The V -filtration and vanishing and nearby cycles

The V -filtration and vanishing and nearby cycles Gus Lonergan Disclaimer: We will work with right D-modules. Any D-module we consider will be at least coherent. We will work locally, choosing etale coordinates x 1,..., x n ; then dx 1... dx n is a generator for K, annihilated by 1,..., n and thus we will regard K as D/( 1,..., n )D, and not bother writing the volume element down. All our constructions will patch together globally in an obvious manner. 1 Introduction: what is the point? The definition of (mixed) Hodge modules is delicate. In fact, it may help psychologically to take the perspective that we are not really trying to define what is a Hodge module per se; rather we will give a definition (inductive) of the category of Hodge modules, and then a Hodge module will just be an object of that category. How might we go about it? Recall that a (rh) D-module M on X is O-coherent on some dense open U. We know what we want an O-coherent Hodge module to be: a variation of Hodge structure. So a Hodge module on X lying over M will consist of a variation of Hodge structure on U lying over M U, plus some sort of similar data on Z = X\U, together with some compatibilities ( gluing ) of these structures. One might naively imagine that the sort of similar data on Z is e.g. a variation of Hodge structure on Z lying over M Z, but this does not work for at least two reasons: 1. (critically) - (or!-) restriction to Z is not exact; 2. (more subtly) - (or!-) restriction to Z loses too much information. In fact, there is a theory of restriction to Z which solves these issues, and is the correct tool for defining Hodge modules: the theory of nearby and vanishing cycles. There are compatible (and intrinsically defined) such theories for both (rh) D-modules and (rational) perverse sheaves. We will first treat the case of D-modules. 2 The V -filtration We are given X a (smooth, connected say) complex variety of dimension n, f a regular function on X, and Z its (smooth) zero set (of codimension 1 in X). Given a D-module M on X, we wish to produce some D-modules on the divisor Z. These will arise as some subquotients of M with respect to a certain filtration, known as the V -filtration. Remark 2.1. 1. The case of irregular (non-zero) f is very important; we will treat it later. 2. To begin with, the V -filtration will be indexed by Z; we will refine it later. 3. V depends very much on f, but we will suppress this from the notation. 4. We will denote Σ j<i V j by V <i, and V i /V <i by gr i. 1

Definition 2.1. The V -filtration on K X is the f-adic filtration: { KX i 1 V i (K X ) = K X.f i 1 i 1 Definition 2.2. The V -filtration on D X is the maximal filtration compatible with that on K X : V i (K X ) = {ξ D X V j K X.ξ V j+i K X for all j.} By definition, the filtration on K X is compatible with that on D X ; moreover if we choose local coordinates x 1, x 2,..., x n 1, f on X (possible since f is regular) then we can write: V i D X = span{o X. a1 1... an 1 f p q f q p i}. (1) In particular, we see that the multiplication by f lowers degrees by 1 and the multiplication by f raises degrees by 1, while the multiplication by the slice operators x i, i, as well as by the Euler operator f f, preserves degrees. In fact, we make the following observations: Example 2.1. 1. V i K X is V 0 D X -coherent (indeed, V 0 D X O X ). 2. The multiplication operator f : V i K X V i 1 K X (resp. f : V i D X V i 1 D X ) is an isomorphism for i < 0. 3. The multiplication operator f f : gr i K X gr i K X is equal to the scalar i. This motivates the definition: Definition 2.3. Let M be a D-module. A V -filtration (along Z) of M is a filtration V M of M by V 0 D X - submodules such that: 1. Each V i M is coherent over V 0 D X ; 2. V is compatible with the filtration V on D; 3. The multiplication operator f : V i M V i 1 M is onto for all i << 0; 4. The multiplication operator f : gr i M gr i M has eigenvalues with real parts in the interval (i 1, i]. Remark 2.2. 1. The filtration V on D X is evidently not a V -filtration according to this definition, but I have abusively called it the V -filtration on D X. No more such abuses will be made. (In fact, this is not so terrible since there is no filtration on D X which satisfies the conditions of the definition). We record the following Theorem 2.1. 1. The V -filtration on M is unique if it exists. 2. If M is holonomic, then the V -filtration always exists. Remark 2.3. Point 1 of the theorem is subsumed in an easy exercise coming up later. Point 2 is much harder. Let us give some more examples. Example 2.2. Let M be and O-coherent D-module; then the V -filtration on M is the f-adic filtration, indexed as for K X ; and the operator f f acts on gr i M as the scalar i. Example 2.3. (Kashiwara s theorem) Let M be a D-module supported on Z, so that we may write M = N[ f ] for some D-module N on Z. Then we have { 0 i < 0 V i M = i j=0 N. j f i 0 and the operator f f acts on gr i M as the scalar i. 2

Example 2.4. Let λ be any complex number with real part in [ 1, 0). We will take X = C with coordinate x = f. Let M be the regular holonomic D-module O C.x λ = j= xλ j generated by x λ. (For λ = 1 this is j O C, the non-split extension of δ 0 by O X ). Then V i M = i j= xλ j, and the operator f f acts on gr i M as the scalar i λ 1. For our purposes, we will restrict attention to regular holonomic D-modules with quasi-unipotent monodromy : Definition 2.4. Let M be a D-module with a V -filtration. We say M has quasi-unipotent monodromy if the eigenvalues of f f on the various gr i M are all rational numbers. Equivalently we will say that M has a rational V -filtration. In light of this definition, given any M with a rational V -filtration, we may refine the V -filtration so that it is indexed by the rational numbers: Redefinition 2.1. Let M be a D-module with a rational V -filtration V. The extension of the indexing of V to Q is characterized as follows: 1. V α V α M V α M 2. V α M/V α M = β α grβ α M gr α M We similarly obtain redefinitions of V <α and gr α. Here gr β i M denotes the β-generalized eigenspace of f f on gr i M. Remark 2.4. 1. This redefinition does not change V α or V <α when α = i is an integer. However, gr i does change in that case: we have replaced V i M/V i 1 M with its i-jordan block for the action of f f. 2. For any M we will consider, the V -filtration will be discrete and right-semicontinuous. This is most neatly summed up in the statements that gr α M is non-zero only for α lying in a discrete subset of Q, while every non-zero section of M has a non-zero symbol in some gr α M. The following is an easy exercise (or you can look at Saito): Lemma 2.1. Suppose V is a rational V -filtration on M. Then 1. f : gr α M gr α 1 M and f : gr α 1 M gr α M are isomorphisms for α 0; 2. f : V α M V α 1 is an isomorphism for α < 0; 3. V α M.V j D X V α+j D X is onto for α, j 0; 4. The forgetful functor {(M, V )} D mod is full; thus {(M, V )} is a full subcategory of D mod, and as such it is closed under sub- and quotient object. 5. The functors V α, V <α, and gr α are exact. Remark 2.5. The assumption of quasi-unipotency is not really essential here. Here {(M, V )} denotes the category of D-modules with rational V -filtrations. For a bit of fun, let us demonstrate the following instructive lemma: Lemma 2.2. Let M be a D-module with a rational V -filtration V. Then V <0 M only depends on M U = j! M. Proof. Let j! j! M M denote the adjunction morphism, and apply the exact functor V <0 to the exact sequence 0 C j! j! M M K 0. C and K are supported on Z, so by Example 2.3, we get the exact sequence 0 V <0 j! j! M M 0. Point 5. (exactness of gr ) is often called strict compatibility with the filtration. In particular it implies that if M is a D-module with a rational V -filtration V, and N M is any D-submodule, then the induced filtration on N is its unique rational V -filtration. In fact, this is the essential point required to prove the lemma. 3

3 Nearby and Vanishing Cycles We have constructed an exact functor gr on {(M, V )}; in particular it makes sense on the category D- mod rh,qu of regular holonomic D-modules with quasi-unipotent monodromy, which is a rather large (and definitely big enough for our purposes) full subcategory of regular holonomic D-modules. What is its target category? Evidently gr M has the structure of a coherent Q-graded gr D X -module. We easily see that D Z [f f ]f α α Z 0 gr α D X = D Z [f f ] f α α Z 0 (2) 0 α / Z Thus gr M is a rationally graded, graded-coherent D-module on the divisor Z, with two endomorphisms f, f, respectively lowering and raising degrees by 1, and whose composition f f is α-potent on gr α M. We make the following Definition 3.1. Let M be a D-module with a rational V -filtration V. Then the two exact functors ψ φ = α [ 1,0) gr αm = α ( 1,0] gr αm and their so-called unipotent versions ψ u φ u = gr 1 M = gr 1 M (3) are called, respectively, nearby cycles, vanishing cycles, unipotent nearby cycles, unipotent vanishing cycles. These functors are valued in D-modules on Z with endomorphisms (given by f f ) with prescribed eigenvalues. There is also a pair of morphisms f : φ u ψ u and f : ψ u φ u. Observe that, following Lemma 2.1, the data of gr M (as a D-module on Z with the actions of f, f ) is equivalent to the data of ψ M, φ M with their actions of f f and the pair of maps f : φ u ψ u and f : ψ u φ u. Remark 3.1. In fact, when we come to define Hodge modules, f will be a map φ u M ψ u M( 1) (Tate twist), while f will still go ψ u M φ u M; let us brush this detail aside for now. There is also a notion of nearby and vanishing cycles for perverse sheaves (with rational coefficients, say) which is compatible with the same-named functors for (rh) D-modules (under base-change to C followed by RH 1 ). Indeed, consider the following diagram: X ẽxp X j X C exp C j C (4) For a constructible complex P on X, we define ψp as i Z j ẽxp ẽxp j P[ 1]. The action of Z on C by deck transformations of exp gives an action of Z on X over ẽxp, and thus by functoriality we get an action of Z on ψp, or equivalently an automorphism (action of 1 Z) T of ψp, known as the monodromy. In fact, on the level of Hodge modules this will be T : ψp ψp( 1). There is an adjunction morphism i Z P[ 1] ψp, which is T -equivariant (where T acts on i ZP as id); we denote the cone by φp, which by functoriality is also equipped with a monodromy operator T. We thus have the T -equivariant exact triangle ψp can φp i ZP +1 (5) on Z. (There is an analogous triangle in the derived category of D-modules with rational V -filtration). Then we have 4

Theorem 3.1. 1. The functors ψ, φ : D b c(x) D b c(z) are exact for the the perverse t-structure. 2. We have canonical equivalences RH 1 (ψp C ) = ψ(rh 1 P C ), RH 1 (φp C ) = φ(rh 1 P C ). 3. The operators T correspond to the operators exp(2πif f ) (hence the name quasi-unipotent monodromy); thus in particular the summand gr α RH 1 P C corresponds to the summand of ψp or ψp where T acts exp(2πi)- potently, denoted ψ exp(2πi) P or φ exp(2πi) P (hence the u for unipotent in ψ u, φ u ). 4. The operators f : ψ u φ u, f : φ u ψ u correspond to the operators can : ψ 1 P φ 1 P and a certain operator var : φ 1 P ψ 1 P. We give three examples which are by now trivial. Example 3.1. Suppose M is O-coherent, with the rational reduction a certain local system L. Then ψm = ψ u M = i Z M[ 1] is an O-coherent D-module on Z (equal to K Z if M = K X ), while φm = 0. Similarly, ψl = ψ 1 L = i ZP[ 1] is a local system on Z, and φl = 0. The monodromy is trivial. Example 3.2. Suppose M = i Z N = N[ f ], with the rational reduction P supported on Z. Then ψ(whatever) = 0 and φ(whatever) = i Z (whatever) = N or P. The monodromy is trivial. Example 3.3. (Exercise) Do the case X = C, f = x, M = C[x, x 1 ]x λ. Before giving a truly interesting and non-trivial example, we need to record the relationship between the V -filtration and the theory of Bernstein-Sato polynomials (or b-functions, for short). 4 b-functions and the case of singular f Suppose we have the above setup, but the function f is singular. Nonetheless we may choose local coordinates x 1,..., x n on X, and consider the graph embedding id f : X X C. Then we may write (id f) M = M[ t ], where t is the coordinate on C, and the structure of D-module is as follows: (ξ. t m ).t = ξf. t m + mξ t m 1 (ξ. t m ). t = ξ. t m+1 (ξ. t m ).x i = ξx i. t m (ξ. t m ). i = ξ i. t m ξf i. t m+1 (6) It is a fun exercise in the chain rule/change of variables to check this. Since the function t is smooth, we may consider the V -filtration with respect to t. Notice that, if it exists for M[ t ], then gr M[ t ] is both scheme theoretically supported on X {0} X C (by definition of V ), and set theoretically supported on the graph Γ f X C, being the associated graded of another such object. Thus, it is a (Q-graded, graded-coherent etc.) D-module on X, set theoretically supported on Z = Z(f). Example 4.1. (Exercise) Check (using uniqueness of the V -filtration) that if f is in fact regular, then the result of the above procedure is the D-module on X corresponding under Kashiwara s theorem with the D-module on Z given by the procedure of the previous sections. Following Equation 6, we have the following amazing/amusing/obvious fact: Observation 4.1. The isomorphism of vector spaces M[ t ] M.f t t [ t ] sending ξ. m t to ξ.f t t. m t is an isomorphism of D-modules on X C, where the LHS has the structure given by Equation 6, while the RHS has the obvious D-module structure. Recall the definition of the b-function of f. Definition 4.1. b f (s) = b f 1 (s) is the monic generator of the non-zero ideal {c(s) C[s] P D X [s] : f s+1.p = c(s)f s }. (7) For a (rh, qu) D-module M on X, and section u of M, we have the more general definition: 5

Definition 4.2. b f u(s) is the monic generator of the non-zero ideal Theorem 4.1. b f u(s) exists, and has negative rational roots. It is a simple observation that {c(s) C[s] P D X [s] : uf s+1.p = c(s)uf s }. (8) V α M[ t ] = {u M[ t ] u.c(t t ) V α 1 M[ t ] for some polynomial c(s) with roots in (α 1, α]}. (9) In fact, it is not so hard to prove (using the Artin-Rees construction for D X, V and some general results on Noetherian rings) that Lemma 4.1. In Equation 9, we can take c so that u.c(t t ) u.v 1 D X C = u.td X [t t ] = ufd X [t t ] V α 1 [t t ]. Writing s = t t and using Observation 4.1, we deduce Proposition 4.1. V α M[ t ] = {u M[ t ] b f u(s) has roots all α}. We end this session with the remark that there is an algorithm for computing b-functions, and thus in principal for computing the V -filtration. Example 4.2. (Exercise/next time)we compute nearby and vanishing cycles in the case X = C 2 with coordinates x, y, and f = x 2 + y 3. 6