = k, then

Ext_{D_{X}}^{i} (F, D_{X}^{Ω}) = 0 for i < k.

We postpone the proof of the theorem until 3.15. Duality functor.

Let us define duality D : D_{coh}(D_{X})^{o} ⟶ D_{coh}(D_{X}) by

D(Ḟ) = R Hom_{D_{X}} (Ḟ , D_{X}^{Ω})[dim X].

It means that we should replace Ḟ by a complex Ṗ of locally projective coherent D-modules Ṗ = {⟶ P_{−1} ⟶ P_{0} ⟶ P_{1} ⟶ … } and put DḞ = DṖ, given by DṖ_{i} = ∗(P_{−dim X−i}), where ∗P_{j} = Hom_{D_{X}} (P_{j}, D_{X}^{Ω}).

Since ∗ ∗ P ≃ P, we have DD = Id. Also by definition

H^{i}(DF) = Ext_{D_{X}}^{dim X+i}(F, D_{X}^{Ω}) for F ∈ μ_{coh}(D_{X}).

Corollary of J.E. Roos’s theorem. Let F be a coherent D_{X}-module. Then

a) complex DF is concentrated in degrees between - dim X and 0, i.e., H^{i}(DF) = 0 for i ∉ [−dim X, 0].

b) F has a locally projective resolution of the length ≤ dim X.

c) F is holonomic iff DF is a module, i.e., H^{i}(DF) = 0 for i ≠ 0.

d) D gives an autoduality D : Hol(D_{X})^{0} ⟶ Hol(D_{X}), i.e., D is a contravariant functor, such that DD = Id_{Hol}. In particular, D is exact.

Proof.

a) Put E^{i} = Ext_{D_{X}}^{i} (F, D_{X}^{Ω}). By Roos’s theorem def(E^{i}) = dim S.S.E^{i} −dim X = dim X −codim S.S.E^{i} is negative if i > dim X. Hence by theorem of defect E^{i} = 0 for these i, and also for i < 0. This means that H^{i}(DF) = 0 for i > 0 and for i < −dim X.

b) We should prove that locally F has a projective dimension ≤ dim X. So we assume that X is affine and F has a finite projective resolution Ṗ. Dual complex DṖ consists of projective modules and by a) is acyclic in degrees i > 0. This means that DṖ = Ṗ′ ⊕ Ṗ′′, where Ṗ′_{i} = 0 for i > 0 and i < −dim X and Ṗ′′ is acyclic. Then D(Ṗ′) gives a resolution of P of the length ≤ dim X.

c) If F is holonomic, then H^{i}(DF) = Ext_{D_{X}}^{dim X+i}(F, D_{X}^{Ω}) = 0 for i < 0 by Roos’s theorem, i.e., DF is a module. Conversely, if F′ = DF is a module, then F = DF′ again is a module, i.e., F = Ext_{D_{X}}^{dim X}(F′, D_{X}^{Ω}) and by Roos’s theorem codim S.S.F ≥ dim X, i.e., dim S.S.F ≤ dim X.

d) follows from c) and DD = Id. Remark. 1. Property c) was the reason for the normalization [dim X] in the definition of duality D.

2. It is clear from d) that D_{hol}(D_{X})) = D_{hol}(D_{X}).

7. Extension lemma. Let F ∈ D_{X}, U must be an open subset of X and H ⊂ F |_{U} a holonomic D_{U}-module. Then there exists a holonomic D_{X}-submodule F′ ⊂ F, such that F′|_{U} = H.

Proof. We can assume that F is coherent and F |_{U} = H (using extension principle for coherent Dmodules). Consider D_{X}-complex DF. It has cohomologies in dimensions ≤ 0. Put G = H^{0}(DF), F′ = DG. By Roos’s theorem dim S.S.G ≤ dim X, i.e., G is a holonomic D_{X}-module. Hence F′ is also harmonic.

Natural morphism DF → G defines a morphism F′ = DG into F = DDF (one can check that this morphism is an imbedding). It is clear that F |_{U} = H = F′|_{U}. Hence F′ (or image of F′ in F) is the holonomic submodule we looked for.

8. Proof of the key lemma. \underline{Step} 1. For closed imbeddings the lemma follows from 2.9. Hence we can assume that i : Y → X is an open imbedding. Also we assume that X is affine and Ḣ = H is a holonomic D-module, generated by one section u. Consider a covering of Y by affine open subsets Y_{α} and replace H by its Čech resolution, consisting of (i_{α})_{+}(H|_{Y_{α}}). This trick reduces the proof to the case when Y is affine, i.e, Y has a form Y = X_{f} = {x|f(x) ≠ 0} for some regular function f on X. In this case i_{∗} = i_{+} is an exact functor.

\underline{Step} 2. Thus we have an affine variety X, a function f ∈ O(X), an open subset i : Y = X_{f} ↪ X and a holonomic D_{Y} -module H, generated by a section u, and we want to prove that D_{X}-module F = i_{+}(H) is holonomic.

The difficult point is to prove that F is coherent. What does it mean?

Since global sections F (X) and H(Y) coincide and D(Y) = ⋃_{n=−∞}^{0}D(X)f^{n}, we see that D_{X}-module F is generated by sections f^{n}u for all n ∈ ℤ. Hence what we really want to prove is the statement:

(*)  for all n ≪ 0 f^{n}u ∈ D(X)(f^{n+1}u).

This follows immediately from the following

Lemma on b-functions. There exists a polynomial in n operator d_{0} ∈ D(X)[n] and a nonzero polynomial b_{0} ∈ k[n] such that

(**)  d_{0}(f^{n+1}u) ≡ b_{0}(n) · (f^{n}u).

Step 3. Proof of the lemma. We extend our situation by extension of scalars k ⟶ K = k(λ) – the field of rational functions. Denote by Ŷ, X̂ extended varieties and define D_{Y}-module Ĥ and D_{X}-module F̂ by

Ĥ = f^{λ} · (K ⊗_{k} H)· and the structure of D_{Y}-module is given by

ξ(f^{λ}h) = \frac{λξ(f)}{f} · f^{λ}hh + f^{λ} · ξh, ξ a vector field on Y, F̂ = i_{+}(Ĥ).

The D_{Y}-module Ĥ is holonomic and by extension lemma 7, F̂ contains a holonomic D_{Y}-module E such that E|_{Y} = Ĥ, i.e., the quotient D_{Y}-module F̂/E is concentrated on X \ Y.

Consider the section û = f^{λ}u ∈ F. Since its image in F̂/E is concentrated on X \ Y, it is annihilated by some power of f, i.e., f^{n_{0}} · û ∈ E. Moreover, since E is holonomic, it has a finite length, that implies that for some n there exists d ∈ D_{X} such that d(f^{n+1}û) = f^{n}ũ. In other words, d(f^{n+1+λ}u) = f^{n+λ}u. Since we can everywhere replace λ by λ + n we have proved the existence of d ∈ D_{X} such that

d(f^{λ+1}u) = f^{λ}u.

Now we can write d = d_{o}/b_{o}, d_{o} ∈ D(X)[λ], b_{o} ∈ k[λ]. Then d_{o}, b_{o} satisfy (**).

Step 4. Now, when we know that F is coherent, let us prove that it is holonomic.

First of all, lemma on b-functions implies that F̃ = E (notations from the step 3), i.e., F̃ is holonomic and is generated by ũ. It means that we can choose operators d_{1}, …, d_{ℓ} ∈ D_{X} such that the set Δ ⊂ T^{∗}(X̃) of common zeroes of their symbols σ_{1}, …, σ_{ℓ} ∈ Σ̃ has dimension dim Δ ≤ dim X.

For almost any n ∈ ℤ we can substitute n ⟼ λ, and we obtain operators d_{i}^{(n)} ∈ D_{X}, their symbols σ_{i}^{(n)} ∈ Σ and the set Δ^{(n)} ∈ T^{∗}(X) of their common zeroes, such that dim Δ ≤ dim X and d_{i}^{(n)}(f^{n}u) = 0. These formulae imply that f^{n}u lies in a holonomic submodule of F. Since F is generated by f^{n}u for any n, which is ≪ 0, it implies that F is holonomic.

9. Functors π_{!}, π^{∗} and their properties.

For any morphism π : Y → X we define functors π_{!} : D_{hol}(D_{Y}) ⟶ D_{hol}(D_{X}) π^{∗} : D_{hol}(D_{X}) ⟶ D_{hol}(D_{Y}) by π_{!} = Dπ_{∗}D π^{∗} = Dπ^{!}D.

This definition makes sense as π_{∗} and π^{!} maps holonomic complexes into holonomic.

Let us list some properties of π_{!} and π^{∗}.

1. There exists the canonical morphism of functors π_{!} → π_{∗} which is an isomorphism for proper π.

2. The functor π_{!} is left adjoint to π^{!}.

3. The functor π^{∗} is left adjoint to π_{∗}. 4, If π is smooth, π^{!} = π^{∗}[2(dim Y − dim X)].

Let us comment on these properties. By definition 3. follows from 2.

Consider in more detail the case when π = j : Y → X is an open imbedding. In this case j^{∗} = j^{!} = restriction on Y, i.e., j^{∗} is left adjoint to j_{∗} and hence j_{!} = Dj_{∗}D is left adjoint to j^{!} = Dj^{∗}D. For any Ḣ ⊂ D_{hol}(D_{Y}) the restriction of j_{!}(Ḣ) on Y coincides with Ḣ, that gives a canonical morphism j_{!}Ḣ → j_{∗}Ḣ, identical on Y.

Thus it remains to prove properties 1 and 2 for proper π and 4 for smooth π. But these properties have nothing to do with holonomicity, since π_{∗} for proper π and π^{!} for smooth π map coherent D-modules into coherent. We will prove them in reasonable generality.

10. The duality theorem for a proper morphism. Theorem. Let π : Y → X be a proper morphism. Then on the category of coherent D-complexes

a) Dπ_{∗} = π_{∗}D and

b) π_{∗} is left adjoint to π^{!}. Proof of the statement a).

Case 1 π is a closed imbedding. Let P be a locally projective D_{Y} -module. I claim that π_{∗}(P) and Dπ_{∗}D(P) are D_{X}-modules and they are canonically isomorphic. It is sufficient to check this locally, so we can assume that P = D_{Y}. In this case it follows from the formula

R Hom_{D_{X}} (D_{X←Y} , D_{X}) = D_{Y →X}[dim Y − dim X].

Case 2 We call a D_{Y} -module P elementary if it has the form P = D_{Y} ⊗_{O_{Y}} π^{Δ}(V) for some locally free O_{X}-module V. Considerations from 2.1b) show that each D_{Y} -module has a resolution, consisting from elementary modules. I claim that for elementary D_{Y} -module P D_{X}-complexes Dπ_{∗}(P)[−dim Y] and π_{∗}(DP)[−dim Y] are sheaves and they are canonically isomorphic.

This fact is local, so I can assume P = D_{Y}. The claim follows from the formulae

R Hom_{D_{Y}} (D_{Y →X}) = D_{X←Y} [dim X − dim Y]

D_{Y →X} = D_{X} ⊗_{k} O_{ℙ}, D_{X←Y} = D_{X} ⊗_{k} Ω_{ℙ},

RΓ(O_{ℙ}) = k, RΓ(Ω_{ℙ}) = k[−dim ℙ]. This proves a).

11. Hom_{D_{X}} and internal Hom.

Usually one can write homomorphisms of 2 sheaves as global sections of the sheaf of homomorphism. Let us look, how to set it for D-modules.

Of course, we have for D_{X}-modules F, F′ the following formula

Hom_{D_{X}} (F, F′) = Γ(Hom_{D_{X}} (F, F′))

or, in derived category,

(*)  RHom_{D_{X}} (Ḟ , Ḟ′) = RΓ(RHom_{D_{X}} (F, F′)).

This formula, by the way, implies that

(**)

Homol. dim Hom_{D_{X}} ≤ Homol. dim Γ + Homol. dim Hom_{D_{X}}
≤ Homol. dim of μ(O_{X}) + dim X ≤ 2 dim X.

But I want to write down RHom in terms of functors, suitable for D-modules.

Definition. Functors “!-tensor product” ⊠ : D(D_{X}) × D(D_{X}) ⟶ D(D_{X}) and “internal Hom” Hom: D_{coh}(D_{X})^{o} × D(D_{X}) ⟶ D(D_{X}) are defined by

Ḟ △^{×} Ḣ = △^{!}(Ḟ ⊠ Ḣ), Hom(Ḟ , Ḣ) = DḞ △^{×} Ḣ

where △ : X ⟶ X × X is the diagonal imbedding, ⊠ is the exterior tensor product over k.

Proposition. RHom_{D_{X}} (Ḟ , Ḣ) = _{∫X}Hom(Ḟ , Ḣ) where _{∫X} : D(D_{X}) ⟶ D(Vect) is the direct image of the projection of X onto a point.

Proof. If F, H are D_{X}-modules, we define D_{X}-module structure on F ⊗_{O_{x}} H by Leibniz rule. It is clear that △^{×} = L(⊗_{O_{X}})[dim X]. (Left derived functor.)

Consider the case when F is coherent and locally projective. Then

Hom(F, H) = Hom_{D_{X}} (F, D_{X}^{Ω}) ⊗_{O_{X}} H = Hom_{D_{X}} (F, D_{X}^{Ω} ⊗_{O_{X}} H).

Let us compute Ω_{X} ⊗_{D_{X}} Hom(F, H). We have

Ω_{X} ⊗_{D_{X}} Hom(F, H) = Ω_{X} ⊗_{D_{X}} Hom_{D_{X}} (F, D_{X}^{Ω} ⊗_{O_{X}} H)
Hom_{D_{X}} (F, Ω_{X} ⊗_{D_{X}} (D_{X}^{Ω} ⊗_{O_{X}} H)) = Hom_{D_{X}} (F, H).

Applying this formula we see that

∫_{X} Hom(F, H) = RΓ(Ω_{X} ⊗_{D_{X}} Hom(F, H) = RΓ(Hom_{D_{X}} (F, H)) = RHom_{D_{X}} (F, H).

General case is proved using resolutions.

12. Proof of the duality theorem, statement b).

Using base change it is easy to check the projection formula

π_{∗}(Ḣ△^{×} π^{!}Ḟ) = π_{∗}(Ḣ)△^{×} Ḟ .

By duality theorem a) we can replace Ḣ and π_{∗}Ḣ on dual complexes and obtain

π_{∗}(Hom(Ḣ, π^{!}Ḟ)) = Hom(π_{∗}(Ḣ), Ḟ).

Now, applying integral ∫, we have

RHom_{D_{Y}} (Ḣ, π^{!}Ḟ) = RHom_{D_{X}} (π_{∗}(Ḣ), Ḟ),

QED.

13. Functor π^{∗} for smooth morphisms.

In order to finish the proof of property 4 in 9 we should check, that for a smooth morphism π : Y → X and a coherent D_{X}-complex Ḟ one has

Dπ^{!}Ḟ = π^{!}DḞ [−2k],

where k = dim Y − dim X.

As in 10. the proof can be reduced to the statement, that

RHom_{D_{Y}} (D_{Y →X}, D_{Y}) = D_{X←Y} [−k].

This statement is proved by studying the resolution DR_{Y/X}(D_{Y}).

14. Classification of irreducible holonomic modules.

Theorem. Let i : Y → X be an affine imbedding with Y -irreducible, E an irreducible O-coherent D_{Y} - module. Put

i_{!∗}E = Im(i_{!}E ⟶ i_{∗}E).

a) i_{!∗}E is an irreducible holonomic module. It is a unique irreducible submodule of i_{∗}E (and unique irreducible quotient of i_{!}E). Also it can be characterized as the unique irreducible subquotient of i_{∗}E (or i_{!}E) which restriction to Y is nonzero.

b) Any irreducible holonomic module F has a form i_{∗!}E for some affine imbedding i : Y → X with irreducible Y and irreducible O-coherent D_{Y} -module E.

We will denote this irreducible holonomic module by L(Y, E).

c) L(Y, E) = L(Y′, E′) if and only if Ȳ = Ȳ′ and restrictions of E and E′ to some subset U ⊂ Y ∩ Y′, open in Y and in Y′ are isomorphic.

Remark. We also will use notation L(Y, E) for nonaffine imbeddings i : Y → X. In this case we should replace i_{!}E and i_{∗}E by their zero components H^{o}(i_{!}E) and H^{o}(i_{∗}E) = i_{+}E, and denote by L(Y, E) the image of i_{!}E → i_{∗}E.

Proof. a) According to theorem A, D_{X}-modules i_{!}E and i_{∗}E are holonomic, and hence have finite lengths.

Let F be any irreducible submodule of i_{∗}E. Then since Hom(F, i_{∗}E) = Hom(i^{!}F, E) ≠ 0 and i^{!}F is irreducible, as well as E, we see that E = i^{!}F. Since i^{!}i_{∗}E = E, there exists only one irreducible subquotient F of i_{∗}E with the property that i^{!}F ≠ 0 and in particular, only one irreducible submodule.

Applying the same arguments to i_{!}E we see that it has a unique irreducible quotient.

Further, Hom(i_{!}E, i_{∗}E) = Hom(E, i^{!}i_{∗}E) = k, and the same is true for Hom(i_{!}E, F), where F is a unique irreducible submodule of i_{∗}E. This shows, that F = Im(i_{!}E → i_{∗}E). b) Let F be an irreducible holonomic D-module, Y an open affine subset of an irreducible component of Supp F. Then sheaf E = i^{!}(F) is irreducible holonomic D_{Y} -module and, decreasing Y, we can assume it is O-coherent. Since Hom(F, i_{∗}E) ≠ 0, F = L(Y, E).

c) The same proof that in a), b).

15. Sketch of the proof of Roos’s theorem.

Step 1. Let Ḟ = {⟶ F^{1} ⟶ F^{2} ⟶ … ⟶ F^{k} ⟶} be a complex of D_{X}-modules, {ϕ_{1}} good filtrations on F_{i}, which are compatible with d. Then it induces a complex of coherent Σ-modules

Ḟ_{Σ} = {0 ⟶ F_{Σ}^{1} ⟶ … ⟶ F_{Σ}^{k} ⟶ 0}.

Lemma. H^{i}(Ḟ)_{Σ} is a subquotient of H^{i}(Ḟ_{Σ}).

In particular, if Ḟ_{Σ} is exact then Ḟ is exact. Also S.S.H^{i}(Ḟ) ⊂ Supp H^{i}(Ḟ_{Σ}).

Step 2. The statement of theorem is local, so I will assume X to be small. Let F be a D_{X}-module, ϕ a good filtration on F, F_{Σ} the associated graded Σ-mdoule.

Since T^{∗}X is regular of dim T^{∗}X = 2n, I can find a free resolution Ċ = {0 → C_{−2n} → … C_{o} → F_{Σ} → o} of Σ-mdoule F_{Σ}. Then it is easy to check that I can lift Ċ to a complex of free D_{X}-modules with a good filtration ϕ Ṗ = {0 → P_{−2n} → … → P_{o} → F → 0} such that Ṗ_{Σ} = Ċ. Then by step 1, P is a free resolution of F.

Step 3. For any locally projective D_{X}-module P, I denote by ∗P the D_{X}-module Hom_{D_{X}} (P, D_{X}^{Ω}). By definition, Ext(F, D_{X}^{Ω}) are calculated as homologies of the complex

∗Ṗ = {0 ⟶ ∗P_{0} ⟶ ∗P_{−1} ⟶ … ⟶ ∗P_{−2n}}.

If we consider the natural filtration on D_{X}^{Ω} and induced filtration on ∗Ṗ, we will get precisely the complex ∗Ċ = {Hom_{Σ}(C_{−i}, Σ)}. (Here I identify Ω_{X} with O_{X}.)

Now we should apply the fact, that the statement of the theorem is true for commutative regular ring Σ. Applying now step 1 we can deduce from this corresponding statement for D_{X}-modules.

4. Holonomic D-modules with regular singularities (RS-modules).

It turns out that class of holonomic D-modules contains a natural subclass, invariant with respect to all operations - - subclass of RS-modules.

1. RS-modules on a curve.

First of all, let us consider the classical case-modules with regular singularities on a curve.

Let C be a curve. Choose a nonsingular curve C^{+}, which contains C as an open dense subset and a point c ∈ C^{+}\C (it plays a role of infinity for C). Let t be a local parameter at c, ∂ = ∂/∂_{t}, d = t∂ ∈ D_{C^{+}}. We denote by D_{C}^{ν} the subsheaf of subalgebras of D_{C^{+}}, generated by d and O_{C^{+}}. It is clear that D_{C}^{ν} and element d in quotient algebra D^{ν}/tD^{ν} do not depend on the choice of a local parameter t. Definition. a) Let F be an O-coherent D_{C}-module. We say that F has a RS at the point c, if its direct image F^{+} = (i_{C→C^{+}})_{+}F is a union of O-coherent D_{C}^{ν}-submodules.

b) We say that an O-coherent D_{C}-module F is RS, if it has RS at all points on infinity (i.e., at all points c ∈ Ĉ \ C of the nonsingular completion Ĉ of the curve C). Definition. Let F be a holonomic D_{C}-module on a curve C. We say that F is RS if its restriction to an open dense subset U ⊂ C is O-coherent RS D_{C}-module. Lemma. Let C, C′ be irreducible curves π : C → C′ a dominant (nonconstant) morphism. Then D_{C′}- module F is RS iff π′(F) is RS; also D_{C}-module H is RS iff π_{∗}(H) is RS.

2. RS D-modules.

Definition. a) Let F be an O-coherent D_{X}-module. Then F is RS if its restriction to any curve is RS.

b) Let ?(Y, E) be an irreducible holonomic D_{X}-module. We say that F is RS if E is RS O-coherent D_{Y} -module.

c) A holonomic D_{X}-module F is RS if all its irreducible subquotients are RS.

d) A holonomic D_{X}-complex Ḟ is RS if all its cohomology sheaves are RS.

We denote by RS (D_{X}) the full subcategory of Hol(D_{X}), consisting of RS-modules, and by D_{RS}(D_{X}) the full subcategory of D(D_{X}) consisting of RS D_{X}-complexes. Proposition. The category RS(D_{X}) is closed with respect to subquotients and extensions. Proof. By definition.

3. Main Theorem B. Main Theorem B.

a) Functors D, π_{∗}, π^{!}, π_{!},π^{∗} preserve subcategory

D_{RS}(D) ⊂ D_{Hol}(D).

b) \underline{RS − criterion} An holonomic D_{X}-complex Ḟ is RS if and only if its restriction i_{C}^{!} Ḟ to any curve C ⊂ X is RS.

Remark. It would be more natural to take b) as a definition of RS D_{X}-complexes. But then it would be difficult to prove “subquotient” properties, like lemma in 2. So we prefer the definition, which makes these properties trivial, and transfers all the difficulties into the “cohomological part”, where we have an appropriate machinery to work with.

The proof of theorem B contains two technical results both due to P. Deligne. The first describes RS property of O-coherent D-modules without referring to curves. The second proves that π_{∗} preserves RS in a simplest case.

4. D-modules with regular singularities along a divisor.

Let X be an algebraic variety. A regular extension of X is a nonsingular variety X^{+}, containing X as an open subset, such that X^{ν} = X^{+} \ X is the divisor with normal crossings. We denote by J ⊂ O_{X^{+}} the ideal of X^{ν}, T^{ν} the subsheaf of vector fields preserving J and D_{X}^{ν} the subalgebra of D_{X^{+}}, generated by T^{ν} and O_{X^{+}}.

Let F be an O-coherent D_{X}-module F^{+} = (i_{X→X^{+}})_{+}F.

Proposition (P. Deligne).. The following conditions are equivalent.

(i) F^{+} is a union of O-coherent D_{X}^{ν} submodules (ii) For any extended curve

σ : (C^{+}, C) ⟶ (X^{+}, X) (i.e., σ : C^{+} → X^{+}, such that σ(C) ⊂ X, σ(c) ∈ X^{+} \ X) F |_{C} has RS at c.

(iii) For each irreducible component W of X^{ν} there is an extended curve σ : (C^{+}, C) ⟶ (X^{+}, X) which intersects W K transversally at c such that F |_{C} has RS at c.

Corollary. Suppose X^{+} is a complete regular extension of X, F and O-coherent D_{X}-module. Then F is RS iff F^{+} is a union of O-coherent D_{X}^{ν}-modules.

5. Proof of theorem B.

Key lemma. Let π_{∗} : Y → X be a morphism, where Y is a surface, X is a curve, X, Y are irreducible. Let H be an O-coherent kRS D_{Y} -module. Then for some open subset

X_{0} ⊂ X  π_{∗}(H)|_{X_{0}}  is RS.

We will prove this lemma in 6.

We also will use the following version of Hironaka’s desingularisation theorem.

Proposition. Let π : Y → X be a morphism. Then there exists a regular extension i : Y → Y^{+} and a morphism π^{+} : Y^{+} → X such that π = π^{+}oi and π^{+} is a proper morphism.

We will call the triple (π^{+}, Y^{+}, i) the resolution of the morphism π.

Now let us start the proof of theorem B. By definition RS is closed with respect to the duality D, and hence D_{RS} is closed with respect to D.

Proof of theorem B for π_{∗}. We have a morphism π : Y → X and an RS D_{Y} -complex Ḣ and we want to prove that π_{∗}(Ḣ) is RS. The proof is by induction on the dimension of S = Supp Ḣ. So we assume that the statement is true for dom S < n. Also we assume that RS-criterion of theorem B is true for dim F < n.

Step 1. Let π = i : Y → Y^{+} be an inclusion into a regular extension of Y, H be an RS O-coherent D_{Y} -module. Then i_{∗}(H) is RS D_{Y^{+}}-module.

Since i is an affine morphism i_{∗}(H) = i_{+}(H). Without loss of generality we can assume Y^{+} to be complete. By Deligne’s proposition i_{+}(H) is a union of O-coherent D_{Y} -modules. Hence arbitrary irreducible subquotient F of i_{+}(H) has this property.

Let AZ^{+} = Supp F. Then it is easy to check that Z^{+} is an irreducible component of an intersection of some components of the divisor X^{ν} and F = L(Z, E), where Z is an open subset of Z^{+}. It is clear that E^{+} = i_{Z→Z^{+}}(E) is a union of O-coherent D_{Z}^{ν}-modules, since D_{Z}^{ν} is a quotient of the algebra D_{Y}^{ν} and E^{+} is a subquotient of H^{+}. Hence E is RS, i.e., F is RS.

6. Sketch of the proof of the key lemma.

We have a smooth morphism π : Y → X with dim Y = 2, dim X = 1. Then, after deleting several points from X, we can find a regular complete extension Y^{+} of Y and a morphism π^{+} : Y^{+} → X^{+}, where X^{+} is the regular completion of X, such that

(i) π^{−1}(X^{ν}) ⊂ Y^{ν}, where X^{ν} = X^{+} \ X, Y^{ν} = Y^{+} \ Y (ii) π^{−1}(X^{ν}) contains all singularities of Y^{ν}.

Denote by T_{Y}^{ν} and T_{X}^{ν} sheaves of vector fields on Y^{+} and X^{+}, which preserve Y^{ν} and X^{ν}. Conditions (i), (ii) imply that each local vector field ξ ∈ T_{X}^{ν} can be lifted locally to a vector field ξ′ ∈ T_{Y}^{ν}. This means that the natural morphism of sheaves on Y^{+}

α : T_{Y}^{ν} ⟶ (π^{+})^{∗}T_{X}^{ν} = O_{Y^{+}} ⨂_{π^{+}·O_{X^{+}}}π^{+} · (T_{X}^{ν})

is epimorphic.

We denote by T_{Y/X}^{ν} the kernel of α. Consider sheaves of algebras D_{Y}^{ν} and D_{X}^{ν} on Y^{+} and X^{+}, generated by T_{Y}^{ν} and by T_{X}^{ν} and denote by M^{R}(D_{Y}^{ν}), M^{R}(D_{X}^{ν}) corresponding categories of right D^{ν}- modules, and by D^{R}(D_{Y}^{ν}), D^{R}(D_{X}^{ν}) derived categories (here I prefer to work with right D-modules as all formulae are simple).

Let us put D_{Y →X}^{ν} = O_{Y^{+}} ⨂_{π^{+}·O_{X^{+}}}π^{+} · (D_{X}^{ν}). This module is D_{Y}^{ν} − π^{+} · (D_{X}^{ν})-bimodule. Using D_{Y →X}

let us define the functor

\begin{matrix}π_{∗}^{ν} : D^{R}(D_{Y}^{ν}) ⟶ D^{R}(D_{X}^{ν}) by \\ π_{∗}^{ν}(E) = R(π^{+}) · (E⨂_{D_{Y}^{ν}}^{L}D_{Y →X}^{ν}). \end{matrix}

Statement. (i) Let H be a right D_{Y} -module, H^{+} = (i_{Y})_{+}H ∈ M^{R}(D_{Y^{+}}). Then, if we consider H^{+} as D_{Y}^{ν} -module, we have

π_{∗}^{ν}(H^{+}) = π_{∗}(H^{+}) as D_{X}^{ν}-module.

(ii) if E is an O-coherent D_{Y}^{ν} -module, then

π_{∗}^{ν}(E) is O-coherent D_{X}^{ν}-module.

This statement implies the key lemma. Indeed, if H is an RS O-coherent (right) D_{Y} -module, then H^{+} is an inductive limit of O_{Y^{+}}-coherent D_{Y}^{ν} -modules and hence π_{∗}(H^{+}) = π_{∗}^{ν}(H^{+}) is an inductive limit of O_{X^{+}}-coherent D_{Y}^{ν} -modules, i.e., it is RS.

Proof of statement. (i) is an immediate consequence of the projection formula and the fact that D_{Y}^{ν} |_{Y} = D_{Y}, D_{Y →X}^{ν}|_{Y} = D_{Y →X}.

(ii) Consider “De Rham” resolution of D_{Y →X}

0 ⟶ D_{Y}^{ν} ⨂_{O_{Y}} T_{Y/X}^{ν} ⟶ D_{Y}^{ν} ⟶ D_{Y →X}^{ν} ⟶ 0.

Using it we see that as O_{X^{+}}-module

π_{∗}^{ν}(E) = R(π^{+}) · (E ⊗ T_{Y/X}^{ν} ⟶ E).

Since π^{+} is a proper morphism, Rπ^{+} maps coherent O_{Y^{+}}-modules into coherent O_{X^{+}}-modules, i.e., π_{∗}^{ν}(E) is O-coherent for O-coherent E.

2. The following statement, due to P. Deligne, is a very useful criterion of RS.

Criterion. Let X^{+} be an irreducible complete normal (maybe singular) variety, X ⊂ X^{+} an open nonsingular subset, E an O-coherent D_{X}-module. Assume that for any component W of X^{ν} = X^{+} \X of codimension 1 in X^{+}, S is RS along W (i.e., E satisfies conditions (i), (ii), (iii) in 4 along W). Then E is RS.

Unfortunately, the only proof of this criterion I know is analytic. I would like to have an algebraic proof.

8. RS-modules with given exponents.

Let us fix some ℚ-linear subspace Λ ⊂ kK, containing 1. Let C be a curve, C^{+} its regular extension c ∈ C^{+} \C, F an RS O-coherent D_{C}-module, F^{+} = (i_{C})_{+}F. For any O-coherent D^{ν} submodule E ⊂ F^{+} we denote by Λ_{c}(E) the set of eigenvalues of the operator d = t∂ in the finite-dimensional space E/tE (t is a local parameter at c, see 1). Now we define

Λ(F) = ⋃_{c,E}Λ(E) for all O-coherent
D^{ν}-submodules of F^{+} and all points c ∈ C^{+} \ C.

The set Λ(F) is called the set of exponents of F. We say that F is RSΛ if Λ(F) ⊂ A. We say that D_{X}-complex Ḟ is RSΛ if for any curve C ⊂ X all cohomology sheaves of i_{C}^{!}(Ḟ) are RSΛ.

It is not difficult to prove that all functors D, π_{∗}, π^{!}, π_{!}, π^{∗} preserve D_{RSΛ(D_{X})} – one should repeat proofs in 1-5 with minor modifications. Apparently criterion 6 is also true for RSΛ (for Λ = ℚ it is proved by Kashiwara). I would like to have an algebraic proof of it.

5. Riemann-Hilbert correspondence.

In this lecture I will work over the field k = ℂ of complex numbers.

1. Constructible sheaves and complexes.

Let X be a complex algebraic variety. We denote by X^{an} the correspondent analytic variety, considered in classical topology.

Let C_{X} be the constant sheaf of complex numbers on X^{an}. We denote by Sh(X^{an}) the category of sheaves of C_{X}-modules, i.e., the category of sheaves of ℂ-vector spaces. Derived category of bounded complexes of sheaves we denote by D(X^{an}). I will call sheaves F ∈ Sh(X^{an}) C_{X}-modules and complexes Ḟ ∈ D(X^{an}) C_{X}-complexes.

I call C_{X}-module F constructible if there exists a stratification X = UX_{i} of X by locally closed algebraic subvarieties X_{i}, such that F |X_{I}^{an} is finite dimensional and locally constant (in classical topology). I call C_{X}-complex Ḟ constructible if all its cohomology sheaves are constructible C_{X}-modules. The full subcategory of D(X^{an}) consisting of constructible complexes I denote by D_{con}(X^{an}).

Any morphism π : Y → X of algebraic varieties induces the continuous map π^{an} : Y^{an} → X^{an} and we can consider functors

\begin{matrix}π_{!}, π_{∗} : D(Y^{an}) ⟶ D(X^{an}) \\ π^{∗}, π^{!} : D(X^{an}) ⟶ D(Y^{an}) \end{matrix}

also we will consider the Verdier duality functor

D; D(X^{an}) ⟶ D(X^{an}).

Theorem. Functors π_{∗}, π_{!}, π^{∗}, π^{!} and Dk preserve subcategories D_{con}( ). On this categories DD = Id and

Dπ^{∗}D = π^{!}, Dπ_{∗}D = π_{!}.

2. De Rham functor.

Denote by O_{X}^{an} the structure sheaf of the analytic variety X^{an}. We will assign to each O_{X}-module F corresponding “analytic” sheaf of O_{X}^{an}-modules F^{an}, which locally is given by

F^{an} = O_{X}^{an} ⨁_{O_{X}} F.

This defines an exact functor

an : M(O_{X}) ⟶ M(O_{X}^{an}).

In particular, sheaf D_{X}^{an} is the sheaf of analytic differential operators on X^{an}, and we have an exact functor

an : M(D_{X}) ⟶ M(D_{X}^{ab}).

Since this functor is exact it induces a functor an : D(D_{X}) ⟶ D(D_{X}^{an}).

Definition. I define the De Rham functor

\begin{matrix}DR : D(D_{X}) ⟶ D(X^{an}) = D(Sh(X^{an})) by \\ DR(Ḟ) = Ω_{X}^{an}⨁_{D_{X}^{an}}Ḟ^{an}. \end{matrix}

Remarks. 1. We know that the complex DR(D_{X}) is a locally projective resolution of the right D_{X}-module Ω_{X}. Hence

DR(Ḟ) = DR_{X}(D_{X}^{an})⨁_{D_{X}^{an}}Ḟ^{an}|n| = DR_{X}(Ḟ^{an})|n|,

where n = dim X.

In particular, if F is an O-coherent D_{X}-module, corresponding to bundle with a flat connection and L = F^{flat} the local system of flat sections of F (in classical topology), then by Poincaré lemma

DR(F) = L|n|.

2. Kashiwara usually uses slightly different functor Sol : D_{coh}(D_{X})^{o} → D(X^{an}),

Sol(Ḟ) = R Hom_{D_{X}^{an}}(F^{an}, O_{X}^{an}).

I claim that Sol(Ḟ) = DR(DḞ)| − dim X|. This follows ¿from the formula

Hom_{D_{X}} (P, O_{X}) = Ω_{X} ⨁_{D_{X}}(∗P),

which is true for any locally projective coherent D_{X}-module P, where ∗P = Hom_{D_{X}} (P, D_{X}^{Ω}).

3. Main Theorem C.

a) DR(D_{hol}(D_{X})) ⊂ D_{con}(X^{an}) and on the subcategory D_{hol} D ∘ DR = DR ∘ D .

If Ḟ ∈ D_{hol}(D_{X}), Ḣ ∈ D(D_{Y}), then

Dr(Ḟ ⊠ Ḣ) ≈ DR(Ḟ) ⊠ DR(Ḣ).

b) On the subcategory D_{RS} functor DR commutes with D, π_{∗}, π^{!}, π_{!}, π^{∗} and ⊠

c) DR : D_{RS}(D_{X}) → D_{con}(X^{an}) is an equivalence of categories.

4. First let us consider some basic properties of the functor DR.

(i) DR commutes with restriction to an open subset. For an étale covering π : Y → X DR commutes with π_{∗} and π^{!}.

(ii) There exists a natural morphism of functors α : DRπ_{∗} → π_{∗} ∘ DR which is an isomorphism for proper π.

In order to prove this let us consider the functor

π_{∗}^{an} : D(D_{Y}^{an}) → D(D_{X}^{an}) on the categories of D^{an}-complexes,

which is given by

π_{∗}^{an}(Ḟ) = Rπ_{.}^{an}(D_{X←Y}^{an} ⨂_{D_{Y}^{an}}Ḟ).

I claim that DRπ_{∗}^{an} = π_{∗} ∘ DR. Indeed,

DR(π_{∗}^{an}(Ḟ) = Ω_{X}^{an}⨁_{D_{X}^{an}}^{L}Rπ_{.}^{an}(D_{X←Y}^{an} ⨁_{D_{Y}^{an}}^{L}Ḟ) =

Rπ_{.}^{an}(π^{.}(Ω_{X}^{an}) ⨂_{π^{.}D_{X}^{an}}^{L}D_{X←Y}^{an} ⨁_{D_{Y}^{an}}^{L}Ḟ) = Rπ_{.}^{an}(Ω_{Y}^{an}⨂_{D_{Y}^{an}}^{L}Ḟ),

since π^{.}Ω_{X} ⨂_{π^{.}D_{X}}D_{X←Y} ≈ Ω_{Y} as D_{Y} -module.

Now there exists in general the natural isomorphism of functors

an ∘ Rπ_{.}(Ḟ) ⟶ Rπ_{.}^{an}(anḞ).

This functor is not an isomorphism in general, since direct image on the left and on the right are taken in different topologies. But according to Serre’s “GAGA” theorem it is an isomorphism for proper π. Combining these 2 observations we obtain (ii). (iii) On the category of coherent D_{X}-complexes there exists a natural morphism of functors

β : DR ∘ D(Ḟ) ⟶ D ∘ DR(Ḟ)

which is an isomorphism for O-coherent Ḟ and which is compatible with the isomorphism π_{∗}DR = DRπ_{∗} for proper π, described in (ii).

By definition of the duality functor D in the category D(X^{an})

D(Ṡ) = RHom_{C_{X}} (Ṡ, C_{X}|ddimX|).

(Here C_{X}|2dimX| is the dualizing sheaf of X^{an}). Hence in order to construct β it is sufficient to construct a morphism

β′ : DR ∘ D(Ḟ) ⊗_{ℂ_{X}} DR(Ḟ) ⟶ cln

where cln is an injective resolution of C_{X}|2 dim X|.

As we saw, DR ∘ D(Ḟ) is naturally isomorphic to Sol(Ḟ)|dim X| = RHom_{D_{X}^{an}}(Ḟ^{an}, O_{X}^{an})|dim X|. Let us realize DR(Ḟ) as DR_{X}(Ḟ^{an}) and DR ∘ D(Ḟ) as Hom_{D_{X}^{an}}(Ḟ^{an}, cl^{an}) where cl is an injective

resolution of kO_{X}|dim X|. Then we have the natural morphism

β′′ : DR ∘ D(Ḟ) ⊗_{C_{X}} DR(Ḟ) ⟶ DR_{X}(cl^{an}).

Since DR_{X}(cl^{an}) ≈ DR_{X}(O_{X}^{an})|dim X| = C_{X}|2dim X|, we have a morphism D_{X}(cl^{an}) ⟶ cln, which composition with β′′ gives us β′. It is easy to check that β is an isomorphism for O-coherent Ḟ. Compatibility condition with π_{∗} it is sufficient to check for imbeddings and projections ℙ × X → X, where it is straightforward. (iv) There is a natural morphism of functors

γ : DR(F ⊠ H) ⟶ DR(F) ⊠ DR(H)

which is an isomorphism for O-coherent F.

Morphism γ is defined by the natural imbedding Ω_{X}^{an} ⊠_{C} Ω_{Y}^{an} ⟶ Ω_{X×Y}^{an}. If F is O-coherent and H is locally projectively is an isomorphism by partial Poincaré lemma. This implies the general statement.

(v) There is a natural morphism of functors δ : DR ∘ π^{!}(Ḟ) → π^{!}DR(Ḟ) which is an isomorphism for smooth π.

Indeed, for smooth π the isomorphism of these functors can be constructed on generators – locally projective modules (for instance if π : Y = T × X → X is the projection, then π^{!}(Ḟ) = O_{T} ⊠ Ḟ |dim T |, π^{!}DR(Ḟ) = C_{T} ⊠ DR(Ḟ)|2dim T | = Dr(O_{T}) ⊠ DR(Ḟ)|dim T |). Consider the case of a closed imbedding i : Y → X. Using i_{∗}, which commutes with DR, we will identify sheaves on Y with sheaves on X, supported on Y. Then i_{∗}i^{!}Ḟ = RΓ_{|Y |}Ḟ in both categories, which gives the natural morphism

δ : DR ∘ i_{∗}i^{!}(Ḟ) = DR(RΓ_{|Y |}Ḟ) ⟶ RΓ_{|Y |}DR(Ḟ) = i_{∗}i^{!}DR(Ḟ).

5. Proof of Theorem C a) (case of holonomic complexes).

Let Ḟ be a holonomic D_{X}-complex. Consider the maximal Zariski open subset U ⊂ X such that DR(Ḟ)|_{U} is constructible. Since F is O-coherent almost everywhere U is dense in X.

Let W be an irreducible component of X \ U. I want to show that DR(Ḟ) is locally constant on some dense Zariski open subset W_{0} ⊂ W.

Claim. I can assume that X = ℙ × W, W = p × W, where p ∈ ℙ, U and V = U ∪ W are open in X. Indeed, consider an étale morphism of some open subset of W onto an open subset of an affine space

𝔸^{k} and extend it to an étale morphism of a neighbourhood of W onto an open subset of 𝔸^{n} ⊃ 𝔸^{k}. By changing base from 𝔸^{k} to W, I can assume that V = U ∪ W is an open subset of X′ = ℙ^{n−k} × W. Then I can extend F to some sheaf of X′.

Now consider the projection pr : X = ℙ × W → W. Since it is a proper morphism DR(pr_{∗}(Ḟ)) = pr_{∗}DR(Ḟ). Since pr_{∗}(Ḟ) is a holonomic D_{W} -complex, it is 0-coherent almost everywhere, i.e., DR(pr_{∗}(Ḟ)) is locally constant almost everywhere.

Put Ṡ = DR(Ḟ) ⊂ D(X^{an}). Replacing W on an open subset, we can assume that pr_{∗}(Ṡ) = DR(pr_{∗}(Ḟ)) is locally constant. We have an exact triangle.

Ṡ_{V} → Ṡ → Ṡ_{X\V} , where Ṡ_{V} = (i_{V}) : (Ṡ/V) is extension by zero.

By the choice of U, Ṡ/_{V} is constructible, i.e., Ṡ_{V} is constructible. Hence pr_{∗}(Ṡ_{X\V}) is constructible and going to an open subset we can assume it is locally constant.

Now Ṡ_{X\V} is a direct sum of 2 sheaves (i_{W})_{!}Ṡ/W and something concentrated on X \ V \ W. This implies that Ṡ/W is a direct summand of the locally constant sheaf pr_{∗}(Ṡ_{X\V}) and hence itself is locally constant. QED

Now let Ḟ be a holonomic complex. Put

Err(Ḟ) = Cone(DR ∘ D(Ḟ) → D ∘ DR(Ḟ)).

This sheaf vanishes on a dense open subset, where Ḟ is 0-coherent. Also function Err commutes with direct image for proper morphisms. Repeating the arguments above we see that Err = 0, i.e., DR commutes with D on D_{hol}(D_{X}).

The same arguments show that DR(Ḟ ⊠ Ḣ) = DR(Ḟ) ⊠ DR(Ḣ) for holonomic Ḟ.

Remark. Of course this proof is simply a variation of Deligne’s proof of “Théorèmes de finitude” in SGA 4 1/2.

6. Proof of theorem C b) for direct image..

Let us prove that the morphism

DR ∘ π_{∗}(Ḣ) → π_{∗} ∘ DR(Ḣ)

is an isomorphism for H ∈ D_{RS}(D_{Y}).

Case 1. π = i : Y → X is a regular extension and H is an RS 0-coherent D_{Y} -module.

In this case the proof is straightforward, using the definition of RS (it was done by P. Deligne). Namely, locally in the neighbourhood of a point x ∈ X \ Y we can choose coordinates x_{1}, …, x_{n} such that X \ Y is given by x_{1}, …, x_{k}. Now we place x by an analytic neighbourhood of x. Then H and H^{+} = i_{+}(H) are determined by monodromy representation of the fundamental group π, (X \ Y). Since this group is commutative, we can decompose H into 1-dimensional subquotients. Using commutativity with ⊠ we can reduce to the case dim Y = 1. Hence as O_{Y} -module H^{+} is generated by one element e, which satisfies the equation x∂(e) = λe. Now direct calculations show that

DR(H^{+}) = (i)_{∗}DR(H).

Case 2. H is an RS 0-coherent D_{Y} -module.

In this case we decompose π = π^{+} ∘ i, where i : Y → Y^{+} is a regular extension and π^{+} : Y^{+} → X is a proper morphism. DR commutes with i by case 1 and with π^{+} by 4 (ii).

General Case. It is sufficient to check the statement on generators. Hence we can assume that Ḣ = i_{∗}(ξ), where i : Z → Y is a locally closed imbedding and ξ an RS 0-coherent D_{Z}-module. Then

DRπ_{∗}(Ḣ) = DR(πi)_{∗}(ξ) =^{case 2} (πi)_{∗} DR(ξ) =

π_{∗}(i_{∗}DR(ξ)) =^{case 2} π_{∗}DR(i_{∗}(ξ)) = π_{∗}DR(Ḣ).

7. Proof of theorem C b).

Functors D, π_{∗} and ⊠ were considered in 5 and 6. \underline{Functor} π^{!}. In 4(v) I have constructed the morphism δ : DRπ^{!} → π^{!}DR which is an isomorphism for smooth π. Hence it is sufficient to check that RS D_{Y} -complexes δ is an isomorphism for the case of a closed imbedding π = i : Y → X. Denote by j : V = X \ Y → X the imbedding of the complementary open set. Then we have the morphism of exact triangles

DR(i_{∗}i^{!}Ḟ)⟶DR(Ḟ)⟶ DR(j_{∗}(Ḟ |_{V}))
↓δ ↓id ↓α
i_{∗}i^{!}DR(Ḟ)⟶DR(Ḟ)⟶j_{∗}(DR(Ḟ)|_{V}).

Since α is an isomorphism by 6, δ is an isomorphism. \underline{Functors π_{!} and π^{∗}.} They commute with DR since π_{!} = Dπ_{∗}D and π^{∗} = Dπ^{!}D.

8. Proof of theorem C c).

First of all, let us prove that DR gives an equivalence of D_{RS}(D_{X}) with a full subcategory of D_{coh}(X^{an}). We should prove that for Ḟ , Ṙ ∈ D_{RS}(D_{X})

DR : Hom_{D_{RS}}(Ḟ , Ḣ) ⟶ Hom_{D_{coh}}(DR(Ḟ), DR(Ḣ))

is an isomorphism.

It turns out that it is simpler to prove the isomorphism of RHom ( ). We have shown in lecture 3 that

RHom(Ḟ , Ḣ) = ∫_{X} Hom(Ḟ , Ḣ) = ∫_{X} Hom(Ḟ , Ḣ) = ∫_{X} DḞ △^{×} Ḣ.

Let us prove that in the category D_{coh}(X^{an}) RHom is given by the same formula we have

R Hom(Ṙ, DS^{.}) = RHom(Ṙ, R Hom(S^{.}, Dual)) =
R Hom(Ṙ ⊗ S^{.}, Dual) = D(R^{.} ⊗ S^{.}) = DR^{.}△^{×} DS^{.}.

Hence

R Hom(Ṙ, S^{.}) = ∫ R Hom(Ṗ , S^{.}) = ∫ D(Ṙ)△^{×} S^{.}.

This proves that DR gives an equivalence of the category D_{RS}(D_{X}) with a full subcategory of D_{coh}(X^{an}).

Now let us prove that this subcategory contains all isomorphism classes of D_{coh}(X^{ab}). Since it is a triangulated full subcategory, it is sufficient to check that it contains generators. As generators we can choose C_{X}-complexes of the form i_{∗}(L) where i : Y → X is an imbedding and L is a local system on Y. Since DR commutes with direct images it is sufficient to check that there exists an RS 0-coherent D_{Y} -module ξ such that DR(ξ) ≈ L|dim Y |, i.e., such that the sheaf of flat sections kof ξ^{an} is isomorphic to L. This is a result by P. Deligne.

9. Perverse sheaves, intersection cohomology and such.

Main theorem C gives us a dictionary which allows to translate problems, statements and notions from D-modules to constructible sheaves and back.

Consider one particular example. The category D_{RS}(D_{X}) of RS-complexes contains the natural full abelian subcategory RS-category of RS-modules.

How to translate it in the language of constructible sheaves.

¿From the description of the functor i^{!} for locally closed imbedding one can immediately get the following Criterion. Let Ḟ be a holonomic D_{X}-complex. Then Ḟ is concentrated in nonnegative degrees (i.e., H^{i}(Ḟ) = 0 for i < 0) if and only if it satisfies the following condition. (∗)_{RS} For any locally closed imbedding i : Y → X there exists an open dense subset Y_{0} ⊂ Y such that i^{!}(Ḟ)|_{Y_{0}} is an 0-coherent D_{Y_{0}}-complex, concentrated in degrees ≥ 0. In terms of constructible complexes this condition can be written as (∗)_{con} For any locally closed imbedding i : Y → X there exists an open dense subset Y_{0} ⊂ Y such that i^{!}(Ṡ)|_{Y} is locally constant and concentrated in degrees ≥ - dim Y.

Thus we have proved the following. Criterion. A constructible complex S^{.} lies in the abelian subcategory

DR(RS(D_{X})) iff Ṡ and DS^{.} satisfy (∗)_{con}.

Now it is easy to recognize this as a definition of a perverse sheaf on X^{an}. Exercise. Let L(Y, ξ) be an irreducible RS D_{X}-module. Then DR(L(Y, ξ))|−dim Y | is the intersection cohomology sheaf, associated to (Y, Loc. syst. ξ).

Thus intersection cohomology sheaves just correspond to irreducible RS d-modules.

10. Analytic criterion of regularity.

For any point x ∈ X I denote by O_{x}^{an} and O_{x}^{form} algebras of convergent and formal power series on X at the point x. For any D_{X}-complex Ḟ the natural inclusion O_{x}^{an} → O_{x}^{form} induces a morphism

ν_{x} : R Hom_{D_{X}} (Ḟ , O_{x}^{an}) ⟶ R Hom_{D_{X}} (Ḟ , O_{X}^{form}).

We say that Ḟ is good at x if ν_{x} is an isomorphism.

Proposition. Let Ḟ be an RS O_{X}-complex. Then Ḟ is good at all points.

Remark. One can show that conversely, if X is a complete variety and Ḟ a holonomic D_{X}-complex good at all points x ∈ X, then Ḟ is RS.

proof. For locally projective D_{X}-module P we have

Hom_{D_{X}} (P, O_{x}^{form}) = Hom_{k}(P/M_{x}P, k) = i_{x}^{Δ}(P)^{∗}.

Hence R Hom_{D_{X}} (Ḟ , O_{x}^{form}) = i′_{x}^{.}(Ḟ)^{∗}|dim X|. If we put Ġ = DḞ and remember that i_{x}^{∗} = Di_{x}^{!}D we see that

R Hom_{D_{X}} (Ḟ , O_{x}^{form}) = i_{x}^{∗}(Ġ)|dim X|.

¿From the other side

R Hom_{D_{X}} (Ḟ , O_{x}^{an}) = fiber at x of Sol(Ḟ) = i_{x}^{∗}DR(Ġ)|dim X|.

Thus we can reformulate our problem, using the DR functor. (*) Holonomic D_{X}-complex Ḟ is good at x iff for Ġ = DḞ the canonical morphism

ν_{x} : i_{x}^{∗}DR(Ġ) ⟶ DR i_{x}^{∗}(Ġ)

is an isomorphism.

Hence the proposition is simply a particular case of theorem C.

The proof of the converse statement is based on the criterion of RS which is discussed in 4.

Lecture 6. D-modules and the proof of the Kazhdan-Lusztig conjecture.

I would like to outline main steps of the proof of the Kazhdan-Lusztig conjecture. Only part of it is connected with D-modules, but somehow it has the same spirit as the theory of D-modules, as I presented it.

The amazing feature of the proof is that it does not try to solve the problem but just keeps translating it in languages of different areas of mathematics (further and further away from the original problem) until it runs into Deligne’s method of weight filtrations which is capable to solve it.

So, have a seat; it is going to be a long journey.

Stop 1. g-modules, Verma modules and such.

Let g be a semisimple Lie algebra over ℂ, f ⊂ g a Carton subalgebra, Σ f^{∗} root system, Σ^{+} the system of positive roots and n ⊂ g corresponding nilpotent subalgebra. To each weight χ ∈ f^{∗} we assign g-module M_{χ} (it is called Verma module) which is a universal g-module, generated by 1 element f_{χ} such that nf_{X} = 0 and f_{χ} is an eigenvector of f with the eigencharacter χ − ρ (here ρ is the halfsum of positive roots). Each Verma module M_{X} has unique irreducible quotient L_{X}, has finite length and all its irreducible subquotients are of the form L_{ψ} for ψ ∈ f^{∗}. Hence we can write in the Grothendieck group

M_{χ} = b_{χψ}L_{ψ}.

Problem. Calculate multiplicities b_{χψ}.

It is usually more convenient to work with the inverse matrix a_{χψ}, such that L_{χ} = Σa_{χψ}M_{ψ}.

Also, using elements of the center z(g) ⊂ U(g) it is easy to show that a_{χψ} ≠ 0 only if χ and ψ lie on one orbit of the Weyl group. The most interesting case is the W -orbit of (−ρ). So let us put for w ∈ W, M_{w} = M_{w(−ρ)}, L_{w(−ρ)} and formulate the Problem A. Calculate matrix a_{ww′}, given by

L_{w} = Σ a_{ww′}M_{w′}.

Stop 2. D-modules, Schubert cells ….

Now we are going to translate Problem A into the language of D-modules.

Let G be an algebraic group corresponding to g, X the flag variety of G, i.e., X = G/B where B is a Borel subgroup of G. The natural action of G, i.e., X = G/B where B is a Borel subgroup of G. The natural action of G on X defines the morphism U(g) → D_{X}. Hence for each D_{X}-module F the space Γ(F) = Γ(X, F) of global sections of F has the natural structure of g-module. Our translation is based on the following Theorem (Beilinson, Bernstein).

The functor Γ : μ(D_{X}) → μ(g), F → Γ(F) gives an equivalence of the category μ(D_{X}) with the category μ_{θ}(g) of g-modules with trivial infinitesimal character θ. Here θ is the character of the center Z(g) ⊂ U(g), i.e., the homomorphism θ : Z(g) ⟶ ℂ, corresponding to the trivial representation of g. We say that g-module M has infinitesimal character θ if Ker θ · M = 0.

The proof of the theorem consists of two parts:

1. We show that the functor Γ is exact and each D_{X}-module F is generated by its global sections. This implies that μ(D_{X}) is equivalent to the category of D(X)-modules, where D(X) = Γ(X, D_{X}) is the algebra of global differential operators. We already saw that this fact is true for projective spaces (see lecture 2); though the proof is different, the effect has the same nature.

2. We show that D(X) = U(g)/Ker θ · U(g).

This is pure luck. The proof is just a direct calculation, which uses Kostant’s theorem on functions on nilpotent cone.

This theorem allows us to translate all the problems of the representation theory, involving modules in μ_{θ}(g) into the language of D-modules. Since M_{w}, L_{w} ∈ M(g) we can translate our problem. Let us indicate how to do it.

It is easy to prove that on any module M = M_{w} or L_{w} the nilpotent algebra n acts locally nilpotent. It means that we can exponentiate this action and define some algebraic action of the corresponding nilpotent subgroup N ⊂ G. Hence on M we have two actions: action κ of the Lie algebra g, and the representation ρ of the Lie group N. It is clear that M is a (g, N)-module, i.e., it satisfies the following conditions:

(i) Representation ρ is algebraic, i.e., M is a union of finite dimensional algebraic representations of the algebraic group N. (ii) Morphism κ : g ⊗ M → M is N-invariant with respect to the adjoint action of N on g and action ρ of N on M. (iii) On Lie algebra n, g actions κ and dρ coincide.

Translating in D_{X}-modules we see that the D_{X}-module F, corresponding to M is really a (D_{X}, N)- module, i.e., it is endowed with an action ρ of the group N such that

(i) ρ is algebraic, i.e., F is a union of coherent O-modules with algebraic action of N (compatible with the natural action of N on X). (ii) Action κ : D_{X} ⊗ F → F is N-invariant. (iii) On Lie algebra n of the group N action κ, given by the natural morphism n → Vect. fields on X, D_{X} coincides with dρ.

In particular, it means that Supp F is N-invariant. Using Bruhat decomposition we see that N haws a finite number of orbits on X. Namely,

X = ⋃_{w∈W} Y_{w}, where Y_{w} = N(wx_{N}),

and x_{N} ∈ X is the point, corresponding to N. If Y is an open orbit of N in the Supp F, then i_{Y}^{!} (F) is an (D_{Y} , N)-module. Now, since N acts transitively on Y it is not difficult to describe all (D_{Y} , N)-modules. They all are direct sums of many copies of the standard (D_{X}, N)-module O_{Y}.

Let us put μ_{Y} = (i_{Y})_{!}(O_{Y}), I_{Y} = (i_{Y})_{∗}(O_{Y}), L_{Y} = Im(μ_{Y} → I_{Y}). Fortunately in this case Y is affine (it is isomorphic to an affine space), so μ_{Y} , I_{Y} are (D_{X}, N)-modules, not complexes.

Lemma. u_{w} = u_{Y_{w}} corresponds to M_{w} L_{w} = L_{Y_{w}} corresponds to L_{w}

It is not quite trivial to establish. But if we are interested only in the images of M, L in the Grothendieck group, then it is easy to prove. Indeed, since each L_{w} is selfdual (since DO_{Y} = O_{Y}), in Grothendieck group μ_{w} ≃ I_{w}. Now it is very easy to directly compute Γ(X, I_{w}) as h-module and to show that it coincides with M_{w}/h. Since an element in the Grothendieck group is determined by its restriction to h, this proves that μ_{w} ∼ M_{w} (in Grothendieck group).

Now we an reformulate the problem.

Problem B. Calculate a_{ww′} given by

L_{w} = Σa_{ww′}μ_{w′}.

Stop 3. Constructible sheaves.

Now we can use Hilbert-Riemann correspondence, I have described in lecture 5, and translate the whole problem into the language of constructible sheaves.

First of all, let us define the Grothendieck group K(D_{RS}) of the category D_{RS}(D_{X}) as a group, generated by RS-complexes and relations [Ḟ] + [Ḣ] = Ġ for any exact triangle Ḟ → Ġ → Ḣ. It is easy to prove that K_{RS} coincides with the Grothendieck group K(RS) of the category RS(D_{X}); isomorphism x : K(D_{RS}) → K(RS) is given by Euler characteristic x([Ḟ]) = Σ(−1)^{i}[H^{i}(Ḟ)]. In the same way K(D_{con}) = K(con). For simplicity we restrict ourselves to the subcategories in K(RS) and K(con) generated by sheaves, which are N-invariant. Functor DR gives us an isomorphism DR : D(D_{RS}) = K(RS) → K(D_{con}) = K(con). Let us look how to translate μ_{w} and L_{w}.

By definition μ_{w} = (i_{Y_{w}})_{!}(O_{y}). Hence DR(μ_{w}) = i_{Y_{w}})_{!}(1_{Y_{w}})[dimY_{w}], where 1_{Y} is the trivial sheaf on Y. If we denote by T_{w} the element (i_{Y_{w}})_{!}(1_{Y_{w}}) ∈ K(con), (extension by zero), we see that DR(μ_{w}) = (−1)^{ℓ(w)}T_{w}, where by definition ℓ(w) = dimY_{w} (it is the usual length function on the Weyl group). As we discussed in lecture 5, DR(L_{w}) = IC(Y_{w})[dimY_{w}], where IC(Y) is the intersection cohomology sheaf of Y. Let us denote by IC_{w} the element of K(con), corresponding to IC(Y_{w}). Then we can reformulate our problem.

Problem C. Find a_{ww′} given by

IC_{w} = Σa_{ww′}(−1)^{ℓ(w)−ℓ(w′)}T_{w′}.

Fast train. Etale cohomologies, changing of the field, ….

What we have done so far is the translation of the very difficult problem A to the not less difficult topological problem C. This problem is essentially the problem of calculating intersection cohomologies of the highly singular varieties \overline{Y}_{w}. The only general method of solving such problems known so far is based on algebraic geometry over finite fields. So we should go this way.

Let us fix the stratification Φ = (X = UY_{w}) and denote by D_{Φ}(X^{an}) the subcategory of D(X^{an}), consisting of C_{X}-complexes, such that their cohomology sheaves are locally constant along each stratum Y_{w} (since Y_{w} is contractible, they in fact are constant along Y_{w}). Corresponding Grothendieck group we denote K_{Φ}. It is clear that K_{Φ} = ⨁_{w∈W} ℤT_{w}, and we just want to find the expression of elements of IC_{w} ∈ K in this basis.

It turns out that we can replace everywhere classical topology by etale topology and all properties of constructible complexes, constructible sheaves, which can be expressed in terms of functors D, π_{∗}, π^{!}, π_{!}, π^{∗} will not change.

Since etale topology is defined purely algebraically, we now can translate the whole situation to arbitrary field.

So, we now consider an algebraically closed field k of arbitrary characteristic p, a flag variety X of a reductive group G over k, and Φ = (X = UY_{w}) the Bruhat stratification. We consider derived category D_{Φ} of complexes with cohomologies, constant along each stratum Y_{w}. In the Grothendieck group K_{Φ} of this category we have a basis T_{w} and elements IC_{w}, corresponding to IC-sheaves, and we want to find an expression of IC_{w} via {T_{w′}}.

There are theorems, which claim that the situation in etale topology over any field will be exactly the same as in classical topology over ℂ.

Remark. In etale topology we are working with ℓ-adic sheaves whose stalks are vector spaces over the algebraic closure \overline{Q}_{ℓ} of the field of ℓ-adic numbers, where ℓ ≠ chark. For simplicity we will identify \overline{Q}_{ℓ} with ℂ.

In fact, ℓ-adic sheaves are not quite sheaves and elements of D_{Φ} are not quite complexes. But it does not matter since we can work with our functors D, π_{∗}, … in the usual way.

Stop 4. Weil sheaves, Tate twist, Lefschetz formula.

Now suppose we are working over the field k which is the algebraic closure of a finite field 𝔽_{q}. Also we assume that our stratification Φ is defined over 𝔽_{q}, i.e., each stratum Y_{w} is given by equations and inequalities with coefficients in 𝔽_{q}. Denote by F r_{q} the automorphism of the field k, given by c ⟼ c^{q}. For any variety Y, defined over F_{q}, F r_{q} induces a bijection F r_{q} : Y (k) → Y (k), which turns out to be a homeomorphism in etale topology.

Let us call Weil sheaf an ℓ-adic sheaf F together with the action of F r_{q} on F. In a similar way we can consider Weil complexes of sheaves. Derived category of Weil complexes, whose cohomologies are constant along strata of stratification Φ we denote D_{Φ}^{W}, and corresponding Grothendieck group K_{Φ}^{W}. These definitions make sense since each stratum Y_{w} is invariant under F r_{q}.

Important example. Let us describe Weil sheaves on the variety pt, consisting of one point. Then any sheaf F is given by a vector space V. Hence Weil sheaf on pt is just a \overline{ℚ}_{ℓ}-vector space V together with a linear transformation F r_{q} : V → V.

Definition. Tate sheaf L over a point p is defined by one-dimensional vector space \overline{ℚ}_{ℓ} together with the morphism F r_{q} : \overline{ℚ}_{ℓ} → \overline{ℚ}_{ℓ}, which is the multiplication by q, i.e., F r_{q}(α) = qα.

If π : Y → X is a morphism of algebraic varieties, which is defined over 𝔽_{q}, it induces functors π_{∗}, π_{!} : D^{W} (Y) → D^{W} (X), π^{∗}, π^{!} : D^{W} (X) → D^{W} (Y). Also there is a functor of Verdier duality D : D^{W} (X) → D^{W} (X). All these functors have the same properties, as we have discussed earlier. But there is one important improvement:

(*) If X is a nonsingular variety, then D(1_{X}) = L^{−dimX} · 1_{X}[2dimX]. Here 1_{X} is the trivial sheaf on X, L we consider as a sheaf on X – this is the Tate sheaf lifted from the point, and L^{−k} means (L^{−1})^{⊗k}.

If we forget the action of F r we have an old formula for dualizing sheaf. So (*) simply means that though dualizing sheaf is essentially isomorphic to the constant sheaf, this isomorphism is not canonical; in particular, F r_{q} changes it in q^{dimX} times.

Exercise. Over a point D(L^{k}) = L^{−k}.

Digression. Weil sheaves and functions.

For each variety X, defined over F_{q} denote by X(q) the finite set, consisting of points of X, which are defined over 𝔽_{q} (i.e., which are fixed points of F r_{q}). To each Weil complex Ḟ I will assign the function f_{F} on the finite set X(q) given by

f_{F} (x) = Σ(−1)^{i}trF rq(stalkH^{i}(Ḟ)_{x})

(it makes sense since x is F r_{q} invariant). It is clear that f_{F} depends only on the class of Ḟ in the Grothendieck group k.

Theorem. Let π : Y → X be a morphism, defined over 𝔽_{q}, and π : Y (q) → X(q) the corresponding map of finite sets. Then

f_{π^{∗}(Ḟ)} = π^{∗}(Ḟ), f_{π_{!}(Ḣ)} = ∫_{π} fḢ,

where Ḟ ∈ D^{W} (X), Ḣ ∈ D^{W} (Y) and operations π^{∗} and ∫_{π} on functions are defined by

π^{∗}(f)(y) = f(π(y)) (∫_{π} f)(x) = ∑_{π(y)=x} f(y).

Here the first statement is triviality and the second is a deep generalization of Lefschetz fixed points theorem.

This theorem claims that all usual operations with functions on finite sets we can rewrite on the level of Weil sheaves (or at least, their Grothendieck group). The importance of this observation can be understood if you realize that starting from one Weil complex Ḟ we can construct the sequence of functions: to any q′ which is a power of q we will assign the function f_{F}^{q′} on the set X(q′); and any natural operation with all these functions can simultaneously be described by one operation with the complex Ḟ. This notion gives the formal definition of the “natural sequence of functions” on sets X(q′), q′ = q^{i}. Example. Consider the projection pr : 𝔸^{k} → pt of the affine space into a point. Then the theorem implies that pr_{!}(a_{𝔸^{k}}) = L^{k}[−2k], (i.e., in K^{W} pr_{!}(1) = L^{k}).

Indeed, comparison with the classical case shows that dimH^{i}(pr_{!}(a_{𝔸^{k}})) = δ_{i,2k′}, and the theorem describes the action of F r_{q} on one-dimensional space H^{2k}(pr_{!}(a_{𝔸^{k}})). Stop 5. Weights and purity.

Let Ḟ be a Weil complex over a point p, which is defined over some field 𝔽_{q′}. We say that w(Ḟ) (weight of Ḟ) is less or equal to ℓ (notation w(Ḟ) ≤ ℓ) if for any i all eigenvalues of F r_{q′} in the space H^{o}(Ḟ) have absolute value ≤ (q′)^{\frac{ℓ+i}{2}} . (Hey, what do you mean? They are supposed to be ℓ-adic numbers.)

Well, if you remember, we have identified \overline{ℚ}_{ℓ} with ℂ, so we consider them as complex numbers, and absolute value is the absolute value. Also Deligne proved that in all interesting cases they are algebraic numbers, so it is all not so bad. And in any case, in what we are going to consider they will always be powers of q. So do not worry).

It is clear that this notion does not depend on the choice of q′, i.e., if we change q′ by q” = (q′)^{ℓ}, it does not affect the condition.

Let now Ḟ be a Weil complex on X. Any point x ∈ S is defined over k = 𝔽_{q′}, i.e., it is defined over some field 𝔽_{q′}. We say that w(Ḟ) ≤ ℓ if for any point x ∈ X the stalk Ḟ_{x} = i_{x}^{∗}(Ḟ) has weight ≤ ℓ.

We say that W (Ḟ) ≥ ℓ if W (DḞ) ≤ −ℓ. We say that Ḟ is pure of the weight ℓ if

W (Ḟ) ≤ ℓ and W (Ḟ) ≥ ℓ.

Deligne’s purity theorem. Let π : Y → X be a morphism, defined over 𝔽_{q}. Then π^{∗} and π_{!} decrease weight, π_{∗} and π^{!} increase weight, i.e.,

if W (Ḟ) ≤ ℓ, then W (π^{∗}Ḟ) ≤ ℓ

if W (Ḣ) ≤ ℓ, then W (π_{!}Ḣ) ≤ ℓ

if W (Ḣ) ≤ ℓ, then W (π_{∗}Ḣ) ≥ ℓ

if W (Ḟ) ≤ ℓ, then W (π^{!}Ḣ) ≥ ℓ .

In particular, proper morphism preserves purity. Gabber’s purity theorem. Let Y be an irreducible algebraic variety, IC(Y) the intersection cohomology Weil sheaf of Y (which coincides with 1_{Y} on the nonsingular part of Y). Then IC(Y) is pure of the weight 0 Stop 6. Hecke algebra.

Get back to flag variety X = UY_{w}. Let us consider only complexes, for which all eigenvalues of all morphisms F r_{q} are powers of q. In general, this category is not invariant with respect to functors, but in our particular case it is.

Let A = K(D^{W} (pt)) be the Grothendieck group of the Weil sheaves over a point. Then A = ℤ[L^{±1}], the algebra of Laurent polynomials.

Denote by H the Grothendieck group K(D_{Φ}^{W} (X)) of Weil sheaves constant along strata of Φ. Then it is clear that H is a free A-module with the basis {T_{w}}.

For any w ∈ W the intersection cohomology sheaf IC_{w} ∈ H satisfies the following relations

(i) D(IC_{w}) = L^{−dimY_{w}} · IC_{w} (ii) IC_{w} = T_{w} + ΣP_{w,w′}T_{w′}, where P_{w,w′} ∈ A satisfy the condition

(*)  P_{w,w′} = 0 ifY_{w′} ⊄ \overline{Y}_{w}  and deg P_{w,w′} < 1/2(ℓ(w) − ℓ(w′)).

Indeed, as a sheaf IC_{w} is selfdual, and since in a neighborhood of Y_{w} it coincides with T_{w} and in this neighbourhood DT_{w} = L^{−dimY_{w}} · T_{w′} we have (i).

In order to prove (ii) let us fix some point x ∈ Y_{w′}. Then by definition of IC_{w} stalks of all cohomology sheaves H^{i}(IC_{w})_{x} equal 0 when i ≥ dimY_{w} − dimY_{w′} = ℓ(w) − ℓ(w′). By Gabber’s theorem w(IC_{w}) ≤ 0, i.e., the action of F r_{q} on H^{i}(IC_{w})_{x} has eigenvalues ≤ q^{i/2}. But it is clear that Σ(−1)^{i}T rF r_{q}(H^{i}(IC_{w})_{x}) = P_{w,w′}(L = q). This proves (ii).

Relations (i) and (ii) gives a hope that if we are able to describe the action of the duality operator D on H, then we would be able to find Kazhdan-Lusztig polynomials P_{w,w′}. After this we can forget about Weil structure (i.e., specialize L → 1) and obtain the formulae for a_{ww′}.

In order to describe the action of D I will introduce on H the structure of an algebra.

The motivation for this came from comparison with functions. Informally H is a space of functions on X(q) constant on N(q) orbits. There is the natural identification of N(q) orbits on X(q) with G(q) orbits on X × X(q), so we can consider elements of H as Q(q)-invariant functions in 2 variables f(x, y), x, y ∈ X(q). But space of functions in 2 variables has the natural operation-convolution, given by

f ∗ h(x, y) = ∫ F (x, z)h(z, y)dz

or, with more details

(f ∗ h)(x, y) = ∫ f(x, u)h(v, y)_{substitute u=v=z}dz.

The discussion on the stop 4 allows us immediately to translate this operation in the derived category, or in the Grothendieck group.

First of all, consider the stratification Ψ of X × X by G-orbits and consider category D_{Ψ}^{W} (X × X) and the corresponding group K_{Ψ}^{W}. This group is naturally isomorphic to H = K_{Ψ}^{W}; isomorphism is given by restriction of the sheaves F on X × X to the fiber x_{0} × X ≃ X. I will identify H and K_{Ψ}^{W} using this isomorphism.

Now, let Ḟ , Ḣ ∈ D_{Ψ}^{W} (X × X). I will define their convolution ∗ by

Ḟ ∗ H = pr_{!}Δ^{∗}(Ḟ ⊠ Ḣ), where

Δ : X × X × X → X × X × X × X, Δ(x, z, y) = (x, z, z, y)

pr : X × X × X → X × X, pr(x, z, y) = (x, y). Proposition. H is an associative A-algebra with respect to convolution ∗ with identity 1 = T_{e}. If ℓ(ww′) = ℓ(w) + ℓ(w′), then T_{w} ∗ T_{w′} = T_{ww′}.

The last statement can be checked straightforwardly. Also it follows from the fact that it is true for usual Hecke algebras, which consist of G(q) invariant functions on X(q) × X(q).

These formulae imply that H as an A-algebra is generated by elements T_{σ′} for simple reflections σ. In order to describe the action of D on H we use the following trick due to Lusztig.

Proposition. Let σ ∈ W be a simple reflection. Then for any h ∈ H we have

D((T_{σ} + 1) ∗ h) = L^{−1}(T_{σ} + 1) ∗ Dh

also (T_{σ} + 1)^{2} = (L + 1)(T_{σ} + 1). Corollary. D is the automorphism of the algebra H. On generators T_{σ} D is given by DT_{σ} = L^{−1}T_{σ} + (L^{−1} − 1).

Indeed, the proposition shows that D((T_{σ} + 1) ∗ h) = D(T_{σ} + 1) ∗ Dh for all h. Since elements T_{σ} + 1 generate H, we have D(f ∗ h) = Df ∗ Dh. The formula D(T_{σ} + 1) = L^{−1}(T_{σ} + 1) gives the action of D on T_{σ}.

The proof of the proposition is based on the following observation. Denote by p_{σ} the parabolic subgroup of G, obtained by adding to the Borel subgroup the simple root, corresponding to σ, and consider the algebraic variety X_{σ} = G/P_{σ}. The natural G-equivariant projection p_{σ} : X → X_{σ} has fibers, isomorphic to the projective line ℙ′. For instance, if we put x_{σ} = p_{σ}(x_{0}), then p_{σ}^{−1} = Y_{e} ∪ Y_{σ} is the projective line with the natural stratification. It means that T_{σ} + 1 correspond to the sheaf R_{σ} which is the trivial sheaf on p_{σ}^{−1}(x_{σ}), extended by zero. After this it is not difficult to prove that for any F ∈ D^{W} we have

(*)  R_{σ} ∗ F = p_{σ}^{∗}(p_{σ})_{!}F.

Now, since p_{σ} is proper, direct image (p_{σ})_{!} = (p_{σ})_{∗} commutes with D. Since p_{σ} is smooth, p_{σ}^{!} = L^{−1}p_{σ}^{∗}, i.e., Dp_{σ}^{∗} = Lp_{σ}^{∗}D (locally X ≃ X_{σ} × ℙ′, so p_{σ}^{∗}(F) = F ⊠ 1_{ℙ′}, i.e., Dp_{σ}^{∗}(F) = DF ⊠ D(1_{ℙ}) = LDF ⊠ 1_{ℙ} = Lp_{σ}^{∗}(DF).

Also, it is clear that (p_{σ})_{!}(R_{σ}) ≃ (L + 1)T_{x_{σ}} (in Grothendieck group) and p_{σ}^{∗}(T_{x_{σ}}) = R_{σ}. This gives the second formula of the proposition.

Last stop. Combinatorial problem.

Proposition. (simple combinatorics).

(i) There exists an A-algebra H which is free with basis T_{w}, such that T_{w} · T_{w′} = T_{ww′} if ℓ(w) − ℓ(w′) = ℓ(ww′). (T_{σ} + 1)^{2} = (L + 1)(T_{σ} + 1) for simple reflections σ ∈ W.

(ii) There exists a unique automorphism D of the algebra H, such that D(L) = L^{−1} D(T_{σ} + 1) = L^{−1}(T_{σ} + 1) for simple reflections σ ∈ W.

(iii) For each w ∈ W there exists a unique element C_{w} ∈ H such that C_{w} = T_{w} + ∑_{w′≨w} p_{w,w′}T_{w′}, where P_{ww′} ∈ A has degree < \frac{1}{2}(ℓ(w) − ℓ(w′)) and DC_{w} = L^{−ℓ(w)}C_{w}. In this case P_{ww′} ∈ ℤ[L].

Example. C_{σ} = T_{σ} + 1.

Polynomials P_{w,w′} are called Kazhdan-Lusztig polynomials. Now, if we summarize our discussion, we will obtain the combinatorial formula for multiplicity matrix a_{w,w′}.

Answer. a_{ww′} = (−1)^{ℓ(w)−ℓ(w′)}P_{ww′}(1).

Some questions.

Question 1. Where is the solution? How can I find these polynomials?

In a sense there was no solution. We have just translated our original problem, adding a new parameter L for rigidity, to a combinatorial problem and proved that this problem has a unique solution. Of course, now we can obtain some recursive formulae for calculation of Kazhdan-Lusztig polynomials, but they are quite complicated.

Whether there exist explicit formulae for p_{ww′}, I think not, i.e., I think that some type of combinatorial complexity is built into the problem.

In some cases one can ge explicit formulae for P. For instance, one can calculate intersection cohomology sheaves for Schubert varieties on usual Grassmannians (see Lascoux and Schutzenberger). But Zelevinsky showed that in this case it is possible to construct small resolutions of singularities. I would say that if you can compute a polynomial P for intersection cohomologies in some case without a computer, then probably there is a small resolution, which gives it.

Question 2. What is the geometrical meaning of other coefficients of p_{ww′}?

Kazhdan and Lusztig showed that all stalks of the sheaves IC_{w} are pure. Hence, if we choose a point x ∈ Y_{w′}, then

dim H^{i}(IC_{w})_{x} = 0 for odd i
= i/2 coefficient of p_{ww′} for even i.

In the proof they used an observation, that transversal section to Y_{w′} of the variety \overline{Y}_{w} is conical, i.e., it has an action of k^{∗} which contracts everything into a point x ∈ Y_{w′}.

In general, stalks of IC sheaves are not pure. But there is one more case, calculated by Vogan and Lusztig, namely the stratification of the flag variety byorbits of complexified maximal compact subgroup, in which stalks always are pure. I do not know why.

Untwisting the situation back we can connect H^{i}(IC_{w}) with

Ext_{μ_{ϕ}(g)}^{i}(M_{w′}, L_{w}) or, if you want, with H^{i}(n, L_{w}).

Question 3. It is all very nice but is it really necessary to go into all this business with varieties over finite fields? How are finite fields connected with g-modules?

In fact, it is not necessary. You can obtain the same results using Hodge theory for constructible sheaves or, even better, directly Hodge theory for D-modules.

One small detail – these theories do not exist yet (there is a Hodge theory for locally constant sheaves – this is Deligne’s theory of variations of Hodge structures – and it is quite powerful, but it is clearly not enough). But at least we know what to think about.