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 M ∼ 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 fXλg 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 fXλg, a sheaf F is called constructible with respect F j 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.