Perverse Sheaves and D-Modules

PERVERSE SHEAVES AND D-MODULES

Maxim Jeffs

January 7, 2019

In recent years, sheaf-theoretic constructions have begun to play an increasingly important role in Floer theory in symplectic topology and gauge theory; in many special cases there now exist versions of Floer homology [AM17, Bus14], Fukaya categories [Nad06, NZ06], and Fukaya-Seidel categories [KS14, KPS17] in terms of sheaves. The purpose of this minor thesis is to give a geometrically-motivated exposition of the central theorems in the theory of D-modules and perverse sheaves, namely: • the Decomposition Theorem for perverse sheaves, • the Riemann-Hilbert correspondence, which relates the theory of regular, holonomic D-modules to the theory of perverse sheaves. For the sake of brevity, proofs are only sketched for the most important theorems when they might prove illuminating, though many examples are provided throughout. Conventions: Throughout all of our spaces X,Y,Z are algebraic varieties over C, and hence paracompact Hausdorff spaces of finite topological dimension; all of our coefficients will for simplicity be taken inthefield C, though the coefficients could just as easily be taken in any field of characteristic zero. Dimension without qualification shall refer to complex dimension. Readers who are unfamiliar with the six-functor formalism for derived categories of sheaves should consult the Appendix, where the rest of the notation used herein is defined. We shall follow the common practice of referring to anobject F of Db(X) as a (derived) sheaf, although the reader is warned that it is really an equivalence class of bounded complexes of sheaves. Acknowledgements: This exposition was written as a Minor Thesis at Harvard University in the Fall of 2018 under the supervision of Dennis Gaitsgory. I have benefited from discussions on this subject with German Stefanich and Yixuan Li.

1 MOTIVATION

Before we motivate the theorems and constructions to be discussed, let us first recall some basic notions about local systems. In this and subsequent section, we shall work in the analytic topology on X. The reader might recall three different notions of a (finite-rank) local system (of complex vector spaces) on X:

• a representation of π1(X) in GLn(C); • a locally constant sheaf of complex vector spaces; • a complex vector bundle on X with a flat connection; All of the constructions we introduce in the following shall be appropriate generalizations of one of these notions of local system which possess superior formal properties. We shall return to the third definition later when we consider D-modules; for now, we regard local systems as finite-rank locally constant sheaves, which form an abelian category we denote by Loc(X). Among other uses, perverse sheaves give us tools for describing the topology of proper maps f : X → Y between algebraic varieties, and provide a useful means of packaging the information about the cohomology of the fibers of the map f at the chain level. To describe the cohomology information of the fibers of a proper map f using sheaf theory, we consider the pushforward (derived) sheaf f∗L on Y where L is a local

1 system on X. Firstly, if L is the constant sheaf CX , then we can resolve by sheaves of singular cochains → C → 0 → 1 → 2 → 0 X CX CX CX ...

i where the sheaves CX are flabby on X and so can be used to compute the right derived functor of f∗. This yields the complex → C → 0 → 1 → 0 f∗ X f∗CX f∗CX ... ⊆ i i −1 C −1 On an open set U Y , we have f∗CX (U) = C (f (U); ), the singular cochains on f (U) with coefficients i in C. Hence when we pass to the cohomology sheaves of the complex, we see that R f∗CX is the sheafification i ⊆ i −1 C of the cohomology presheaf Hf , which associates to U Y the cohomology H (f (U); ) of the inverse i ∼ i −1 image. Furthermore, by proper base-change we see that the stalks are given by (R f∗CX )y = H (f (y); C). More generally, if L is any local system on X, then we can tensor the resolution above with L, since L and i Z CX are (left and right, respectively) modules over the group ring [π1(X)]: → → 0 → 1 → 2 → 0 L CX (L) CX (L) CX (L) ...

Hence when we pass to cohomology we get Hi(f −1(U); L) over open sets U: the singular cohomology −1 of f (U) with coefficients in the local system L. In other words, the sheaf f∗L carries the chain-level information of the cohomology with coefficients in L of all of the fibers of f: describing this sheaf in detail shall be the principal goal of our study of perverse sheaves, and will come to a conclusion with the decomposition theorem. Understanding the same computation using the resolution instead in terms of differential forms shall be an important part of our discussion of D-modules.

Let us make a first observation about the sheaf f∗L; when f is smooth and proper, then by Ehresmann’s theorem, f is a fibration and hence the cohomology groups Hi(f −1(U); L) are locally constant. This means i that the cohomology sheaves H (f∗L) are local systems on Y . Let us recall how one makes use of this local system in algebraic topology to understand the topology of the map f; when f : X → Y is a smooth fiber i i bundle with compact fiber F , we have a monodromy representation of π1(Y ) on H (F ) that turns H (F ) into a local system on Y . Then we have a Leray spectral sequence

ij i j ⇒ i+j E2 = H (Y ; H (F )) = H (X) where the cohomology of Y with coefficients in the local system Hj(F ) converges to the cohomology of X. Since X,Y are in fact algebraic varieties, then we can say more: the following classic theorem of Deligne is perhaps in some sense the simplest incarnation of the decomposition theorem: THEOREM 1. (Deligne) Suppose f : X → Y is a proper map between non-singular projective varieties; then the Leray spectral sequence degenerates at the E2 page and we have ⊕ ∼ Hi(X) = Ha(Y ; Hb(F )) a+b=i

b Moreover, the monodromy representations of π1(Y ) on H (F ) are semisimple. The question we hope to answer in the subsequent sections is: how does this generalize to maps f that might not be smooth? To do so, we shall need to understand sheaves like f∗L in the general case (called constructible sheaves), and come up with an appropriately adapted form of cohomology (called intersection cohomology) in which to express the theorem. We begin with the former, while in later sections we shall finally return to the third notion of local system discussed above to gain a new perspective on thisproblem, in terms of D-modules.

2 CONSTRUCTIBLE SHEAVES

In order to study a proper map f that may not be smooth, we may wish to decompose the relevant spaces into subspaces over which f is smooth, so we may apply the discussions above.

DEFINITION 2. A stratification of a variety X is a finite collection {Xλ} of locally closed, smooth connected subvarieties Xλ ⊆ X called strata such that

2 • X is the disjoint union of the Xλ;

• the closure of each stratum X¯λ is a union of strata.

Note that each Xλ is pure-dimensional, and we use dλ to denote the dimension of a stratum Xλ. Other conditions are often imposed upon stratifications to provide a kind of ‘homogeneity’ of each stratum, for instance, the Whitney condition, or local normal triviality. Such stratifications may always be obtained by refinement: THEOREM 3. Any stratification of a variety admits a refinement to a stratification where theself- homeomorphisms act transitively on each stratum. Moreover, given a proper map f : X → Y of algebraic varieties, Y has a such a stratification such that f is a smooth fiber bundle (possibly in singular varieties) over each stratum. When our space has a stratification, there is a natural extension of the notion of a local system.

DEFINITION 4. On a variety X with a stratification {Xλ}, a sheaf F is called constructible with respect F | F to this stratification if the restriction Xλ of to every stratum is a local system. More generally, we say a sheaf F on X is constructible if there exists some stratification so that F is constructible with respect to this stratification. Finally, we say that a derived sheaf F is constructible if all of the cohomology sheaves Hi(F ) of the complex F are constructible. From our discussion in §1 and the theorem above, the following is immediate:

COROLLARY 5. When f is proper, the derived sheaf f∗L is constructible on Y . A few remarks shall be necessary at this point. Remark 6. Alternatively, in the definition of a stratification, we could have allowed thestrata Xλ to be analytic submanifolds; this would have resulted in a class of constructible sheaves larger than that considered here. We shall return to this point later. Remark 7. It is a non-obvious result that the bounded derived category of constructible sheaves should be equivalent to the category of constructible derived sheaves (the reader should consider this statement): b both happen to be equivalent for complex algebraic varieties, and we denote the resulting category Dc(X) (see [Dim04, Theorem 4.1.4]). Remark 8. Observe that we may, by suitably refining our stratification, always assume that a constructible sheaf is constructible with respect to a ‘nicer’ stratification, for instance, a Whitney stratification. The following example shall be important throughout. Example 9. Let X = C and consider the stratification with strata given by U = C∗ and Z = {0}, and inclusions j : U → C, i : Z → C. Examples of constructible sheaves we shall consider throughout are the skyscraper sheaf i∗C, as well as j∗L and j!L for L a local system on U. Constructible sheaves still possess suitably good formal properties, as evinced by the following quite difficult result: b THEOREM 10. All of Grothendieck’s six functors (see Apppendix) preserve Dc(X). Another desirable aspect of constructible sheaves is their tame geometric behaviour. PROPOSITION 11. If F is a derived sheaf constructible with respect to a Whitney stratification, then for every point x ∈ Xλ in a stratum Xλ, there is an open neighbourhood U of x in X such that there is a natural isomorphism RΓ(U, F ) → Fy in the derived category for all points y ∈ U ∩ Xλ. b Let us now turn to the consideration of the categorical structure of Dc(X).

3 PERVERSE SHEAVES

b Recall that a triangulated category such as Dc(X) may arise as the derived category of several non-equivalent abelian subcategories: we shall see a remarkable example of this when we discuss the Riemann-Hilbert correspondence below. The question of how we may reconstruct an abelian category A from its derived category D is answered by the formalism of trunctation structures (or t-structures for short).

3 DEFINITION 12. The data of a t-structure on a triangulated category D consists of a pair of full subcategories D≤0,D≥0, where we denote D≤i = D≤0[−i] and D≥i = D≥0[−i]. These are required to satisfy the conditions 1. Hom(D≤0,D≥1) = 0; 2. D≤−1 ⊆ D≤0 and D≥1 ⊆ D≥0; 3. for every X in D, there is a distinguished triangle A → X → B → with A ∈ D≤0 and B ∈ D≥1. When D has a t-structure, we say that M = D≤0 ∩D≥0 is the heart of the t-structure. The functors X 7→ A and X 7→ B defined by the triangle in (3) are called the truncation functors and are denoted τ ≤0 and τ >0 respectively. The most important example of a t-structure to carry in mind is the following; when D arises as the derived category of an abelian category A, then D≥0,D≤0 are those subcategories consisting of complexes having only cohomology in non-negative (non-positive, respectively) degrees. In this case, the conditions above are evidently met, and the intersection D≤0 ∩ D≥0 recovers the original category A in its embedding as the subcategory of complexes with cohomology only in degree 0. Thus the following is expected: THEOREM 13. The heart of a t-structure is an abelian category, and there is an exact embedding r : M → D. However, note that in general we need not have an equivalence of the bounded derived category Db(M) with Db (though this will turn out to be the case in the examples we consider). The crucial insight of the categorical study of perverse sheaves is that the derived category of constructible b sheaves Dc(X) admits several alternative t-structures, characterized by their perversity: DEFINITION 14. A perversity function is a decreasing map p : 2N → Z such that 0 ≤ p(n) − p(m) ≤ m − n for all n ≤ m. The dual perversity function p∗ is defined by the formula p∗(n) = −n − p(n). Then for every perversity p we have a t-structure:

THEOREM 15. Suppose X has a Whitney stratification {Xλ}, and define two subcategories by • F ∈ pD≤0 if and only if for every stratum X we have Hj(i∗ F ) = 0 for all j > p(2d ); λ Xλ λ • F ∈ pD≥0 if and only if for every stratum X we have Hj(i! F ) = 0 for all j < p(2d ). λ Xλ λ b then this gives a t-structure on Dc(X) which we call the perverse t-structure for the given stratification. DEFINITION 16. We call an object of the heart of this t-structure a perverse sheaf of perversity p with respect to the given stratification. Refining a stratification gives an inclusion of categories of perverse sheaves; the direct limit over all stratifications is defined to be the (abelian) category of perverse sheaves (of perversity p) and is denoted Pervp(X). Before proceeding to a proof of the above theorem, let us note one very important formal consequence that might serve as motivation for considering the above t-structure: p − 1 COROLLARY 17. The Verdier duality functor preserves Perv (X) when p(m) = 2 m is the self-dual perversity.

Proof. (of Corollary) For any perversity p, we have the following series of equivalences: • F ∈ pD≤0 iff • Hj(i∗ F ) = 0 for all j > p(2d ) iff Xλ λ • Hj(Di! DF ) = 0 for all j > p(2d ) iff Xλ λ − − • H 2dλ j(i! DF ) = 0 for all j > p(2d ) iff Xλ λ • Hk(i! DF ) = 0 for all k < p∗(2d ) iff Xλ λ ∗ • DF ∈ p D≥0

4 and similarly for pD≥0. Hence when p = p∗, that is, p is self-dual, then Verdier duality interchanges pD≥0 and pD≤0 and thus preserves Pervp(X). ■

Proof. (of Theorem) We prove this theorem by induction on gluing in a recollement situation (see Appendix), as expressed in the following lemma: LEMMA 18. Suppose i : Z → X is a closed inclusion and let j : U → X be the inclusion of the complement. b b ≥0,≤0 ≥0,≤0 Suppoose both Dc(Z) and Dc(U) have t-structures, denoted DU and DZ . Then the subcategories of Db(X) defined by gluing: c { } ≤0 F ∗F ∈ ≤0 ∗F ∈ ≤0 • DX = : i DZ , j DZ ; { } ≥0 F !F ∈ ≥0 !F ∈ ≥0 • DX = : i DZ , j DU .

b give a t-structure on Dc(X). This theorem is an elementary consequence of the recollement relations described in the appendix. Now, for each stratum Xλ of X, we will define a (perverse) t-structure on the derived category of locally constant b b sheaves Dc(Xλ) = D Loc(Xλ) for this stratification. This is defined by { } p ≤0 F Hi F • Dλ = : ( ) = 0 for i > p(2dλ) { } p ≥0 F Hi F • Dλ = : ( ) = 0 for i < p(2dλ) p ≤0 ∩ p ≥0 so that the heart of the t-structure is given by Dλ Dλ = Loc(Xλ)[p(2dλ)], the local systems in degree p(2dλ). Applying the above gluing lemma inductively to the stratification yields exactly the perverse b ■ t-structure defined above on Dc(X).

Because of the Corollary above, perverse sheaves of self-dual perversity are most useful in applications. Hence we shall follow the common practice that a perverse sheaf without further qualification refers to a perverse sheaf of self-dual perversity. In this case, we have the following alternative characterization of perverse sheaves that may shed more light on the geometry of the definition PROPOSITION 19. A constructible sheaf F is perverse on X if and only if the following two dual conditions hold • dimsupp H−i(F ) ≤ i for all i ∈ Z; • dimsupp H−i(DF ) ≤ i for all i ∈ Z; In other words, we can think of a perverse sheaf as one whose cohomology sheaves in more negative degrees are supported on larger subsets. But once i > dim(X), they must all be zero, due to the following proposition: PROPOSITION 20. Any perverse sheaf F can be represented by a complex of constructible sheaves with non-zero terms only in degrees − dim X ≤ m ≤ 0. This proposition will become immediately clear following our discussion of D-modules later. Now let us have some examples. Example 21. If X is a smooth variety of dimension d, then for any local system L, the complex L[d] is a perverse sheaf. Indeed, −d is the only degree in which local systems are preserved by Verdier duality. More generally, if j : U → X is an open stratum for X, then any local system L[d] on U will also define a perverse sheaf on X. Example 22. Returning to our example of X = C with strata given by j : C∗ → C, i : {0} → C, examples of perverse sheaves we shall consider later are: the skyscraper sheaf i∗C (in degree 0), as well as j∗L[1] and ∗ j!L[1] for L a local system on C . One of the most important things to note about perverse sheaves is that they behave like sheaves, in the sense that perverse sheaves can be glued together from open covers, and that morphisms can be defined locally (although the reader is warned that a morphism of perverse sheaves may be zero on stalks while not being the zero morphism). In other words, perverse sheaves form a stack. In particular, this implies

5 that the preceding example for perverse sheaves on C is enough to understand perverse sheaves on curves in general; we shall have more to say about this shortly. Another respect in which perverse sheaves behave like sheaves is a result of Beilinson that the (bounded) b derived category of perverse sheaves is equal to all of Dc(X); that is, we can think of an arbitrary (derived) constructible sheaf instead as a complex of perverse sheaves. We shall have more to say on this when we discuss the Riemann-Hilbert correspondence.

4 INTERSECTION COHOMOLOGY

How do we understand a general perverse sheaf? The crucial result here is the following THEOREM 23. The category Perv(X) is both Noetherian and Artinian. In other words, every perverse sheaf has a finite increasing filtration by simple perverse sheaves, which uniquely determine theperverse sheaf. This essentially reduces the question of understanding the general perverse sheaf to one of finding and classifying the simple perverse sheaves; fortunately, this is not so hard to do by an inductive procedure. Firstly, let us understand how to relate perverse sheaves defined on different subspaces. However, unless we are in dimension 1 or are considering an affine inclusion, then the usual push and pull functors on sheaves will not preserve perverse sheaves. We shall need to know how these functors interact with the t-structure. DEFINITION 24. The perverse cohomology functors pHi are defined by pHi(F ) = (pτ ≥i pτ ≤iF )[−i]. b b This is a cohomological functor that maps the category Dc(X) to Perv(X). If we instead imagine Dc(X) as the derived category of perverse sheaves (as above), then these functors would simply be given by the operation of taking cohomology of a complex of perverse sheaves in the ith place. By a similar analogy, we say that a functor between categories with t-structures is t-exact if it preserves D≤0 and D≥0. A t-exact functor applied to a distinguished triangle will then induce a long exact sequence in perverse cohomology sheaves, which we can use to define what it means for a functor tobe left t-exact or right t-exact.. We can use this framework to talk about functors between categories of perverse sheaves. Suppose we are in a recollement situation, with the usual notation as explained in the Appendix, with i : Z → X the inclusion of a closed subset and j : U → X the inclusion of an open subset. ∗ ! PROPOSITION 25. The functors i , j! are right t-exact, and the functors i , j∗ are left t-exact. If we define perverse pullbacks pi∗, pi!, pj∗, pj! by composing with perverse cohomology, i.e. pi∗ = pH0(i∗), then these take perverse sheaves to perverse sheaves and are adjoint to the appropriate pushforward functors. Particularly important is the following simple proposition, which allows us to regard a perverse sheaf on Z as being a perverse sheaf on X supported on Z.

PROPOSITION 26. (Kashiwara’s equivalence) If PervZ (X) denotes the perverse sheaves on X with sup- p ! p ∗ port in Z, then i = i induces an exact equivalence of PervZ (X) and Perv(Z), with inverse given by p p i∗ = i!. Similarly, we can view perverse sheaves on U as those supported away from Z: LEMMA 27. Suppose F is a perverse sheaf on U; then: p • j!F has no quotients supported on Z; p • j∗F has no subobjects supported on Z. Because of this result, there is another functor on perverse sheaves that is more important than either of these pushforward functors; it is called the minimal extension functor, or sometimes the intermediate extension functor because in some sense it lies ‘in between’ these two functors. This functor is defined as p p the image of the natural map j!F → j∗F , and is typically written using the slightly unfortunate notation j!∗F . The importance of this functor derives from the following:

PROPOSITION 28. If F is a perverse sheaf on U, then j!∗F has no quotients or subobjects supported on Z. In particular, j!∗ takes simple perverse sheaves to simple perverse sheaves. Moreover, j!∗ commutes

6 with Verdier duality. Hence we see that the simple objects of Perv(X) split into • minimal extensions of simple perverse sheaves in Perv(U); • pushforwards of simple perverse sheaves in Perv(Z); This opens the way to an inductive procedure for finding all of the simple perverse sheaves on X. DEFINITION 29. Suppose j : V → Y is a smooth Zariski open dense subset of some variety Y , and L a local system on V . Then the intersection cohomology sheaf ICY (L) (with coefficients in L) is defined to be the minimal extension j!∗L[dY ]. Recall that a local system is simple if it is irreducible as a representation of the fundamental group. Then we finally have the following result characterizing the simple perverse sheaves:

THEOREM 30. If X has a stratification {Xλ}, then the simple perverse sheaves with respect to this stratification are exactly IC ¯ (L) where L is a simple local system on X . In general, the simple perverse Xλ λ sheaves in Perv(X) are the intersection cohomology sheaves of smooth algebraic locally closed subsets with simple local system coefficients. As motivation for the terminology intersection cohomology, we make the following definition: DEFINITION 31. The intersection cohomology of X with coefficients in a local system L is defined to be i i IH (X; L) = R Γ(X,ICX (L)[−dX ]) where ICX (L) = j!∗L[dX ] for j : Xreg → X the inclusion of the regular locus.

In the case when X is smooth, then ICX (L)[−dX ] = L and we recover the singular cohomology of X. When X is not smooth, then the intersection cohomology gives a form of cohomology appropriately adapted to singular spaces. Most importantly, since minimal extension commutes with Verdier duality, we see that the intersection cohomology sheaf is fixed by Verdier duality; we obtain Poincaré duality immediately if X is proper: ∼ − ∨ IHi(X) = IH2dX i(X) Intersection cohomology has other properties desirable for algebraic geometry, such as a Hodge decomposition, though the reader is warned that is not a cohomology theory in the usual topological sense of the term as it fails to be homotopy invariant. In historical terms, this definition (by Goresky-MacPherson) preceded the introduction of perverse sheaves; perverse sheaves were defined initially to axiomatize the properties of the intersection cohomology sheaves, which can be defined purely in terms of support conditions for simplices in stratified spaces. Example 32. Returning to our example of X = C with stratification C = C∗ ∪ {0}, we see that the simple perverse sheaves are:

• the skyscraper sheaf i∗C at {0}; ∗ • local systems j!∗L[−1] for L a simple local system on C . This example is enough to describe all perverse sheaves on curves, because of the local nature of perverse sheaves disucssed above. As an exercise, express j∗C and j!C in terms of these perverse sheaves.

5 THE DECOMPOSITION THEOREM

Now we are finally in a position to state the decomposition theorem (in varying levels of abstraction) and derive some corollaries. Recall that we wished to give a description of the sheaf f∗L for f : X → Y a morphism that is not necessarily smooth, and L a local system on X. Firstly, we now know that instead of using ordinary singular cohomology, we should instead use a form of cohomology appropriately adapted to singular spaces such as X, namely intersection cohomology. Thus we should instead consider f∗ICX (L); now, what would it mean to ‘describe’ such a sheaf? The content of the decomposition theorem is essentially that f∗ICX (L) takes the simplest possible form: it is the direct sum of shifts of simple perverse sheaves.

7 THEOREM 33. (Decomposition Theorem I) Suppose f : X → Y is a proper map of algebraic varieties; then there exists a collection of finitely many disjoint locally closed smooth subvarieties Yα with simple local systems Lα such that we have isomorphisms ⊕ ⊕ ∼ p i ∼ ∗ H ∗ − − − f ICX (L) = (f ICX (L))[ i] = ICY¯α (Lα)[dX dY dYα ] i∈Z α

The reader should note the subtle point that while the second direct sum decomposition respects the first pHi (in the sense that each is canonically a sum over ICY¯α (L)), the first decomposition is not canonical. The reader should also observe that the simplicity of the local systems Lα corresponds to the semisimplicity of the monodromy representations of π1(Y ) in Theorem 1. Taking sections of these sheaves over an open subset U yields the decomposition theorem for intersection cohomology groups: THEOREM 34. (Decomposition Theorem II) Suppose f : X → Y is a proper map of algebraic varieties; then there exists a collection of finitely many disjoint locally closed smooth subvarieties Yα with simple local systems Lα such that we have isomorphisms: ⊕ i −1 ∼ i−dY IH (f (U); L) = IH α (U ∩ Y¯α; Lα) α

If X is smooth, we can make more reductions, since then ICX is just L[−dX ]; moreover, if U is open and non-singular and Hi(F ) are local systems, then we can express the perverse cohomology in terms of the usual cohomology sheaves with a degree shift by:

p i m−dU H (F )|U = H (F )[dU ]|U

This finally yields Deligne’s result on the degeneration of the Leray spectral sequence intheform COROLLARY 35. (Deligne) Suppose f : X → Y is a proper submersion of smooth varieties; then ⊕ i f∗C = R f∗C[−i] i∈Z

One remarkable fact about the decomposition theorem is that, despite its essentially geometric nature, its proof depends fundamentally on the arithmetic properties of ℓ-adic sheaves, the description of which would take us too far afield here. Philosophically, the idea of the intersection theorem is that the topology of the smooth locus of f sub- stantially determines the topology of the singular locus of f. As an example of this, we shall deduce the invariant cycle theorem: THEOREM 36. If f : X → Y is a family of projective varieties, then the image of the restriction map ∗ ∗ IH (X) → IH (Fy) from the total space to the fiber Fy is precisely equal to the monodromy invariants of π1(Y, y). i This follows from the decomposition theorem by taking an open set U ⊆ Y over which R f∗ICX is locally constant (i.e. away from the singularities of f); then we can deduce from the degeneration of the Leray- i 0 i i π (Y,y) Serre spectral sequence that the restriction map IH (X) → H (U, R f∗ICX ) = IH (Fy) 1 for y ∈ U is surjective, which is the invariant cycle theorem. Example 37. We can now use our previous examples and the decomposition theorem to understand how local systems behave under maps between curves. The essential result follows immediately from the decomposition theorem:

PROPOSITION 38. Suppose f : X → Y is a surjective morphism of smooth curves, then f∗C[1] is j!∗L[1] where L is the local system x 7→ H0(f −1(x); C) on the smooth locus of Y . k ∗ As an example, consider the map z 7→ z ; then f∗C comes from the local system L on C given by the ∗ ∼ representation of π1(C ) = Z by the permutation matrix in GLn(C).

8 6 LOCAL SYSTEMS REVISITED: D-MODULES

Returning now to our discussion in the introduction, another way of thinking about local systems is as vector bundles with flat connections. Just as we motivated the theory of perverse sheaves by starting withthe definition of a local system as a locally constant sheaf, we shall now take an equivalent definition interms of (analytic) flat connections and generalize in a process that is in many ways parallel to that forperverse sheaves. The resulting theory is that of regular holonomic D-modules, and in the end, the Riemann-Hilbert correspondence will tell us that these are the same as perverse sheaves. The new perspective will shed light on many constructions that we perform with perverse sheaves. In particular, this formalism will have the advantage that instead of thinking of a perverse sheaf as an equivalence class of complexes of sheaves, we will be able to regard it as a single D-module. A more precise way of stating these equivalent definitions of local systems is the following THEOREM 39. (Frobenius) If X is a smooth algebraic variety, then there is an equivalence of categories between analytic vector bundles (E, ∇), and locally constant sheaves on X. This equivalence is given by the functor associating to (E, ∇) its sheaf of analytic flat sections with respect to the connection ∇. This equivalence is the simplest instance of the Riemann-Hilbert correspondence, and it is constructed by looking at the sheaf of solutions of a certain differential equation, ∇s = 0. The question that shall be our starting point for D-modules is: how do we formulate the notion of the ‘sheaf of solutions’ to a partial differential equation? Firstly, we will need to reformulate the notion of a partial differential equation sheaf-theoretically. From this point onwards, all of our algebraic varieties X will be smooth. We will now work with sheaves in the Zariksi topology, but the reader is advised to pay close attention to which constructions are analytic, and which algebraic.

DEFINITION 40. The sheaf of differential operators DX on a smooth variety X is the C-subalgebra of E nd(OX ) generated by TX (that is, the tangent sheaf acting by derivations) and OX (that is, regular functions acting by multiplication). A left D-module is an OX -module that also has an action of the sheaf DX on the left. Similarly, a right D-module is an OX -module with an action of DX on the right. More concretely, supposing we have a coordinate system given locally on an open set U by regular functions {x1, . . . , xn}, then we can express DX |U as the algebra ⊕ αx | O 1 ··· αxn DX U = U ∂1 ∂n α∈Nn

with commutation relations [∂xi , ∂xj ] = 0 and [∂xi , xj] = δij. Note that these constructions are algebraic and use the sheaf of regular algebraic functions on X. The reader is also warned that since DX is a sheaf of non-commutative rings, some care shall be required to distinguish between left and right D-modules in what follows; we shall follow the convention that D-module without further qualification always means left D-module. Suppose we wish to study locally the solutions u to the partial differential equation P u = 0 where P is some partial differential operator. In place of the equation P u = 0, let us consider the left D-module DX /DX P , which will ‘represent’ the spaces of solutions, in the following sense. Firstly, homomorphisms O O ∈ O HomDX (DX , X ) of D-modules from DX to X are locally determined by the image of 1 X and hence O ∼ O O HomDX (DX , X ) = X locally. Therefore, if we look at Hom(DX /DX P, X ), then it consists of those D- module homomorphisms ϕ : DX → OX that have ϕ(P ) = 0, or equivalently, such that ϕ(P 1) = P ϕ(1) = 0. But since ϕ(1) ∈ OX specifies the homomorphism ϕ, locally we have

Hom(DX /DX P, OX ) = {u ∈ OX : P u = 0}

More generally, if S is any D-module, then Hom(DX /DX P, S) should describe the solutions of P u = 0 in the class S. We may apply the same construction to any system of partial differential equations P1u = ··· = Pku = 0; we may represent the spaces of solutions by the D-module M = DX /(DX P1 + ··· + DX Pk), or equivalently, by the complex (P ,...,P ) Dk −−−−−−→1 k D → M → 0

9 This passing to complexes suggests that we should really be studying the derived category of D-modules instead, as we shall see. Another way of thinking about D-modules is provided by the following proposition, which makes clear the relationship of D-modules to the flat connections discussed earlier.

PROPOSITION 41. If M is an OX -module, then it is has a left DX -module structure if and only if there is a connection on M: a C-linear map ∇ : TX → E nd(M) satisfying:

• ∇fv(s) = f∇v(s);

• ∇v(fs) = v(f)s + f∇v(s); ∇ ∇ ∇ • [v1,v2]s = [ v1 , v2 ]s for every f ∈ OX , s section of M, and v section of TX .

Moreover, if M is coherent over OX then it is locally free, and hence M is a vector bundle with a flat connection, sometimes referred to simply as an integrable connection. C m ··· Example 42. Consider an ordinary differential equation on given by P = am(z)∂z + + a0(z) and the { ̸ } | ⊕m−1O (i) associated D-module given by M = D/DP . Then on U = am(z) = 0 , we can see that M U = i=0 U u where u(i) = ∂iu is the initial condition for the ordinary differential equation. By rewriting this system of equations as m first-order ODE, then we can view M as a vector bundle of rank m with flat connection on U. Example 43. For instance, the differential equation z∂z −λ can be viewed as coming from the flat connection ∗ ∇ = ∂z + λdz on the trivial line bundle over C . The flat sections correspond to the solutions of this ODE and are given by zλ on suitably small open sets. If we look at the monodromy of such a solution around the ∗ ∼ origin, we see that the corresponding local system is that coming from the representation of π1(C ) = Z in U(1) by 1 7→ e2πiλ. Example 44. The structure sheaf OX has an obvious action of DX on the left by differentiation that makes it into a left D-module. We shall sometimes refer to this D-module as the trivial D-module, since it will play a role in D-module theory analogous to the constant sheaf in the theory of constructible sheaves. In the case where X = C, then we can think of the structure sheaf as in the previous examples as OX = DX /DX (∂z), or the trivial line bundle with the connection simply given by differentiation. The flat sections of OX , or alternatively the solutions of this D-module, are exactly the constant functions, those with ∂zu = 0, and hence the sheaf of flat sections is the constant sheaf.

7 FUNCTORS ON D-MODULES

Let us now consider how we can relate D-modules on different smooth varieties X and Y . We shall b see that the resulting functors are most naturally defined on Dqc(DX ), the bounded derived category of quasi-coherent (over OX ) left DX -modules on X, and we shall aim towards developing an analogue of the six-functor formalism for constructible sheaves, though we shall not be able to attain this aim at present.

DEFINITION 45. Suppose that f : X → Y is a smooth map, and define the transfer bimodule DX→Y −1 −1 O ⊗ −1 by X f OY f DY , which has the structure of a left DX -module and a right f DY -module. −1 −1 If M is a DY module, then the naïve pullback f M is a f DY -module; taking the tensor product with −1 −1 → ⊗ −1 the transfer bimodule DX Y f DY f M cancels out these f DY actions on the left and right and yields a left DX -module. The resulting functor turns out to be right exact, and we denote the resulting derived functor as follows: → † b → b DEFINITION 46. If f : X Y is a smooth map, there is an exact functor f : Dqc(DY ) Dqc(DX ) called the pullback defined on quasi-coherent (over OY ) DY -modules, by † L −1 f M = (D → ⊗ −1 f M)[d − d ] X Y f DY X Y

b In order to define derived functors on Dqc(DX ), we really ought to check that there are enough injective and flat objects; we will not do this here, but see[HTT08]. The dimension shift used above will ensure that later this functor will correspond to the exceptional pullback on constructible sheaves.

10 Pushforward functors for D-modules are more difficult to define. In light of the above, we might naïvely hope to define ⊗ f∗M = f∗(M DX DX→Y ) but we can see that this is only defined for right DX modules, and yields a right DY -module. A simple analogy might illuminate the situation: while left D-modules can be thought of as sheaves of functions, and hence can naturally be pulled back, right D-modules should instead be thought of as distributions (think of integrating by parts) and so are more naturally pushed forward, or integrated. Therefore we should understand how to relate left and right D-modules. Continuing with our analogy of considering right D- dim X modules as distributions, the canonical sheaf KX = ΩX has the structure of a right DX -module, where we can see from Stokes’ theorem that a vector field v should act on KX by the Lie derivative −Lv. In a local coordinate system {x1, . . . , xn}, this action will be given by the formula

t (fdx1 ∧ · · · ∧ dxn) · P = (P f)dx1 ∧ · · · ∧ dxn

t where P is the formal adjoint operator in these coordinates. Now, as a subring of DX , OX has the structure of both a left and a right DX -module, and so the tensor product ⊗ KX OX M takes a left DX -module M to a right DX -module. The good news is ⊗ · THEOREM 47. The functor KX OX gives an equivalence of categories between left DX -modules and −1 ⊗ · right DX -modules, and has quasi-inverse given by KX OX

Applying this chain of equivalences to the functor f∗ defined on right D-modules as above leads us to define −1 DEFINITION 48. The transfer bimodule DY ←X is the right DX -module and left f DY -module defined by −1 −1 DY ←X = KX ⊗O DX→Y ⊗f −1O ⊗f K ∫ X Y Y b → b THEOREM 49. There is an exact functor f : Dqc(DX ) Dqc(DY ) called the pushforward defined on quasi-coherent left DX -modules by ∫ L M = Rf∗(D ← ⊗ M) Y X DX f

This definition may seem somewhat abstruse, so it will be helpful to understand it in some special cases;so let j : U → X be an open inclusion, and i : Z → X be the inclusion of the smooth complement. Then

∫PROPOSITION 50. The transfer bimodule of an open inclusion j is given by DX←U = DU and so O j = Rj∗, the usual pushforward functor on quasi-coherent U -modules. For a closed inclusion, the transfer bimodule can be thought of as describing the normal directions of Z in X: { } ··· PROPOSITION 51. Suppose we take local coordinates xk, ∂xk on X near Z such that xr+1 = = xn = 0 locally defines Z. Then we have local isomorphisms describing the transfer bimodules as ∼ ⊗ C DZ→X = DZ C [∂xr+1 , . . . , ∂xn ], ∼ DZ←X = C[∂x , . . . , ∂x ] ⊗C DZ ∫ r+1 n so that i M is concentrated in degree 0 and is given locally by ∫ ∼ C ⊗ M = [∂xr+1 , . . . , ∂xn ] C i∗M i

From consideration of the above formulas one can prove the following important result, parallel to that for perverse sheaves:

THEOREM 52. (Kashiwara’s∫ Equivalence) Suppose i : Z → X is an closed embedding of smooth varieties. Then the pushforward functor i induces an equivalence of categories between quasi-coherent DZ -modules † and quasi-coherent DX modules supported on Z, and has quasi-inverse given by i .

11 Example 53. Let us return to the example of C = C∗ ∪ {0} with j and i as before, and let us perform some calculations that will be analogues of the previous results for perverse sheaves. We may compute that ∫ −1 OU = j∗OC∗ = OC[z ] j

Since the modules DC/DCP are cyclic, generated by the image of 1 ∈ DX in the quotient, we can see that −1 −1 we may instead regard OC[z ] as the quotient DC/DC(∂zz), since z is the solution to the equation ∂zz on open sets not containing {0}. We may also compute ∫

O{0} = C[∂z] = DC/DC(z) i

The equation zu = 0 of this D-module has solution u supported entirely at 0, which we call δ0, the Dirac δ-distribution at 0. Just as with constructible sheaves, we might also hope to have a duality functor that would allow us to define ∗ analogues of the other functors f! and f on D-modules by conjugating with duality. This cannot be realized on merely quasi-coherent D-modules, since their spaces of sections are still very ‘infinite-dimensional’ and so would not admit duality relations; our goal in the next section will be to find a class of suitably ‘finite- dimensional’ D-modules that are preserved under the functors we have already defined. Let us first note however that we have a candidate for a duality functor defined as follows:

DEFINITION 54. Suppose M is a quasi-coherent (left) DX -module; then we define its dual DM by the formula D H ⊗ −1 M = R omDX (M,DX ) OX KX [dX ]

The tensor product with KX is there to ensure that the result is indeed another left DX -module; one can D2 show that the dual will indeed be another quasi-coherent DX -module and that X = 1.

8 SINGULAR SUPPORT

In order to have good functorality properties for our D-modules, we wish to look for a class of D-modules that (locally) have finite-dimensional spaces of solutions. They should be in some sense ‘maximally overde- termined’, and we shall describe this condition in terms of the singular support, firstly in a heuristic version. When our D-module has the form M = DX /(DX P1+···+DX Pk) for partial differential operators P1,...,Pk, then we may define the singular support as follows. Associated to each partial differential operator Pi is a ∗ function σ(Pi) ∈ OT ∗X on the cotangent bundle T X called the principal symbol; roughly speaking, this is given in local coordinates by replacing every ∂xi by the corresponding cotangent coordinate function ξi and keeping only the highest degree terms of the resulting polynomial. The singular support of M is then defined to be the vanishing locus V (P1,...,Pk). A more important interpretation of the principal symbol is that if P (x, ∂) is a partial differential operator on Rn acting on the Fourier transform of a test function ϕ(ξ), then we can make a formal calculation ∫ ∫ P F(ϕ) = P (x, ∂) dξϕ(ξ)eix·ξ = [P (x, iξ) · ϕ(ξ)]eix·ξ = F(P (x, iξ)ϕ) Rn Rn so that, after the Fourier transform, P is simply given by multiplication by the polynomial P (x, iξ), the highest-order term of which is the principal symbol σ(P )(x, ξ). Therefore, we can imagine that, away from the set where the highest-order term σ(P ) is zero, we could find solutions to the equation P ϕ = f simply by inverting P (x, iξ) and taking the inverse Fourier transform ( ) ϕ ∼ F P (x, iξ)−1F −1(f) The set σ(P ) = 0 therefore plays a special role in the study of the solutions of the equation P ϕ = f: it determines those directions in which solutions ϕ do not propagate. This provides motivation for the notion of singular support, but to define the singular support for a more general D-module, we shall use a standard technique in the theory of D-modules: using filtrations to pass from the non-commutative sheaf ofrings DX to the commutative sheaf of rings OT ∗X on the cotangent bundle, thus simplifying our problem by allowing us to apply the standard tools from commutative algebra and scheme theory.

12 DEFINITION 55. The order filtration on DX is defined inductively by F0DX = OX and

FℓDX = {P ∈ End(OX ):[P, f] ∈ Fℓ−1DX }

This is just the filtration of DX by degrees of differential operators, with FℓDX being the differential operators of order ℓ. This is compatible with the ring structure in the sense that FℓDX · FkDX = Fℓ+kDX , and has the property that if F ∈ FℓDX and G ∈ FkDX , then [F,G] ∈ Fℓ+k−1. The latter suggests that the associated graded ring should be commutative; the following proposition gives an explicit description.

THEOREM 56. When we form the associated sheaf of graded rings for the filtration F of DX ∑∞ F gr DX = FℓDX /Fℓ−1DX ℓ=0 ∗ then it is isomorphic to π∗OT ∗X where π : T X → X is the projection. { } Under this identification, in an open set U with local coordinate system xi, ∂xi the differential operators F O ∂xi are identified with the functions ξi on the cotangent fiber and gr DU is simply equal to U [ξ1, . . . , ξn]. Furthermore, under this identification, a differential operator P ∈ FℓDX of order ℓ in DX is taken to its F principal symbol σ(P ) in grℓ DX = FℓDX /Fℓ−1DX . ∗ Now we can use this filtration to turn DX -modules equipped with a filtration into OT ∗X -modules on T X:

DEFINITION 57. A good filtration of a quasi-coherent (over OX ) DX -module M is given by an increasing filtration FiM of M by quasi-coherent OX -modules, satisfying:

• we have FiM = 0 for all i sufficiently small;

• for all j ≥ 0, FjDX · FiM ⊆ Fi+jM and we have equality: FjDX · FiM = Fi+jM for all i sufficiently large.

PROPOSITION 58. Every coherent (over DX ) D-module has a good filtration, and any two good filtrations are comparable (in the sense that each is contained in the other after some shift of indexing).

Now, given a coherent DX -module, we may take a good filtration F and pass to the associated graded F F module gr M; this clearly has a natural structure of a graded module over gr DX , and because of the F ∗ theorem above we may pull back gr M to T X and regard it as a graded module over OT ∗X . The most important invariant of this construction is: ∗ DEFINITION 59. The singular support SS(M) ⊆ T X of a coherent DX -module M is the support of the coherent sheaf π∗grF M obtained from any good filtration F of M.

The singular support of a DX -module carries a lot of geometric structure, according to the following difficult theorem

THEOREM 60. (Gabber) Let M be a coherent DX -module; then the singular support SS(M) is independent of the choice of good filtration on M and is a closed subvariety of T ∗X that is coisotropic with respect to the canonical symplectic structure on T ∗X and conical with respect to the multiplication action on the fibers. Example 61. To see that this construction recovers our earlier notion of singular support, let M = DX u = DX /DX I for some solution u, where I is the annihilating ideal of u, with generators P1,...Pk as before. Define a good filtration on M by FiM = (FiDX )u and on I by FiI = FiDX ∩I. Then the associated graded F F F gr M will be the quotient gr DX /gr I, and hence the singular support of M will be the vanishing locus V (π∗grF I). But the ideal grF I will be generated by the principal symbols σ(P ) for P ∈ I and so we recover our earlier definition of the singular support.

We can see from the above example that as the number of equations P1,...,Pk we impose on the solutions of a D-module is increased, the singular support gets smaller, so that if we are looking for those D-modules M that are most likely to have finite-dimensional spaces of solutions, we should look tomake SS(M) as small as possible. However, from Gabber’s theorem, we see that the smallest possible dimension of the singular support of M is dim X. We single such D-modules out as a class worthy of our attention: DEFINITION 62. A coherent D-module M is holonomic if dim SS(M) = dim X.

13 Example 63. Suppose M is a flat connection; then can define a good filtration on M by FiM = 0 for i < 0 ≥ F ∗ F ∼ Or and FiM = M for i 0. Then the associated graded gr M is simply M, and locally π gr (M) = X , so ∗ F ∗ that π gr (M) is supported exactly on the zero section TX X = SS(M). The converse is in fact also true: any D-module with singular support given by the zero section is necessarily a flat connection. In particular, all flat connections are holonomic D-modules. Example 64. Suppose i : Z → X is a closed embedding of∫ a smooth closed subvariety; then∫ one can easily O ∗ deduce from our description of the pushfoward functor i that the singular support of i Z is TZ X, the cotangent bundle of Z in X. Hence these D-modules are holonomic also. In fact, the above examples are in some sense the most general. The following alternative definition sheds more light on the analogy between constructible sheaves as stratified local systems and holonomic D- modules: PROPOSITION 65. A coherent D-module M is holonomic if and only if there is a finite decreasing sequence of closed subsets Xj of X such that the inclusion ij : Xj \ Xj+1 → X is a smooth embedding and i † the cohomology sheaves H (ijM) are all flat connections. The next theorem is the main reason we wish to single out holonomic D-modules in particular: b THEOREM∫ 66. The category Dh(DX ) of complexes of DX -modules with holonomic cohomology is preserved † D by f and f ; moreover, the duality functor takes holonomic D-modules to holonomic D-modules (not D b complexes), that is, preserves the heart of the category Dh(DX ). This may provide belated motivation for the shift introduced in the definition of the duality functor; hope- fully the analogy with perverse sheaves has become apparent to the reader. By applying this proposition we may now define the remaining two of the six functors: DEFINITION 67. For a smooth map f : X → Y between smooth varieties X,Y , let ∫ ∫ D D • f ! = Y f X ; ∗ † • f = DX f DY

All of the relevant adjunctions are then purely formal consequences of this definition. Indeed,∫ the∫ analogy → runs further; we may define a minimal extension functor as the image of the natural map f ! f just as for perverse sheaves, and the alternative characterization of holonomic D-modules suggests that: THEOREM 68. The category of holonomic D-modules is Noetherian and Artinian, so that every holonomic D-module has a finite composition series of simple D-modules; these are all minimal extensions of integrable connections on smooth algebraic locally closed subsets. Example 69. Returning to our example of C = C∗ ∪ {0}, we may compute that ∫

OU = DC/DC(z∂z) j!

from the usual open-closed exact sequence in the six-functor formalism. Indeed, z∂z is precisely the adjoint equation of ∂zz. An an exercise, decompose this into simple holonomic D-modules.

9 ALGEBRAIC VS. ANALYTIC

Restricting to holonomic D-modules, however, will still not be enough for our purposes. Throughout our previous discussions, the D-modules we have been considering have been algebraic, that is, defined in terms of the sheaf OX of regular algebraic functions. This leads to strange situations such as the following: Example 70. Consider the two differential operators given on C by ∂z and ∂z − 1. The first has solutions given by the constant functions, but the second has no non-trivial algebraic solutions at all, since the only z solution to the equation ∂zf − f = 0 is the analytic function e . As analytic flat vector bundles, these two are analytically isomorphic under the holomorphic change of coordinate given by multiplication by ez, and one can see that they both give rise to the constant local system on C. However, as algebraic flat vector bundles, the two are inequivalent.

14 The problem we encountered in this example is that algebraic D-modules do not have ‘enough’ solutions. Moreover we see that algebraic D-modules that we wish to regard as equivalent as they correspond to the same local system will be algebraically non-isomorphic, unless we restrict the class of D-modules we consider still further. In other words, we shall have to pass through the analytic realm to hope to understand D-modules through their solutions.

So far we have considered D-modules with respect to the sheaf of subalgebras DX of the endomorphism sheaf of OX of regular algebraic functions. One can instead consider analytic D-modules that are defined in exactly the same fashion but instead using the sheaf of analytic differential operators: DEFINITION 71. The sheaf of analytic differential operators is defined to be the sheaf of subalgebras DXan inside the endomorphism sheaf OXan of regular holomorphic functions on the underlying complex an manifold X generated by TXan , the holomorphic vector fields, and OXan . The resulting theory is entirely analogous in almost every respect (the only minor change being with respect to the properties of the pushforward functor). We shall still reserve the term D-module to refer to algebraic D-modules. Any (algebraic) D-module can be alternatively regarded as an analytic D-module as follows. Since every Zariski closed subset of X is closed in the analytic topology, there is a continuous map i : Xan → X that moreover extends to a morphism of locally ringed spaces (from the fact that every algebraic function is −1 holomorphic). This morphism gives DXan the natural structure of a left i DX -module and hence we have: b b b DEFINITION 72. The analytification functor D (DX ) → D (DXan ) is defined for M ∈ D (DX ) by

an −1 an ⊗ −1 M = DX i DX i M

The analytification functor has some promising properties: b → b PROPOSITION∫ 73. The analytification functor an : D (DX ) D (DXan ) is exact, and commutes with † f and f for f proper. However, because of our earlier example, there is of course no hope that this functor will be faithful. Thus we would like to restrict to an appropriate class of (algebraic) D-modules that are related most closely to their analytification; since this topic can be rather technical, we shall try to give an impressionistic outlineof the main ideas. The general philosophy is that once we are on a proper variety X, we can apply GAGA-type results, and in particular conclude that all regular functions are algebraic; thus in general we should single out those D-modules that extend ‘nicely’ over compactifications without picking up ‘bad’ singularities, so that then we will have a means to relate algebraic and analytic solutions. To understand this better, let us return to our example from above. The problem with the differential equation ∂zf − f = 0 is that it fails to be regular at ∞ in the sense of classical Fuchsian differential equations. That is, if we change coordinates to study the equation in a coordinate z around ∞, then the equation becomes f ∂ f − = 0 z z2 and so we see that the coefficients in the equation have a higher-order poleat ∞; we say that ∞ is an irregular singularity for the differential equation. We can formalize this notion in the following manner. Suppose ∇ is a flat connection on some algebraic vector bundle over an open disk D ⊆ C around the origin; then we may trivialize this flat vector bundle to write ∇ = ∂z + A(z) where A is some matrix-valued 1-form. Then we say that the integrable connection ∇ is regular at 0 if zA(z) is a matrix of regular 1-forms on D. More generally, we say that an integrable connection on a smooth curve C is regular if at every point x on a compactification C¯ of C, for any choice of regular local coordinates near x, the resulting connection is regular at x. The notion of regularity of (algebraic) D-modules is a straightforward generalization of this notion: DEFINITION 74. We say that an integrable connection on X is regular if its restriction to any immersed smooth curve in X is regular. In general, we say that a DX -module is regular if all of its composition factors are minimal extensions of regular integrable connections.

15 The significance of this definition lies in the following fundamental result of Deligne, sometimes described as Deligne’s Riemann-Hilbert theorem: THEOREM 75. (Deligne) The analytification functor give an equivalence of categories between regular algebraic integrable connections and analytic integrable connections. The proof of this theorem involves applying Serre’s GAGA theorem by passing to a compactification of X obtained by adding a divisor. Describing this proof would involve introducing many notions that are not necessary at this point, and shall be omitted.

10 THE RIEMANN-HILBERT CORRESPONDENCE

b Now we have an appropriate category of D-modules to consider, namely Drh(DX ), the bounded derived category of D-modules with regular, holonomic cohomology, we shall take up again our motivating question of the relationship between local systems and solutions of differential equations. This equivalence was supposed to take place by studying sheaves of solutions, and we now know from the previous section that we should really study these solutions analytically. Hence, following our motivation from §6, we define

DEFINITION 76. The solutions functor is defined for DX -modules M by an H O an Sol(M) = R omDXan (M , X ) and the de Rham functor by L an DR(M) = K an ⊗ M X DXan both of which yield sheaves on Xan.

The de Rham functor simply arises as the dual DR(M) = Sol(DM)[dX ] of the solutions functor, and is more appealing for the following reason. We may take a resolution of the right DX -module KX by differential forms: → 0 ⊗ → 1 ⊗ ⊗ → · · · → ⊗ → → 0 ΩX OX DX ΩX OX DX KX OX DX KX 0

Tensoring this with any DX -module M yields the explicit description ∗ ⊗ DR(M) = (ΩX OX M)[dX ] where the shift puts the complex in degrees i with −dX ≤ i ≤ 0. This should seem similar to the description of perverse sheaves in §3.

Let us recall how to relate integrable connections M to local systems. The degree −dX part of the complex computing DR(M) is given by → 0 ⊗ ∼ → 1 ⊗ 0 ΩX OX M = M ΩX OX M and this differential is given exactly by the connection ∇. Therefore analytifying and then taking cohomology in degree −dX will simply give the kernel of ∇, or the sheaf of analytic flat sections, while the cohomology in other degrees will simply be the cohomology of the usual twisted de Rham complex and so will be trivial on all suitably small open subsets by Poincaré’s Lemma. By combining this with Frobenius’ theorem and Deligne’s Riemann-Hilbert correspondence, we therefore have COROLLARY 77. The cohomology of the de Rham functor H∗(DR) induces an equivalence of categories between the regular integrable connections on X and the local systems Loc(X)[dX ]. In this section, we wish to outline the proof of the Riemann-Hilbert correspondence which extends this result still further: THEOREM 78. (Riemann-Hilbert Correspondence) The de Rham functor gives an equivalence of b b categories of Drh(DX ) with the derived category Dc(X) of constructible sheaves. b Essentially, the proof of the correspondence consists of deducing that the two categories Drh(DX ) and b Dc(X) share sufficiently many of the same functorial properties, at which point it reduces to showingthat the two categories have the same generating objects, which is the substance of the above Corollary. Thus we return again to the question of which functors we can define on holonomic D-modules; these formal properties are captured in the following:

16 b THEOREM 79. All six functors are defined on Drh(DX ) and satisfy all adjunction and duality relations (see the Appendix). We will then need to know that the de Rham functor preserves these formal properties: THEOREM 80. The deRham functor DR commutes with all pushforward and pullback functors, as well as the tensor product. None of these proofs are particularly difficult: they principally consist in formal manipulations withad- junctions, combined with some simple reduction tricks. The proofs that DR commutes with pushforward illustrate best the main ideas of the proof and the important role of regularity. That DR commutes with D D duality is essentially the easy statement that X DR(M) = SolX (M)[dX ], which may simply be∫ checked for integrable connections, and so would suffice to show that the de Rham functor commutes with f . From the previous theorem, we can immediately deduce: LEMMA 81. The de Rham functor commutes with the Hom functor also. This follows from the general fact that Hom can be expressed in terms of the other five functors as

! RHom(F,G) = π2∗(∆ (DX F ⊠C G)) where ⊠ is the exterior tensor product, and ∆ : X → X × X is the diagonal map. Since we know that DR commutes with all of these other functors, DR commutes with Hom also. Another way of stating this result is that DR is a fully faithful functor! Therefore, we have reduced the proof of the Riemann- Hilbert correspondence to showing that DR is essentially surjective onto constructible sheaves. The fact that this functor hits a collection of generating objects will follow from Beilinson’s theorem, combined with the fundamental: THEOREM 82. (Kashiwara Constructibility) If M is a holonomic D-module, then DR(M) is a perverse sheaf. Moreover, every simple perverse sheaf is the image of a simple holonomic D-module under the de Rham functor.

Proof. Because of our earlier work, the proof of this theorem is almost trivial, the point being that the definitions are essentially engineered so that we have this consequence. We simply observe thatsimple objects in the category of holonomic D-modules are minimal extensions of integrable connections on smooth algebraic locally closed subsets Y . Since we saw above that DR commutes with both pushforward functors, it therefore commutes with minimal extension also. But the de Rham functor gives an equivalence between the category of regular integrable connections on Y and Loc(Y )[dY ] by our Corollary above; the minimal extensions of these shifted local systems are exactly the simple objects in the category of perverse sheaves. ■

APPENDIX: THE SIX FUNCTOR FORMALISM

Throughout this text, any space X will be a second-countable paracompact Hausdorff space of finite topological dimension n. Let AX denote the A-valued constant sheaf on X. The six functor formalism concerns relations between a collection of six different functors defined on the (bounded) derived category of sheaves on X, denoted simply by Db(X). We shall follow the practice of referring to objects of Db(X) simply as (derived) sheaves; the (underived) sheaf given by taking the cohomology in the kth place of such a derived sheaf is denoted Hk(F ) and is referred to as the kth cohomology sheaf.

Recall that for any continuous map f : X → Y between spaces as above, we have a pushforward functor f∗ that takes Sh(X) → Sh(Y ) and a pullback functor f ∗ : Sh(Y ) → Sh(X). These functors are only left-exact and right-exact respectively; we can replace them with exact functors by passing to the derived category Db(X). In order to do this, we have to be able to take a resolution of any sheaf in a suitable class of sheaves. Fortunately, it is always possible to resolve by flasque sheaves in Sh(X). Hence we may always define the b i right derived functor of f∗ on D (X), which we shall continue to denote by f∗, while we reserve R f∗ to denote the cohomology functors. In particular, if f is the constant map on any space X, then f∗ = Γ, the global sections functor, and hence we can define the sheaf cohomology groups, which we denote Hi(X, F ).

17 For a continuous map f : X → Y as above, we have another functor f! on sheaves, called the proper pushforward, defined as { } −1 f f!(F )(U) = s ∈ F (f (U)) : supp(s) −→ U is proper

This is a subsheaf of f∗F and if f is proper, then clearly f! = f∗. When f is an open inclusion then f! is sometimes referred to as extension by zero. When f : X → ∗ is the map to a point, then f! = Γc, the −1 compactly supported sections functor. More generally, the fibers (f!F )y are given by Γc(f (y); Ff −1(y)), the compactly supported sections on the fiber.

Just as with f∗, the functor f! is left exact, and by taking a soft resolution of any sheaf on X, we can define the right derived functor Rf!, which we will also continue to denote by f!. This functor also has an adjoint on the level of derived categories: ! b b THEOREM 83. (Verdier Duality) There is a functor f : D (Y ) → D (X) that is adjoint to f!. The depth of this theorem arises from the fact that it is not true simply on the level of categories of sheaves; this functor can only be constructed on the derived category by using a representability theorem and resolving by soft, flat sheaves. This result therefore gives us a canonical sheaf we can defineonany space X: ! DEFINITION 84. The dualizing sheaf ωX on X is defined to be f Z where f : X → ∗ is the map to a point.

When X is a topological manifold, then ωX is simply the shifted orientation sheaf orX [n]; if X is also orientable, then it is simply the constant sheaf ZX . b DEFINITION 85. We define the Verdier dual functor DX on D (X) by DX F = RHom(F , ωX ).

For instance, if X is smooth, then the Verdier dual of the constant sheaf ZX is simply the shift: DX ZX = ZX [2n]. b D2 Verdier duality defines an automorphism of D (X) with the property that X is the identity. This duality functor exchanges the different pushforward and pullback functors above: ! ∗ • f = DX f DY ;

• f! = DY f∗DX The categorical interactions of the functors defined above constitute the six functor formalism. One of the most important of these is THEOREM 86. (Base Change) If we have a pullback diagram

f ′ X′ X g′ g f Y ′ Y ∗ ∼ ′ ′ ∗ then f g! = (g )!(f )

RECOLLEMENT

Let us consider the six functors above in the following special case, called a recollement situation. Let j : U → X be an open subset, and i : Z → X its complement. In this case, several of the functors above become identified: • for an open inclusion j, we have j! = j∗;

• for a closed inclusion i, we have i! = i∗; ! • for a closed inclusion i, we also have i = ΓZ , the functor of sections with support on Z; Then we have a diagram of functors and categories, where every pair of opposite arrows is an adjunction:

18 i∗ j∗ ∗ i∗=i j =j Db(Z) ! Db(X) ! Db(U)

! ! i =ΓZ j

∗ ∗ ! as well as the composition relations j i∗ = i j! = i j∗ = 0. These come with the following Verdier dual exact triangles: ! ∗ j!j F → F → i∗i F → · · · ! ∗ i!i F → F → j∗j F → · · · which represent the decomposition of a sheaf F on X into sheaves supported on Z and U. Finally, we also know that i∗, j∗ and j! are fully faithful, so that we have isomorphisms coming from adjunction:

∗ ! j j∗F → F → j j!F

∗ ! i i∗F → F → i i!F

These relations shall be important for our discussion of t-structures on the category of constructible sheaves. When we combine the above exact triangles in the case where we have a filtration Y ⊆ Z ⊆ X of closed sets, then we get the grand octahedron, from which we can deduce all of the usual long exact sequences in cohomology.

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