A topological construction of the weight filtration

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1 manuscripta math. 133, (2010) Springer-Verlag 2010 Fouad El Zein, D. T. Lê, Luca Migliorini A topological construction o the weight iltration Received: 21 October 2009 / Revised: 1 April 2010 Published online: 22 June 2010 Abstract. Let j : X \ Y X be the embedding o the complement o a Cartier divisor Y in a complex algebraic variety X, and let K beaperversesheaonx \ Y. With the aid o the specialization unctor introduced by Verdier in Analyse et Topologie sur les espaces singuliers vol. II, III, Astérisque : [13], we deine a iltration W o topological origin on the perverse complex Rj K which should play the role o the weight iltration when K is a local system underlying a polarised variation o Hodge structures. 1. The monodromy iltration Let D be an eective principal divisor in a complex algebraic variety U, with deining equation Ɣ(U, O U ), denote by D i j U U \ D =: U the corresponding embeddings, and denote by D U (resp. D D, D U ) the derived category o bounded complexes o Q-sheaves on U (resp. D, U ) with constructible cohomology sheaves. Associated with the deining equation there is a nearby cycle unctor : D U D D ([5]), o which we recall the deinition in the complex analytic setting: Let e : C C be the map e(ζ ) = exp(ζ ), setu := U e C and consider the ollowing diagram ẽ U U j U i D C C e C {0} A la mémoire de Jean Louis Verdier F. El Zein: UFR de Mathématiques, Equipe Géométrie et Dynamique, 2, Place Jussieu, Paris Cedex 05, France. elzein@math.jussieu.r D. T. Lê: The Abdus Salam International Centre or Theoretical Physics, Strada Costiera 11, Trieste, Italy. ledt@ictp.it L. Migliorini (B): Dipartimento di Matematica Università di Bologna, Piazza di Porta S. Donato 5, Bologna, Italy. migliori@dm.unibo.it Mathematics subject classiication (2000): 14D06, 14D07, 32S35, 32S40, 32G20 DOI: /s y

2 174 F. El Zein et al. The nearby cycle shea (K) D D is deined as: (K) := i R ẽ ẽ (K). Suppose rom now on that K D U is a perverse shea (with rational coeicients) with respect to the middle perversity t-structure (see [3], 4.0, and [10], 10.3). In view o [3], Corollaire , Rj K and j! K are perverse sheaves on U. It is known that (K)[ 1] is a perverse shea on D ([4], Théorème 1.2, [9], 6.13 and [10], Corollary ), endowed with the monodromy automorphism T. Since the category o perverse sheaves, being the core o a t-structure, is abelian ([3], Théorème 1.3.6), the usual linear algebra notions such as kernel, image, cokernel o a map, make perectly good sense. In particular, i T = T st u is the decomposition o T into its semisimple and unipotent part, the nilpotent endomorphism N := T u I, determines an increasing iltration, the monodromy weight iltration W (N ) o (K), uniquely characterized by the property that N W l (N ) W l 2 (N ) and N l : Grl W isomorphism or all l 0, (N ) Gr W l (N ) is an see [6], 1.6., where Grl W (N ) denotes the l-th graded object associated with the iltration. Explicitly, the iltration is given by: W j (N ) = i Im N i Ker N i + j + 1. (1.0.1) From the deinition o the intermediate extension unctor j! K := Im{ j! K Rj K},(see[3] Déinition ), it ollows that and Coker{ j! K Rj K} Coker{ j! K Rj K}=:Rj K/j! K i i! j! K, Ker{ j! K Rj K} Ker{ j! K j! K} i i j! K[ 1]. Let p τ 0, p τ 0 be the truncation unctors ([3] Proposition 1.3.3) associated with the middle perversity t-structure on D U. Then p H 0 := p τ p 0 τ 0 is a cohomological unctor rom D U to the abelian category P U o perverse sheaves on U([3] Théorème 1.3.6), in particular a distinguished triangle B C A in D U givesrisetothelong exact sequence o perverse cohomology sheaves... p H a (A) p H a (B) p H a (C) p H a + 1 (A)...,

3 A topological construction o the weight iltration 175 where p H a := p H 0 [a]. Taking the long exact sequences associated with the two distinguished triangles, (see or instance [4] Prop.1.1), we ind T I i (K) i (K) i i Rj K Rj K j! K Rj K/j! K p H 0 (i i Rj K) i Coker{ (K)[ 1] T I (K)[ 1]}. (1.0.2) The iltration W (N ) induces a iltration W l (Coker N ) on Coker N Rj K/j! K. It ollows rom (1.0.1) that ( W l (Coker N ) = Ker N l Im N )/ImN and W l (CokerN ) ={0} i l < 0. (1.0.3) We thus obtain a unique iltration W (Rj K) on the perverse shea Rj K satisying W l (Rj K) = 0il 1, W 0 (Rj K) = j! K and Grl W (Rj K) = Grl 1 W (CokerN ) i l > 0. Similarly, W (N ) induces a iltration W l (KerN ) on Ker N, given by ( ) W l (Ker N ) = Ker N Im N l i l 1, and W l (Ker N ) = Ker N i l 0. (1.0.4) Using the dual isomorphisms Ker{ j! K j! K} Ker{ j! K Rj K} p H 1 (i i Rj K) Ker{ (K)[ 1] T I (K)[ 1]}, we obtain a iltration W ( j! K) on j! K such that W l ( j! K) = j! K or l 0 and Gr W 0 ( j!k) j! K. Remark In the ormulæ above, one can use the smaller unipotent nearby cycle u (K), deined by the condition that u (K)[ 1] is the biggest perverse subshea o (K)[ 1] on which T acts unipotently. In act, there is a direct sum decomposition o perverse sheaves (K)[ 1] = u (K)[ 1] nu (K)[ 1],

4 176 F. El Zein et al. where nu (K)[ 1] is the non-unipotent part o nu (K)[ 1]. Since the restriction o T I to nu (K) is an isomorphism, Cone{ (K) T I (K)} =Cone{ u (K) T I u (K)}. Notice in particular that i u (K) = 0, then j!k j! K Rj K. For an useul and elegant construction o u (K) done entirely within the category o perverse sheaves, see and [2], 4.2. Remark From the point o view o Hodge Theory, it is the endomorphism N := log T u, rather than N, which would be natural to consider. However, since log T u = (T u I )Q, with Q invertible and commuting with T u, the two corresponding iltrations coincide. Example Let L be a unipotent local system o rank r on U := C U := C. LetV denote the stalk o L at a base-point, let T : V V be the corresponding monodromy, and set N := T I. I (z) = z : C C is the identity map, then (L) is isomorphic to V 0, the constant shea with stalk V at 0 C shited in degree 1, and the monodromy action on it is exactly given by T. We have the isomorphism j! (L) (τ 0 Rj L) j L, the non-derived direct image, rom which we obtain: Ker{ j! L j! L} i i j L i ((Ker N) 0 ), where (Ker N) 0 denotes the shea concentrated at 0 C with stalk Ker N. The iltration on j! L is thus given by: j! L i l 0 W l ( j! L) = (Ker N) 0 i l = 1 ( Ker N ImN l 1 ) (1.0.5) i l < 1. 0 Example Let U = C 2 and D = D 1 D 2 ={(z,w) C 2, s.t. zw = 0}, and consider the perverse shea Q U [2]. Wehave j! Q U [2] Q U [2], and Ker{ j! Q U [2] j! Q U [2]} i i j! Q U =i Q D.I ν : D := D 1 D2 D = D 1 D 2 denotes the normalization map, we have IC D ν Q D [2]. The iltration o i Q D coming rom the isomorphism i Q D Ker T I, deines a iltration W ( j! Q U [2]) o j! Q U [2] which is easily seen to satisy Gr W 0 ( j!q U [2]) IC D and Gr W 1 ( j!q U [2]) Q 0. This example can be generalized replacing the constant shea Q U with a local system L on U, to give a iltration o the perverse shea j! L. The determination o this iltration and o its graded objects, though, becomes substantially more complicated, see [8].

5 A topological construction o the weight iltration Statement o the main result The main drawback o the nearby cycle unctor is its dependence on the deining unction and not only on the eective divisor D, as the ollowing example shows (see Remark or an explanation o this act): Example Let X = {(z,w) C 2 such that w = 0} and Y = {z = 0}. Consider (z,w) = z and g(z,w) = z/w. IL is a non trivial rank r local system on C, deine K = L[2]. Then (K) is a (shited) trivial local system, i.e. it is isomorphic to the constant shea Q r D [2], while g(k) is not. In the l-adic set-up, Gabber proved that i K is pure, the iltration W (N ) o the mixed perverse shea (K) equals, up to a renumbering, the iltration by weights o Frobenius (see [2], Theorem 5.1.2). When K is mixed, the relation between the weight iltration o (K) and its monodromy iltration is more complicated, and involves the so called relative iltration, introduced in [6], Proposition The corresponding theorems or mixed Hodge modules are due to Saito [11]. These results imply in particular that the monodromy iltration does not depend on the local equation o D. The aim o this note is to provide a topological construction o the iltration which avoids the choice o deining equations o D, thus giving another proo that the monodromy iltration does not depend on such choice. In particular, this allows us to extend the deinition o the iltration on Rj K and j! K rom the case o a principal divisor to that o a Cartier divisor, namely we prove: Theorem Let Y be an eective Cartier divisor in a complex algebraic variety X, and let K be a perverse shea on X \ Y. There exists a iltration W (Coker) o Coker{ j! K Rj K} by perverse subsheaves with the ollowing property: For every Zariski open subset U X over which Y U is a principal divisor, and or every local deining equation Ɣ(U, O U ) or Y U, the restriction to U o W 1 (Coker) on Coker{ j! K Rj K} U coincides with the iltration induced by W (CokerN ) via the isomorphism (1.0.2). A similar statement holds or Ker{ j! K Rj K}. Theorem aords the ollowing deinition: Deinition Let q : Rj K Rj K/j! K be the canonical map. The iltration W (Rj K), o Rj K, deined by W 0 (Rj K) = j! K, and W l (Rj K) = q 1 (W l 1 (Coker)) is called the weight iltration o K. Dually, the iltration W (Ker) deines a unique iltration W ( j! K) o j! K, such that Gr W 0 ( j!k) j! K, and W l ( j! K) = W 0 ( j! K) = j! K or l > 0. Remark In terms o the Deinition 2.0.7, the natural map j! K Rj K has W <0 ( j! K) as kernel, and W >0 (Rj K) as cokernel. Example Let us go back to the Example 2.0.5, with L a unipotent rank r local system on C as considered in Example Then (K) is isomorphic to the constant shea Q r [2] on D, and the monodromy action on it is exactly

6 178 F. El Zein et al. given by T. We have the isomorphisms j! K := j! L[2] = j! L[2] and j! K = j! ( L[2]) j L[2], rom which we obtain: Ker{ j! K j! K } i ((Ker N) 0 ), and the iltration o this kernel is just the pull-back by o the one deined in (1.0.5). Theorem implies that we obtain the same iltration i we consider g. Example Consider the iltration o j! Q U [2] deined in Example By Theorem 2.0.6, this iltration can be deined globally on j! Q X\Y,iX is a nonsingular projective surace and Y X is a normal crossing divisor. O course in this case, taking cohomology, we just get the weight iltration or the Mixed Hodge structure on the relative cohomology groups H (X, Y, Q). Remark More generally, i Y is a normal crossing divisor in a nonsingular variety X, and L is a local system underlying a polarized variation o Hodge structures, the above iltration can be eectively constructed together with a Hodge iltration and it can be shown that the weight iltration coincides, up to a shit, with the one constructed here. In the general case the construction o the iltration can be reduced to the normal crossing case by Hironaka s desingularization, see [7,8]. Moreover, in this case, the iltration has the ollowing properties: 1. The weight iltrations deined by two Cartier divisors with the same support coincide. 2. The spectral sequence associated with the weight iltration degenerates at E 2. The proo o Theorem relies on the specialization unctor, which was introduced in [13] by Verdier. We recall its deinition and main properties in 4.In this paper, given an algebraic variety Z, we denote by D Z the derived category o bounded complexes o sheaves on Z with coeicients in Q and with constructible cohomology sheaves. 3. The deormation to the normal cone Let Bl Y {0} (X C) X C be the blow up o X C along Y {0}. The ibre π 1 (0) o the composition π : Bl Y {0} (X C) X C p 2 C is the union o the blow-up Bl Y X o X along Y and the projective completion P(C Y X O Y ) o the normal cone C Y X o Y in X, glued identiying the ininity locus in P(C Y X O Y ) with the exceptional divisor in Bl Y X. We set We have (see [12]): M := Bl Y {0} (X C) \ Bl Y X, π := π M. 1. The map π is lat, 2. π 1 (C ) = X C, 3. π 1 (0) = C Y X, 4. there is a closed embedding Y C M.

7 A topological construction o the weight iltration 179 We also set M := M \ (Y C), and C Y X := M π 1 (0) = C Y X \ Y. The map p : C Y X Y is a principal C -ibration. Remark Since the shea o ideals I Y deining Y is invertible, the normal cone C Y X is just the total space o the line bundle (I Y /IY 2), and CY X is the associated principal C -bundle. A deining equation or Y on a Zariski open set U deines a trivialization s : Y U p 1 (Y U) CY X. The ollowing diagram summarize the various spaces introduced, together with some maps which will be used in the next section: X \ Y j X i Y p (X \ Y ) C j ˆp M i p CY X j id X C M C Y X p 2 C π C {0} 4. Specialization Associated with the unction π : M C deined above is the nearby cycle unctor π : D M\CY X D CY X. The specialization unctor Y : D X D CY X is deined by Y (K) := π (p 1 K), where p 1 is the projection X C X. Remark I K is deined only on X \ Y, we deine Y (K) := Y ( j! K).The restriction to C Y X o Y (K) does not depend on the choice o the extension o K to X. We will denote the corresponding unctor D X\Y D C Y X by Y. A complex K D CY X is said to be monodromic i its cohomology sheaves H i (K) restrict to local systems on the ibres o p. The specialization unctor has the ollowing properties, see [13]: 1. Y (K) is monodromic, hence it is endowed with a monodromy automorphism T : Y (K) Y (K). 2. Y (K) is t-exact, in particular, i K is perverse, then Y (K) is perverse. 3. Let be a deining equation or Y on a Zariski open set U. Set Y := Y U, and let s be the trivialization o the normal bundle deined in Sect. 3. There is a natural isomorphism in D Y : s ( Y (K) p 1 (Y U)) (K U ). (4.0.6)

8 180 F. El Zein et al. The ollowing generalization o the Wang exact sequence will be needed in the next section: Lemma There is a distinguished triangle in D Y : s Y (K) s (T I ) (Rp Y (K)) Y s Y (K). Proo. Since s deines a local trivialisation o CY X Y, the exactness can be checked ibre by ibre. As Y (K) is monodromic, we are reduced to checking that, or a local system L on C with monodromy T, RƔ(C, L) L s (y) T I L s (y) (4.0.7) is a distinguished triangle. 5. The weight iltration From the deinition o Y (K), it ollows that there is a map i Rj p K Y (K). Looking at the diagram at the end o Sect. 3 we have a base-change map and, by adjunction, p i Rj K = i ˆp Rj K i Rj p K, i Rj K Rp i Rj p K. Proposition The map I : i Rj K Rp Y (K) given by the composition i Rj K Rp p i Rj K Rp i Rj p K Rp Y (K) is an isomorphism. Proo. It is enough to check the statement locally. Thus we pick a local deining equation on an open set U. By Lemma , and the isomorphism (4.0.6) in Sect. 4, we have a map o distinguished triangles in D Y : i Rj K Y (K U ) T I (K U ) I Y (Rp Y (K)) Y s Y (K) s (T I ) s Y (K) Since the two morphisms are isomorphisms, the third must be an isomorphism as well.

9 A topological construction o the weight iltration 181 Since Y (K) is perverse on C Y X, and p is a C - ibration, it ollows that p H i (Rp Y (K)) = 0ori = 1, 0. The perverse truncation triangle thus becomes p H 1 (Rp Y (K)) Rp Y (K) p H 0 (Rp Y (K)) The ollowing lemma ollows immediately rom the long exact sequence o perverse cohomology and the act that p is a C -ibration: Lemma The unctor, rom the category o perverse sheaves on C Y Xtothe category o perverse sheaves on Y, deined by K p H 1 (Rp K), is let exact, in particular it sends monomorphisms to monomorphisms. Similarly, the unctor K p H 0 (Rp K) is right exact and it sends epimorphisms to epimorphisms. Proo o Theorem The iltration W (N) on Y (K) associated with the monodromy operator N := T u I induces, by , the iltration p H 1 (Rp W (N)) on p H 1 (Rp Y (K)) I p H 1 (i Rj K) Ker :{j! K j! K} and, dually, the kernels o the epimorphisms p H 0 (Rp Y (K)) p H 0 (Rp ( Y (K)/W (N))) give a iltration on p H 0 (Rp Y (K)) I p H 0 (i Rj K) Coker :{j! K Rj K}. The distinguished triangles in the proo o give isomorphisms p H 1 I Y (Rp Y (K)) Y p H 1 (i Rj K) Y Ker { N : (K U )[ 1] (K U )[ 1] } p H 0 (Rp Y (K)) Y I Y p H 0 (i Rj K) Y Coker { N : (K U )[ 1] (K U )[ 1] }. showing that this iltration coincides, on the open set U, with the one induced by the monodromy weight iltration o (K U ). Remark Just as in the case o principal divisors, see Remark 1.0.1, one deines the unipotent specialization Y u (K). Notice that the map Rp Y u (K) Rp Y (K) is an isomorphism. This can be checked on a ibre, where it boils down to the analogous act or the cohomology o a local system on C, immediate rom (4.0.7). Acknowledgments. This paper grew rom a series o discussions while the irst-named and the third-named authors were guests o the ICTP Abdus Salam in Trieste, which they thank or support, hospitality and excellent working conditions. The authors thank the reeree or pointing out several inaccuracies and suggesting improvements in the exposition.

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