SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES

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1 Journal of Singularities Volume , received: 18 Decemer 2013 in revised form: 20 May 2015 DOI: /jsing e SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI, WITH AN APPENDIX BY JÖRG SCHÜRMANN Astract. We study symmetries and stailization properties of perverse sheaves of vanishing cycles U,f of a regular function f : U C on a smooth C-scheme U, with critical locus X = Critf. We prove four main results: a If Φ : U U is an isomorphism fixing X and compatile with f, then the action of Φ on U,f is multiplication y det dφ X red = ±1. U,f depends up to canonical isomorphism only on X3, f 3, for X 3 the third-order thickening of X in U, and f 3 = f X 3 : X 3 C. c If U, V are smooth C-schemes, f : U C, g : V C are regular, X = Critf, Y = Critg, and Φ : U V is an emedding with f = g Φ and Φ X : X Y an isomorphism, there is a natural isomorphism Θ Φ : U,f Φ X V,g Z/2Z P Φ, for P Φ a principal Z/2Z-undle on X. d If X, s is an oriented d-critical locus in the sense of Joyce [23], there is a natural perverse sheaf P X,s on X, such that if X, s is locally modelled on Critf : U C then P X,s is locally modelled on U,f. We also generalize our results to replace U, X y complex analytic spaces, and U,f y D-modules or mixed Hodge modules. We discuss applications of d to categorifying Donaldson Thomas invariants of Calai Yau 3-folds, and to defining a Fukaya category of Lagrangians in a complex symplectic manifold using perverse sheaves. Contents 1. Introduction Background on perverse sheaves Constructile complexes on C-schemes Perverse sheaves on C-schemes Neary cycles and vanishing cycles on C-schemes Perverse sheaves of vanishing cycles on C-schemes Summary of the properties we use in this paper Perverse sheaves on complex analytic spaces D-modules on C-schemes and complex analytic spaces Mixed Hodge modules: asics Monodromic mixed Hodge modules Mixed Hodge modules of vanishing cycles Action of symmetries on vanishing cycles Proof of Proposition Part a: detdψ 1 dφ X red X red = ± Part : Φ = det dψ 1 dφ X red X red Ψ D-modules and mixed Hodge modules

2 86 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI 4. Dependence of U,f on f Proof of Proposition Proof of Theorem 4.2 for C-schemes D-modules and mixed Hodge modules Stailizing vanishing cycles Theorem 5.4a: the isomorphism Θ Φ Theorem 5.4: Θ Φ depends only on Φ X : X Y Theorem 5.4c: composition of the Θ Φ D-modules and mixed Hodge modules Perverse sheaves on oriented d-critical loci Background material on d-critical loci The main result, and applications Proof of Theorem 6.9 for C-schemes D-modules and mixed Hodge modules Appendix A. Compatiility results, y Jörg Schürmann References Introduction Let U e a smooth C-scheme and f : U C a regular function, and write X = Critf, as a C-suscheme of U. Then one can define the perverse sheaf of vanishing cycles U,f on X. Formally, X = c fx X c, where X c X is the open and closed C-suscheme of points x X with fx = c, and U,f Xc = φ p f c A U [dim U] Xc for each c fx, where A U [dim U] is the constant perverse sheaf on U over a ase ring A, and φ p f c : PervU Pervf 1 c is the vanishing cycle functor for f c : U C. See 2 for an introduction to perverse sheaves, and an explanation of this notation. This paper will prove four main results, Theorems 3.1, 4.2, 5.4 and 6.9. The first three give properties of the U,f, which we may summarize as follows: a Let U, f, X e as aove, and write X red for the reduced C-suscheme of X. Suppose Φ : U U is an isomorphism with f Φ = f and Φ X = id X. Then Φ induces a natural isomorphism Φ : U,f U,f. Theorem 3.1 implies that dφ T : T U U X red X T U red Xred has determinant det dφ X red : X red C \ {0}, which is a locally constant map X red {±1}, and Φ : U,f U,f is multiplication y det dφ X red. In fact Theorem 3.1 proves a more complicated statement, which only requires Φ to e defined étale locally on U. Let U, f, X e as aove, and write I X O U for the sheaf of ideals of regular functions U C vanishing on X. For each k = 1, 2,..., write X k for the k th order thickening of X in U, that is, X k is the closed C-suscheme of U defined y the vanishing of the sheaf of ideals I k X in O U. Write f k := f X k : X k C.

3 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 87 Theorem 4.2 says that the perverse sheaf U,f depends only on the third-order thickenings X 3, f 3 up to canonical isomorphism. As in Remark 4.5, étale locally, U,f depends only on X 2, f 2 up to noncanonical isomorphism, with isomorphisms natural up to sign. c Let U, V e smooth C-schemes, f : U C, g : V C e regular, and X = Critf, Y = Critg as C-suschemes of U, V. Let Φ : U V e a closed emedding of C- schemes with f = g Φ : U C, and suppose Φ X : X Y is an isomorphism. Then Theorem 5.4 constructs a natural isomorphism of perverse sheaves on X: Θ Φ : U,f Φ X V,g Z/2Z P Φ, 1.1 where π Φ : P Φ X is a certain principal Z/2Z-undle on X. Writing N UV for the normal undle of U in V, then the Hessian Hess g induces a nondegenerate quadratic form q UV on N UV X, and P Φ parametrizes square roots of detq UV : KU 2 X Φ X K2 V. Theorem 5.4 also shows that the Θ Φ in 1.1 are functorial in a suitale sense under compositions of emeddings Φ : U V, Ψ : V W. Here c is proved y showing that étale locally there exist equivalences V U C n identifying ΦU with U {0} and g : V C with f z z 2 n : U C n C, and applying étale local isomorphisms of perverse sheaves U,f = U,f L C n,z = z2 n U C n,f z1 2+ +z2 n = V,g, using C n,z = A z2 n {0} in the first step, and the Thom Seastiani Theorem for perverse sheaves in the second. Passing from f : U C to g = f z zn 2 : U C n C is an important idea in singularity theory, as in Arnold et al. [1] for instance. It is known as stailization, and f and g are called staly equivalent. So, Theorem 5.4 concerns the ehaviour of perverse sheaves of vanishing cycles under stailization. Our fourth main result, Theorem 6.9, concerns a new class of geometric ojects called d-critical loci, introduced in Joyce [23], and explained in 6.1. An algeraic d-critical locus X, s over C is a C-scheme X with a section s of a certain natural sheaf SX 0 on X. A d-critical locus X, s may e written Zariski locally as a critical locus Critf : U C of a regular function f on a smooth C-scheme U, and s records some information aout U, f in the notation of aove, s rememers f 2. There is also a complex analytic version. Algeraic d-critical loci are classical truncations of the derived critical loci more precisely, 1-shifted symplectic derived schemes introduced in derived algeraic geometry y Pantev, Toën, Vaquié and Vezzosi [42]. Theorem 6.9 roughly says that if X, s is an algeraic d-critical locus over C with an orientation, then we may define a natural perverse sheaf P X,s on X, such that if X, s is locally modelled on Critf : U C then P X,s is locally modelled on U,f. The proof uses Theorem 5.4. These results have exciting applications in the categorification of Donaldson Thomas theory on Calai Yau 3-folds, and in defining a new kind of Fukaya category of complex Lagrangians in complex symplectic manifold, which we will discuss at length in Remarks 6.14 and Although we have explained our results only for C-schemes and perverse sheaves upon them, the proofs are quite general and work in several contexts: i Perverse sheaves on C-schemes or complex analytic spaces with coefficients in any wellehaved commutative ring A, such as Z, Q or C. ii D-modules on C-schemes or complex analytic spaces. iii Saito s mixed Hodge modules on C-schemes or complex analytic spaces.

4 88 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI We discuss all these in 2, efore proving our four main results in 3 6. Appendix A, y Jörg Schürmann, proves two compatiility results etween duality and Thom Seastiani type isomorphisms needed in the main text. This is one of six linked papers [6, 9 11, 23], with more to come. The est logical order is that the first is Joyce [23] defining d-critical loci, and the second Bussi, Brav and Joyce [9], which proves Daroux-type theorems for the k-shifted symplectic derived schemes of Pantev et al. [42], and defines a truncation functor from 1-shifted symplectic derived schemes to algeraic d-critical loci. This paper is the third in the sequence. Comining our results with [23,42] gives new results on categorifying Donaldson Thomas invariants of Calai Yau 3-folds, as in Remark In the fourth paper Bussi, Joyce and Meinhardt [11] will generalize the ideas of this paper to motivic Milnor fires we explain the relationship etween the motivic and cohomological approaches elow in Remark 6.10, and deduce new results on motivic Donaldson Thomas invariants using [23,42]. In the fifth, Ben-Bassat, Brav, Bussi and Joyce [6] generalize [9,11] and this paper from derived schemes to derived Artin stacks. Sixthly, Bussi [10] will show that if S, ω is a complex symplectic manifold, and L, M are complex Lagrangians in S, then the intersection X = L M, as a complex analytic suspace of S, extends naturally to a complex analytic d-critical locus X, s. If the canonical undles K L, K M have square roots K 1/2 L, K1/2 M then X, s is oriented, and so Theorem 6.9 elow defines a perverse sheaf P L,M on X, which Bussi also constructs directly. As in Remark 6.15, we hope in future work to define a Fukaya category of complex Lagrangians in X, ω in which HomL, M = H n P L,M. Conventions. All C-schemes are assumed separated and of finite type. All complex analytic spaces are Hausdorff and locally of finite type. Acknowledgements. We would like to thank Oren Ben-Bassat, Alexandru Dimca, Young-Hoon Kiem, Jun Li, Kevin McGerty, Sven Meinhardt, Pierre Schapira, and especially Morihiko Saito and Jörg Schürmann for useful conversations and correspondence, and Jörg Schürmann for a very careful reading of our manuscript, leading to many improvements, as well as providing the Appendix. This research was supported y EPSRC Programme Grant EP/I033343/1. 2. Background on perverse sheaves Perverse sheaves, and the related theories of D-modules and mixed Hodge modules, make sense in several contexts, oth algeraic and complex analytic: a Perverse sheaves on C-schemes with coefficients in a ring A usually Z, Q or C, as in Beilinson, Bernstein and Deligne [5] and Dimca [14]. Perverse sheaves on complex analytic spaces with coefficients in a ring A usually Z, Q or C, as in Dimca [14]. c D-modules on C-schemes, as in Borel [8] in the smooth case, and Saito [48] in general. d D-modules on complex manifolds as in Björk [7], and on complex analytic spaces as in Saito [48]. e Mixed Hodge modules on C-schemes, as in Saito [45, 47]. f Mixed Hodge modules on complex analytic spaces, as in Saito [45, 47]. All our main results and proofs work, with minor modifications, in all six settings a f. As a is argualy the simplest and most complete theory, we egin in with a general introduction to constructile complexes and perverse sheaves on C-schemes, the neary and

5 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 89 vanishing cycle functors, and perverse sheaves of vanishing cycles P V U,f on C-schemes, following Dimca [14]. Several important properties of perverse sheaves in a either do not work, or ecome more complicated, in settings f. Section 2.5 lists the parts of that we will use in proofs in this paper, so the reader can check that they do work in f. Then give rief discussions of settings f, focussing on the differences with a in A good introductory reference on perverse sheaves on C-schemes and complex analytic spaces is Dimca [14]. Three other ooks are Kashiwara and Schapira [27], Schürmann [50], and Hotta, Tanisaki and Takeuchi [21]. Massey [36] and Rietsch [43] are surveys on perverse sheaves, and Beilinson, Bernstein and Deligne [5] is an important primary source, who cover oth Q-perverse sheaves on C-schemes as in a, and Q l -perverse sheaves on K-schemes as in g elow. Remark 2.1. Two further possile settings, in which not all the results we need are availale in the literature, are the following. g Perverse sheaves on K-schemes with coefficients in Z/l n Z, Z l, Q l, or Ql for l char K 2 a prime, as in Beilinson et al. [5]. h D-modules on K-schemes for K an algeraically closed field, as in Borel [8]. The issue is that the Thom Seastiani theorem is not availale in these contexts in the generality we need it. Once an appropriate form of this result ecomes availale, our main theorems will hold also in these two contexts, sometimes under the further assumption that char K = 0, needed for the results quoted from [9, 42]. We leave the details to the interested reader Constructile complexes on C-schemes. We egin y discussing constructile complexes, following Dimca [14, 2 4]. Definition 2.2. Fix a well-ehaved commutative ase ring A where well-ehaved means that we need assumptions on A such as A is regular noetherian, of finite gloal dimension or finite Krull dimension, a principal ideal domain, or a Dedekind domain, at various points in the theory, to study sheaves of A-modules. For some results A must e a field. Usually we take A = Z, Q or C. Let X e a C-scheme, always assumed of finite type. Write X an for the set of C-points of X with the complex analytic topology. Consider sheaves of A-modules S on X an. A sheaf S is called algeraically constructile if all the stalks S x for x X an are finite type A-modules, and there is a finite stratification X an = j J Xan j of X an, where X j X for j J are C- suschemes of X and Xj an X an the corresponding susets of C-points, such that S X an is an j A-local system for all j J. Write DX for the derived category of complexes C of sheaves of A-modules on X an. Write DcX for the full sucategory of ounded complexes C in DX whose cohomology sheaves H m C are constructile for all m Z. Then DX, DcX are triangulated categories. An example of a constructile complex on X is the constant sheaf A X on X with fire A at each point. Grothendieck s six operations on sheaves f, f!, Rf, Rf!, RHom, L act on DX preserving the sucategory DcX. That is, if f : X Y is a morphism of C-schemes, then we have two different pullack functors f, f! : DY DX, which also map DcY DcX. Here f is called the inverse image [14, 2.3], and f! the exceptional inverse image [14, 3.2]. We also have two different pushforward functors Rf, Rf! : DX DY

6 90 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI mapping D cx D cy, where Rf is called the direct image [14, 2.3] and is right adjoint to f : DY DX, and Rf! is called the direct image with proper supports [14, 2.3] and is left adjoint to f! : DY DX. We need the assumptions from 1 that X, Y are separated and of finite type for Rf, Rf! : D cx D cy to e defined for aritrary morphisms f : X Y. For B, C in D cx, we may form their derived Hom RHomB, C [14, 2.1], and left derived tensor product B L C in D cx, [14, 2.2]. Given B D cx and C D cy, we define B L C = π X B L π Y C in D cx Y, where π X : X Y X, π Y : X Y Y are the projections. If X is a C-scheme, there is a functor D X : D cx D cx op with D X D X = id : D c X D cx, called Verdier duality. It reverses shifts, that is, D X C [k] = D X C [ k] for C in D cx and k Z. Remark 2.3. Note how Definition 2.2 mixes the complex analytic and the complex algeraic: we consider sheaves on X an in the analytic topology, which are constructile with respect to an algeraic stratification X = j X j. Here are some properties of all these: Theorem 2.4. In the following, X, Y, Z are C-schemes, and f, g are morphisms, and all isomorphisms = of functors or ojects are canonical. i For f : X Y and g : Y Z, there are natural isomorphisms of functors Rg f = Rg Rf, Rg f! = Rg! Rf!, g f = f g, g f! = f! g!. ii If f : X Y is proper then Rf = Rf!. iii If f : X Y is étale then f = f!. More generally, if f : X Y is smooth of relative complex dimension d, then f [d] = f! [ d], where f [d], f! [ d] are the functors f, f! shifted y ±d. iv If f : X Y then Rf! = DY Rf D X and f! = D X f D Y. v If U is a smooth C-scheme then D U A U = A U [2 dim U]. If X is a C-scheme and C D cx, the hypercohomology H C and compactly-supported hypercohomology H cc, oth graded A-modules, are H k C = H k Rπ C and H k c C = H k Rπ! C for k Z, 2.1 where π : X is projection to a point. If X is proper then H C = H cc y Theorem 2.4ii. They are related to usual cohomology y H k A X = H k X; A and H k c A X = H k c X; A. If A is a field then under Verdier duality we have H k C = H k c D X C Perverse sheaves on C-schemes. Next we review perverse sheaves, following Dimca [14, 5]. Definition 2.5. Let X e a C-scheme, and for each x X an, let i x : X map i x : x. If C D cx, then the support supp m C and cosupport cosupp m C of H m C for m Z are supp m C = { x X an : H m i xc 0 }, cosupp m C = { x X an : H m i! xc 0 },

7 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 91 where { } means the closure in X an. If A is a field then cosupp m C = supp m D X C. We call C perverse, or a perverse sheaf, if dim C supp m C m and dim C cosupp m C m for all m Z, where y convention dim C =. Write PervX for the full sucategory of perverse sheaves in D cx. Then PervX is an aelian category, the heart of a t-structure on D cx. Perverse sheaves have the following properties: Theorem 2.6. a If A is a field then PervX is noetherian and artinian. If A is a field then D X : D cx D cx maps PervX to PervX. c If i : X Y is inclusion of a closed C-suscheme, then Ri, Ri! which are naturally isomorphic map PervX to PervY. Write PervY X for the full sucategory of ojects in PervY supported on X. Then Ri = Ri! are equivalences of categories PervX PervY X. The restrictions i PervY X, i! PervY X which map PervY X to PervX, are naturally isomorphic, and are quasi-inverses for Ri, Ri! : PervX PervY X. d If f : X Y is étale then f and f! which are naturally isomorphic map PervY to PervX. More generally, if f : X Y is smooth of relative dimension d, then f [d] = f! [ d] map PervY to PervX. e L : D cx D cy D cx Y maps PervX PervY to PervX Y. f Let U e a smooth C-scheme. Then A U [dim U] is perverse, where A U is the constant sheaf on U with fire A, and [dim U] means shift y dim U in the triangulated category D cx. Note that Theorem 2.4v gives a canonical isomorphism D U AU [dim U] = AU [dim U]. When A = Q, so that PervX is noetherian and artinian y Theorem 2.6a, the simple ojects in PervX admit a complete description: they are all isomorphic to intersection cohomology complexes IC V L for V X a smooth locally closed C-suscheme and L V an irreducile Q-local system, [14, 5.4]. Furthermore, if f : X Y is a proper morphism of C-schemes, then the Decomposition Theorem [5, 6.2.5], [14, Th ], [45, Cor. 3] says that, in case IC V L is of geometric origin, Rf IC V L is isomorphic to a finite direct sum of shifts of simple ojects IC V L in PervY. The next theorem is proved y Beilinson et al. [5, Cor , , & Th ]. The analogue for DcX or DX rather than PervX is false. One moral is that perverse sheaves ehave like sheaves, rather than like complexes. Theorem 2.7i will e used throughout 3 6. Theorem 2.7ii will e used only once, in the proof of Theorem 6.9 in 6.3, and we only need Theorem 2.7ii to hold in the Zariski topology, rather than the étale topology. Theorem 2.7. Let X e a C-scheme. Then perverse sheaves on X form a stack a kind of sheaf of categories on X in the étale topology. Explicitly, this means the following. Let {u i : U i X} i I e an étale open cover for X, so that u i : U i X is an étale morphism of C-schemes for i I with i u i surjective. Write U ij = U i ui,x,u j U j for i, j I with projections π i ij : U ij U i, π j ij : U ij U j, u ij =u i π i ij =u j π j ij : U ij X. Similarly, write U ijk = U i X U j X U k for i, j, k I with projections π ij ijk : U ijk U ij, π ik ijk : U ijk U ik, π jk ijk : U ijk U jk, π i ijk : U ijk U i, π j ijk : U ijk U j, π k ijk : U ijk U k, u ijk : U ijk X,

8 92 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI so that πijk i = πi ij πij ijk, u ijk = u ij π ij ijk = u i πijk i, and so on. All these morphisms u i, πij i,..., u ijk are étale, so y Theorem 2.6d u i = u! i maps PervX PervU i, and similarly for πij i,..., u ijk. With this notation: i Suppose P, Q PervX, and we are given α i : u i P u i Q in PervU i for all i I such that for all i, j I we have π i ij α i = π i ij α j : u ijp u ijq. Then there is a unique α : P Q in PervX with α i = u i α for all i I. ii Suppose we are given ojects P i PervU i for all i I and isomorphisms in PervU ij for all i, j I with α ii = id and α ij : π i ij P i π j ij P j π jk ijk α jk π ij ijk α ij = π ik ijk α ik : π i ijk P i π k ijk P k in PervU ijk for all i, j, k I. Then there exists P in PervX, unique up to canonical isomorphism, with isomorphisms β i : u i P P i for each i I, satisfying for all i, j I. α ij π i ij β i = π j ij β j : u ijp π j ij P j, We will need the following proposition in 3.3 to prove Theorem 3.1. Most of it is setting up notation, only the last part α X = β X is nontrivial. Proposition 2.8. Let W, X e C-schemes, x X, and π C : W C, π X : W X, ι : C W morphisms, such that π C π X : W C X is étale, and π C ι = id C : C C, and π X ιt = x for all t C. Write W t = π 1 C t W for each t C, and j t : W t W for the inclusion. Then π X Wt = π X j t : W t X is étale, and ιt W t with π X Wt ιt = x, so we may think of W t for t C as a 1-parameter family of étale open neighourhoods of x in X. Let P, Q PervX, so that y Theorem 2.6d as π X is smooth of relative dimension 1 and π X Wt is étale, we have πx [1]P PervW and π X W t P = j t [ 1] π X[1]P PervW t, and similarly for Q. Suppose α, β : P Q in PervX and γ : π X [1]P π X [1]Q in PervW are morphisms such that π X W 0 α = j 0[ 1]γ in PervW 0 and π X W 1 β = j 1[ 1]γ in PervW 1. Then there exists a Zariski open neighourhood X of x in X such that α X = β X : P X Q X. Here we should think of j t [ 1]γ for t C as a family of perverse sheaf morphisms P Q, defined near x in X locally in the étale topology. But morphisms of perverse sheaves are discrete to see this, note that we can take A = Z, so as j t [ 1]γ depends continuously on t, it should e locally constant in t near x, in a suitale sense. The conclusion α X = β X essentially says that j 0[ 1]γ = j 1[ 1]γ near x. If P X is a principal Z/2Z-undle on a C-scheme X, and Q PervX, we will define a perverse sheaf Q Z/2Z P, which will e important in 5 6. Definition 2.9. Let X e a C-scheme. A principal Z/2Z-undle P X is a proper, surjective, étale morphism of C-schemes π : P X together with a free involution σ : P P, such that the orits of Z/2Z = {1, σ} are the fires of π. We will use the ideas of isomorphism of principal

9 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 93 undles ι : P P, section s : X P, tensor product P Z/2Z P, and pullack f P W under a C-scheme morphism f : W X, all of which are defined in the ovious ways. Let P X e a principal Z/2Z-undle. Write L P D cx for the rank one A-local system on X induced from P y the nontrivial representation of Z/2Z = {±1} on A. It is characterized y π A P = A X L P. For each Q D cx, write Q Z/2Z P D cx for Q L L P, and call it Q twisted y P. If Q is perverse then Q Z/2Z P is perverse. Perverse sheaves and complexes twisted y principal Z/2Z-undles have the ovious functorial ehaviour. For example, if P X, P X are principal Z/2Z-undles and Q D cx there is a canonical isomorphism Q Z/2Z P Z/2Z P = Q Z/2Z P Z/2Z P, and if f : W X is a C-scheme morphism there is a canonical isomorphism f Q Z/2Z P = f Q Z/2Z f P Neary cycles and vanishing cycles on C-schemes. We explain neary cycles and vanishing cycles, as in Dimca [14, 4.2]. The definition is complex analytic, X an, C in 2.2 do not come from C-schemes. Definition Let X e a C-scheme, and let f : X C e a regular function. Define X 0 = f 1 0, as a C-suscheme of X, and X = X \ X 0. Consider the commutative diagram of complex analytic spaces: X an 0 i f X an f j π X an f p X an {0} C C C. Here X an, X0 an, X an are the complex analytic spaces associated to the C-schemes X 0, X, X, and i : X0 an X an, j : X an X an are the inclusions, ρ : C C is the universal cover of X an C = C \ {0}, and = X an C f,c,ρ the corresponding cover of X an, with covering map p : X an X an, and π = j p. As in 2.6, the triangulated categories DX, DcX and six operations f, f!, Rf, Rf!, RHom, L also make sense for complex analytic spaces. So we can define the neary cycle functor ψ f : DcX DcX 0 to e ψ f = i Rπ π. Since this definition goes via X an which is not a C-scheme, it is not ovious that ψ f maps to algeraically constructile complexes DcX 0 rather than just to DX 0, ut it does [14, p. 103], [27, p. 352]. There is a natural transformation Ξ : i ψ f etween the functors i, ψ f : D cx D cx 0. The vanishing cycle functor φ f : D cx D cx 0 is a functor such that for every C in D cx we have a distinguished triangle ρ f 2.2 i C ΞC ψ f C φ f C [+1] i C in D cx 0. Following Dimca [14, p. 108], we write ψ p f, φp f for the shifted functors ψ f [ 1], φ f [ 1] : D cx D cx 0. The generator of Z = π 1 C on C induces a deck transformation δ C : C C which lifts to a deck transformation δ X : X X with p δ X = p and f δx = δc f. As in [14, p. 103,

10 94 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI p. 105], we can use δ X to define natural transformations MX,f : ψ p f ψp f and M X,f : φ p f φp f, called monodromy. Alternative definitions of ψ f, φ f in terms of specialization and microlocalization functors are given y Kashiwara and Schapira [27, Prop ]. Here are some properties of neary and vanishing cycles. Parts i,ii can e found in Dimca [14, Th & Prop ]. Part iv is proved y Massey [37]; compare also Proposition A.1 in the Appendix. Theorem i If X is a C-scheme and f : X C is regular, then the functors ψ p f, φp f : D cx D cx 0 oth map PervX to P ervx 0. ii Let Φ : X Y e a proper morphism of C-schemes, and g : Y C e regular. Write f = g Φ : X C, X 0 = f 1 0 X, Y 0 = g 1 0 Y, and Φ 0 = Φ X0 : X 0 Y 0. Then we have natural isomorphisms RΦ 0 ψ p f = ψ p g RΦ and RΦ 0 φ p f = φ p g RΦ. 2.3 Note too that RΦ = RΦ! and RΦ 0 = RΦ0!, as Φ, Φ 0 are proper. iii Let Φ : X Y e an étale morphism of C-schemes, and g : Y C e regular. Write f = g Φ : X C, X 0 = f 1 0 X, Y 0 = g 1 0 Y, and Φ 0 = Φ X0 : X 0 Y 0. Then we have natural isomorphisms Φ 0 ψ p f = ψ p g Φ and Φ 0 φ p f = φ p g Φ. 2.4 Note too that Φ = Φ! and Φ 0 = Φ! 0, as Φ, Φ 0 are étale. More generally, if Φ : X Y is smooth of relative complex dimension d and g, f, X 0, Y 0, Φ 0 are as aove, then we have natural isomorphisms Φ 0[d] ψ p f = ψ p g Φ [d] and Φ 0[d] φ p f = φ p g Φ [d]. 2.5 Note too that Φ [d] = Φ! [ d] and Φ 0[d] = Φ! 0[ d]. iv If X is a C-scheme and f : X C is regular, then there are natural isomorphisms ψ p f D X = D X0 ψ p f and φp f D X = D X0 φ p f Perverse sheaves of vanishing cycles on C-schemes. We can now define the main suject of this paper, the perverse sheaf of vanishing cycles U,f for a regular function f : U C. Definition Let U e a smooth C-scheme, and f : U C a regular function. Write X = Critf, as a closed C-suscheme of U. Then as a map of topological spaces, f X : X C is locally constant, with finite image fx, so we have a decomposition X = c fx X c, for X c X the open and closed C-suscheme with fx = c for each C-point x X c. Note that if X is non-reduced, then f X : X C need not e locally constant as a morphism of C-schemes, ut f X red : X red C is locally constant, where X red is the reduced C-suscheme of X. Since X, X red have the same topological space, f X : X C is locally constant on topological spaces. For each c C, write U c = f 1 c U. Then as in 2.3, we have a vanishing cycle functor φ p f c : PervU PervU c. So we may form φ p f c A U [dim U] in PervU c, since A U [dim U] PervU y Theorem 2.6f. One can show φ p f c A U [dim U] is supported on the closed suset X c = Critf U c in U c, where X c = unless c fx. That is, φ p f c A U [dim U] lies in PervU c Xc.

11 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 95 But Theorem 2.6c says PervU c Xc and PervX c are equivalent categories, so we may regard φ p f c A U [dim U] as a perverse sheaf on X c. That is, we can consider φ p f c A U [dim U] Xc = i X c,u c φ p f c A U [dim U] in PervX c, where i Xc,U c : X c U c is the inclusion morphism. As X = c fx X c with each X c open and closed in X, we have PervX = PervX c. c fx Define the perverse sheaf of vanishing cycles U,f of U, f in PervX to e U,f = φ p f c A U [dim U] Xc. c fx That is, U,f is the unique perverse sheaf on X = Critf with U,f Xc = φ p f c A U [dim U] Xc for all c fx. Under Verdier duality, we have A U [dim U] = D U A U [dim U] y Theorem 2.6f, so φ p f c A U [dim U] = D Uc φ p f c A U [dim U] y Theorem 2.11iv. Applying i X c,u c and using D Xc i X c,u c = i! Xc,U c D Uc y Theorem 2.4iv and i! X c,u c = i Xc,U c on PervU c Xc y Theorem 2.6c also gives φ p f c A U [dim U] Xc = DXc φ p f c A U [dim U] Xc. Summing over all c fx yields a canonical isomorphism σ U,f : U,f For c fx, we have a monodromy operator = D X U,f. 2.6 M U,f c : φ p f c A U [dim U] φ p f c A U [dim U], which restricts to φ p f c A U [dim U] Xc. Define the twisted monodromy operator y τ U,f : U,f U,f τ U,f Xc = 1 dim U M U,f c Xc : φ p f c A U [dim U] Xc φ p f c A U [dim U] Xc, 2.7 for each c fx. Here twisted refers to the sign 1 dim U in 2.7. We include this sign change as it makes monodromy act naturally under transformations which change dimension without it, equation 5.15 elow would only commute up to a sign 1 dim V dim U, not commute and it normalizes the monodromy of any nondegenerate quadratic form to e the identity, as in The sign 1 dim U also corresponds to the twist 1 2 dim U in the definition 2.24 of the mixed Hodge module of vanishing cycles HV U,f in The compactly-supported hypercohomology H U,f, H c U,f from 2.1 is an important invariant of U, f. If A is a field then the isomorphism σ U,f in 2.6 implies that H k U,f = H k c U,f, a form of Poincaré duality.

12 96 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI We defined U,f in perverse sheaves over a ase ring A. Writing U,f A to denote the ase ring, one can show that U,f A = U,f Z L Z A. Thus, we may as well take A = Z, or A = Q if we want A to e a field, since the case of general A contains no more information. There is a Thom Seastiani Theorem for perverse sheaves, due to Massey [35] and Schürmann [50, Cor ]. Applied to U,f, it yields: Theorem Let U, V e smooth C-schemes and f : U C, g : V C e regular, so that f g : U V C is regular with f gu, v := fu+gv. Set X = Critf and Y = Critg as C-suschemes of U, V, so that Critf g = X Y. Then there is a natural isomorphism T S U,f,V,g : U V,f g L U,f V,g 2.8 in PervX Y, such that the following diagrams commute: U V,f g D σ X Y U V,f g U V,f g T S U,f,V,g L U,f V,g σ U,f L σv,g D X U,f L D Y V,g = D X Y T S U,f,V,g U V,f g τ U V,f g D X Y U,f L V,g, U V,f g 2.9 U,f T S U,f,V,g L V,g T S U,f,V,g L τ U,f τv,g U,f L V,g The next example will e important later. Example Define f : C n C y fz 1,..., z n = z zn 2 for n > 1. Then Critf = {0}, so C n,z = z2 φp n f A C n[n] {0} is a perverse sheaf on the point {0}. Following Dimca [14, Prop , Ex & Ex ], we find that there is a canonical isomorphism C n,z z2 n = H n 1 MF f 0; A A A {0}, 2.11 where MF f 0 is the Milnor fire of f at 0, as in [14, p. 103]. Since fz = z z 2 n is homogeneous, we see that MF f 0 = { z 1,..., z n C n : fz 1,..., z n = 1 } = T S n 1, so that H n 1 MF f 0; A = H n 1 S n 1 ; A = A. Therefore we have C n,z z2 n = A{0} This isomorphism 2.12 is natural up to sign unless the ase ring A has characteristic 2, in which case 2.12 is natural, as it depends on the choice of isomorphism H n 1 S n 1, A = A, which corresponds to an orientation for S n 1. This uncertainty of signs will e important in 5 6. We can also use Milnor fires to compute the monodromy operator on C n,z. There z2 n is a monodromy map µ f : MF f 0 MF f 0, natural up to isotopy, which is the monodromy

13 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 97 in the Milnor firation of f at 0. Under the identification MF f 0 = T S n 1 we may take µ f to e the map d 1 : T S n 1 T S n 1 induced y 1 : S n 1 S n 1 mapping 1 : x 1,..., x n x 1,..., x n. This multiplies orientations on S n 1 y 1 n. Thus, µ f : H n 1 S n 1, A H n 1 S n 1, A multiplies y 1 n. By [14, Prop ], equation 2.11 identifies the action of the monodromy operator M C n,f {0} on C n,z with the action of µ z2 f on H n 1 S n 1, A. So M n C n,f {0} is multiplication y 1 n. Comining this with the sign change 1 dim U in 2.7 for U = C n shows that the twisted monodromy is τ C n,z z2 n = id : C n,z z2 n C n,z z2 n Equations also hold for n = 0, 1, though 2.11 does not. Note also that these results are compatile with the Thom Seastiani Theorem 2.13, and can e deduced from it and the case n = 1. We introduce some notation for pullacks of V,g y étale morphisms. Definition Let U, V e smooth C-schemes, Φ : U V an étale morphism, and g : V C a regular function. Write f = g Φ : U C, and X = Critf, Y = Critg as C-suschemes of U, V. Then Φ X : X Y is étale. Define an isomorphism Φ : U,f Φ X V,g in PervX 2.14 y the commutative diagram for each c fx gy : U,f Xc =φ p f c A U [dim U] Xc α φ p f c Φ A V [dim V ] Xc Φ Xc Φ X c V,g β Φ 0 φ p g c A V [dim V ] Xc Here α is φ p f c applied to the canonical isomorphism A U Φ A V, noting that, as Φ is étale, dim U = dim V and β is induced y 2.4. By naturality of the isomorphisms α, β in 2.15 we find the following commute, where σ U,f, τ U,f are as in : U,f D σ X U,f U,f Φ X Φ V,g Φ X σ V,g D X Φ Φ X DY V,g = D X Φ X V,g, 2.16 U,f τ U,f Φ Φ X V,g Φ X τ V,g U,f Φ Φ X V,g If U = V, f = g and Φ = id U then idu = id U,f. If W is another smooth C-scheme, Ψ : V W is étale, and h : W C is regular with g = h Ψ : V C, then composing 2.15 for Φ with Φ X c of 2.15 for Ψ shows that Ψ Φ = Φ X Ψ Φ : U,f Ψ Φ X W,h That is, the isomorphisms Φ are functorial.

14 98 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI Example In Definition 2.15, set U = V = C n and fz 1,..., z n = gz 1,..., z n = z z 2 n, so that Y = Z = {0} C n. Let M On, C e an orthogonal matrix, so that M : C n C n is an isomorphism with f = g M and M {0} = id {0}. As M Y = id Y, Definition 2.15 defines an isomorphism M : C n,z z2 n C n,z z2 n Equation 2.11 descries C n,z in terms of MF z2 f 0 = T S n 1. Now n multiplies orientations on S n 1 y det M, so M MFf 0 : MF f 0 MF f 0 M MFf 0 : H n 1 MF f 0; A H n 1 MF f 0; A is multiplication y det M. Thus 2.11 implies that M in 2.19 is multiplication y det M = ± Summary of the properties we use in this paper. Since parts of do not work for the other kinds of perverse sheaves, D-modules and mixed Hodge modules in , we list what we will need for 3 6, to make it easy to check they are also valid in the settings of i There should e an A-linear aelian category PX of P-ojects defined for each scheme or complex analytic space X, over a fixed, well-ehaved ase ring A. We do not require A to e a field. ii There should e a Verdier duality functor D X with D X D X = id, defined on a suitale sucategory of P-ojects on X which includes the ojects we are interested in. We do not need D X to e defined on all ojects in PX. iii If U is a smooth scheme or complex manifold, then there should e a canonical oject A U [dim U] PU, with a canonical isomorphism D U A U [dim U] = A U [dim U]. iv Let f : X Y e a closed emedding of schemes or complex analytic spaces; this implies f is proper. Then f, f! : PX PY should exist, inducing an equivalence of categories PX P X Y as in Theorem 2.6c, where P X Y is the full sucategory of ojects in PY supported on X. v Let f : X Y e an étale morphism. Then the pullacks f, f! : PY PX should exist. More generally, if f : X Y is smooth of relative dimension d, then there should e pullacks f [d], f! [ d] mapping PY PX. If X, Y are smooth, there should e a canonical isomorphism f [d]a Y [dim Y ] = A X [dim X]. We do not need pullacks to exist for general morphisms f : X Y, though see xi elow. vi An external tensor product L : PX PY PX Y should exist for all X, Y. vii If X is a scheme or complex analytic space, P X a principal Z/2Z-undle, and Q PX, the twisted perverse sheaf Q Z/2Z P PX should make sense as in Definition 2.9, and have the ovious functorial properties. viii A vanishing cycle functor φ p f : PU PU 0 and a monodromy transformation M U,f : φ p f φp f in 2.4 should exist for all smooth U and regular/holomorphic f : U A 1.

15 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 99 ix The functors D X, f, f!, f, f!, φ p f should satisfy the natural isomorphisms in Theorems 2.4 and 2.11, provided they exist. They should have the ovious compatiilities with L, and restriction to Zariski open sets. x There should e suitale sucategories of P-ojects which form a stack in the étale or complex analytic topologies, as in Theorem 2.7. In the algeraic case we only need Theorem 2.7ii to hold for Zariski open covers, not étale open covers. xi Proposition 2.8 must hold. This involves pullacks jt y a morphism j t : W t W which is not étale or smooth, as in v aove. But on ojects we only consider j t π X P = π X W t P which exists in PW t y v as π X Wt is étale, so j t is defined on the ojects we need. xii There should e a Thom Seastiani Theorem for P-ojects, so that the analogue of Theorem 2.13 holds. Remark The existence of a ounded derived category of P-ojects will not e assumed, or used, in this paper. On the other hand, in all the cases we consider, there will e a realization functor from the category of P-ojects to an appropriate category of constructile complexes, and the notation used aove reflects this. So in iii,v aove, [1] does not stand for a shift in any derived category; the notation means a P-oject or morphism whose realization is the appropriate constructile oject or morphism. See Remark 2.20 elow Perverse sheaves on complex analytic spaces. Next we discuss perverse sheaves on complex analytic spaces, as in Dimca [14]. The theory follows , replacing smooth C- schemes y complex analytic spaces complex manifolds, and regular functions y holomorphic functions. Let X e a complex analytic space, always assumed locally of finite type that is, locally emeddale in C n. In the analogue of Definition 2.2, we fix a well-ehaved commutative ring A, and consider sheaves of A-modules S on X in the complex analytic topology. A sheaf S is called analytically constructile if all the stalks S x for x X are finite type A-modules, and there is a locally finite stratification X = j J X j of X, where now X j X for j J are complex analytic suspaces of X, such that S Xj is an A-local system for all j J. Write DX for the derived category of complexes C of sheaves of A-modules on X, exactly as in 2.1, and DcX for the full sucategory of ounded complexes C in DX whose cohomology sheaves H m C are analytically constructile for all m Z. Then DX, DcX are triangulated categories. When we wish to distinguish the complex algeraic and complex analytic theories, we will write DcX alg, PervX alg for the algeraic versions in with X a C-scheme, and DcX an, PervX an for the analytic versions. Here are the main differences etween the material of for perverse sheaves on C- schemes and on complex analytic spaces: a If f : X Y is an aritrary morphism of C-schemes, then as in 2.1 the pushforwards Rf, Rf! : DX DY also map D cx alg D cy alg. However, if f : X Y is a morphism of complex analytic spaces, then Rf, Rf! : DX DY need not map D cx an D cy an without extra assumptions on f, for example, if f : X Y is proper. The analogue of Theorem 2.7 says that perverse sheaves on a complex analytic space X form a stack in the complex analytic topology. This is proved in the suanalytic context

16 100 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI in [27, Th ]; the analytic case follows upon noting that a sheaf is complex analytically constructile if and only if is locally at all points, as proved in [14, Prop ]. See also [21, Prop ]. The analogues of i xii in 2.5 work for complex analytic perverse sheaves, and so our main results hold in this context. If X is a C-scheme, and X an the corresponding complex analytic space, then DX in 2.1 for X a C-scheme coincides with DX an for X an a complex analytic space, and D cx alg D cx an an, PervX alg PervX an an are full sucategories, and the six functors f, f!, Rf, Rf!, RHom, L for C-scheme morphisms f : X Y agree in the algeraic and analytic cases D-modules on C-schemes and complex analytic spaces. D-modules on a smooth C-scheme or smooth complex analytic space X are sheaves of modules over a certain sheaf of rings of differential operators D X on X. Some ooks on them are Borel et al. [8], Coutinho [12], and Hotta, Takeuchi and Tanisaki [21] in the C-scheme case, and Björk [7] and Kashiwara [26] in the complex analytic case. For a singular complex C-scheme or complex analytic space X, the definition of a well-ehaved category of D-modules is given y Saito [48], via locally emedding X into a smooth scheme or space. The analogue of perverse sheaves on X are called regular holonomic D-modules, which form an aelian category Mod rh D X, the heart in the derived category D rh ModD X of ounded complexes of D X -modules with regular holonomic cohomology modules. The whole package of works for D-modules. Our next theorem is known as the Riemann Hilert correspondence [7, V.5], [21, Th ], see Borel [8, 14.4] for C-schemes, Kashiwara [25] for complex manifolds, and Saito [48, 6] for complex analytic spaces, and also Maisonoe and Mekhout [34]. Theorem Let X e a C-scheme or complex analytic space. Then there is a de Rham functor DR : Drh ModD X DcX, C, which is an equivalence of categories, restricts to an equivalence Mod rh D X PervX, C, and commutes with f, f!, Rf, Rf!, RHom,, L and also with ψ p f, φp f for X smooth. Here D cx, C, PervX, C are constructile complexes and perverse sheaves over the ase ring A = C. Because of the Riemann Hilert correspondence, all our results on perverse sheaves of vanishing cycles on C-schemes and complex analytic spaces in 3 6 over a well-ehaved ase ring A, translate immediately when A = C to the corresponding results for D-modules of vanishing cycles, with no extra work Mixed Hodge modules: asics. We write this section in the minimal generality needed for our applications. The statements made work equally well in the category of algeraic C- schemes and the category of complex analytic spaces. By space, we will mean an oject in either of these categories. The theory of mixed Hodge modules works with reduced spaces; should a space X e non-reduced, the following constructions are taken y definition on its reduction. For a space X, let HMX denote Saito s category [45] of polarizale pure Hodge modules, locally a direct sum of sucategories HMX w of pure Hodge modules of fixed weight w. On a smooth X, a pure Hodge module M consists of a triple of data: a filtered holonomic D-module M, F, a Q-perverse sheaf, and a comparison map identifying the former with the complexification of the latter under the Riemann Hilert correspondence; see [45, 5.1.1, p. 952] and [47, 4]. This triple has to satisfy many other properties; in particular, the underlying holonomic D- module is automatically regular, and algeraic Hodge modules are asked to e extendale to an algeraic compactification. Thus there is a forgetful functor HMX Mod rh D X from Hodge

17 SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES 101 modules to regular holonomic algeraic D-modules. Hodge modules on singular spaces are defined, similarly to D-modules, via emeddings into smooth varieties; see Saito [47] and also Maxim, Saito and Schürmann [39, 1.8]. There is a duality functor D H X : HMX HMX. Pure Hodge modules also admit a Tate twist functor M M 1, see [45, 5.1.3, p. 952]. This functor shifts the filtration and rotates the rational structure on the underlying perverse sheaf: the D-module filtration M, F is shifted to M, F [n] with F [n] i = F i n ; the underlying perverse sheaf is tensored y Zn = 2πi n Z C, as in [45, 2.0.2, p. 876]. A polarization of weight w on a pure Hodge module M HMX w is a morphism of pure Hodge modules σ : M D H XM w, satisfying the extra conditions using vanishing cycles descried on [45, on p. 956 and on p. 968], as well as the condition that on points it should correspond to the classical notion of a polarization of a pure Hodge structure including positive definiteness. Next, let MHMX denote the category of graded polarizale mixed Hodge modules [45, 47]. A graded polarizale mixed Hodge module carries a functorial weight filtration W, with graded pieces eing polarizale pure Hodge modules, see [45, , p ]. The forgetful functor rat : MHMX PervX to the appropriate category of perverse Q-sheaves on X is faithful and exact; faithfulness in particular means that a morphism in MHMX is uniquely determined y the underlying morphism of perverse sheaves. The Tate twist functor extends to MHMX; under this functor, the weight filtration W of the mixed Hodge module is changed to W [2n] with W [2n] i = W i+2n as on [45, p. 855]. The duality functor D H X also extends to MHMX and is compatile with Verdier duality on the perverse realization. There is also a forgetful functor MHMX Mod rh D X to regular holonomic D-modules, even for singular spaces. Theorem The categories of graded polarizale mixed Hodge modules have the following properties: i By [47, Th. 3.9, p. 288], the category of mixed Hodge modules for X a point is canonically equivalent to Deligne s category of graded polarizale mixed Hodge structures. ii For a smooth space U, we have a canonical oject of weight dim U Q H U [dim U] HMU MHMU, which y [45, Prop , p. 971] possesses a canonical polarization σ : Q H U [dim U] D H U Q H U [dim U] dim U. iii For an open inclusion f : Y X of spaces, there is a pullack functor f = f! : MHMX MHMY. More generally, y [47, Prop. 2.19, p. 258], for an aritrary morphism f : Y X, there exist cohomological pullack functors L j f, L j f! : MHMX MHMY compatile with perverse cohomological pullack on the perverse sheaf level. iv For a closed emedding i : X Y, there is a pushforward functor i = i! : MHMX MHMY, whose essential image is the full sucategory MHM X Y of ojects in MHMY supported on X. Its inverse is i =i! : MHM X Y MHMX. More generally, y [45, Th , p. 977] and [47, Th. 2.14, p. 252], for a projective map f : X Y there are cohomological pushforward functors R j f : MHMX MHMY.

18 102 C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDRŐI v There is an external tensor product functor L : MHMX MHMY MHMX Y, which is compatile with duality in the sense that for all M MHMX and N MHMY, there is a natural isomorphism D H XM L D H Y N = D H X Y M L N. Remark We will not need to use any derived category D? MHMX of mixed Hodge modules in this paper, which is just as well since on singular analytic X, the appropriate oundedness conditions do not appear to e well understood, and the general pullack and pushforward functors of Theorem 2.19iii,iv are not known to exist as derived functors outside of the algeraic context of [47, 4]. Hence, in part ii aove, [1] does not stand for a shift in the derived category; Q H U [dim U] just denotes a mixed Hodge module whose realization is the perverse sheaf Q U [dim U] on U. Compare Remark 2.17 aove. Using the functors aove, we can now define the twist of a mixed Hodge module y a principal Z/2Z-undle. In the setup of Definition 2.9, given a Z/2Z-undle π : P X, and an oject M MHMX on a space X, we have a natural map M π π M, which is an injection y faithfulness of the realization functor and the fact that it is an injection on the perverse sheaf level. The quotient oject will e denoted, y ause of notation, y M Z/2Z P in MHMX Monodromic mixed Hodge modules. To discuss neary and vanishing cycle functors in a way consistent with monodromy, we need an extension of the category of mixed Hodge modules. For a space X, following Saito [49, 4.2] denote y MHMX; T s, N the category of mixed Hodge modules M on X with commuting actions of a finite order operator T s : M M and a locally nilpotent operator N : M M 1. There is an emedding of categories MHMX MHMX; T s, N defined y setting T = id and N = 0. As proved y [49, 4.6.2], the category MHMX; T s, N is equivalent to the category MHMX C mon,! of monodromic mixed Hodge modules on X C extended y zero to X C; compare also [33, 4.2]. Every oject M MHMX; T s, N decomposes into a direct sum M = M 1 M 1 of the T s- invariant part and its T s -equivariant complement. The Tate twist, and appropriate cohomological pullack and pushforward functors continue to exist. There is a duality functor D T X : MHMX; T s, N MHMX; T s, N defined y D T XM = D H XM 1 D H T M 11, equipped with the finite-order operator D X T s 1 and the nilpotent operator D X N. This duality functor still satisfies D T X D T X = id. Saito [49, 5.1] also defines an external tensor product T : MHMX 1 ; T s, N MHMX 2 ; T s, N MHMX 1 X 2 ; T s, N. defined on the monodromic category as follows. The addition map on fires induces the additive convolution π : X 1 C X 2 C X 1 X 2 C π : MHMX 1 C mon,! MHMX 2 C mon,! MHMX 1 x 2 C mon,!. One can translate this external tensor product T to the MHMX; T s, N defined y concrete data M, T s, N. On the underlying D-modules and perverse sheaves, it is just the usual product