LOGARITMIC COMPARISON WIT SMOOT BOUNARY IISOR IN MIXE OGE MOULES arxiv:1710.07407v1 [math.ag] 20 Oct 2017 CUANAO WEI 1. Introduction The main goal of this paper is to use filtered log--modules to represent the dual localization of Saito s Mixed odge Modules along a smooth hypersurface, and show that they also behave well under the direct image functor and the dual functor in the derived category of filtered log--modules. We will apply the results of this paper to prove a generalization of the results of M. Popa and C. Schnell [PS14] in the log setting. This generalization will appear in [Wei17]. We first generalize the logarithmic comparison theorem of Grothendieck-eligne into the language of Mixed odge Modules: Theorem. Let M be a right X -module that represents a Mixed odge Module on a smooth variety X. Let be a normal crossing divisor on X. Assuming that M is of normal crossing type respect to a normal crossing divisor which contains, then we have Sp X, 0 M[ ] Sp X M[ ] Sp X, <0 M[!] Sp X M[!] in G CX, the derived category of graded C X -modules. Note that when we are taking the <0 piece, it only depends on M X\. In particular, <0M[!] = <0M. Further, if M X\ is an admissible variation of mixed odge structures, after a right-to-left side changing functor, replacing the Spencer functor by the de Rham functor, and forgetting the filtration, this is just the classical logarithmic comparison theorem: R X, Ri R X,resp. Ri! R X, where i : X \ X is the natural embedding,, is eligne s canonical extension that admits a logarithmic connection with and Re{eigenvalues of Res i } 1,0], resp. Re{eigenvalues of Res i } [ 1,0, R X, := [0 Ω 1 X log... 1... Ω dx X log 0].
2 CUANAO WEI is the log-de Rham complex of. We refer to [Wu16] for a survey on this topic. See also [Sai90, 3.] for an exposition of the extensions in terms of Mixed odge Modules of normal crossing type. One of the main theorems in this paper that will immediately imply the theorem above is: Theorem. Let be a normal crossing divisor on a smooth variety X. Given two sub-divisor S and a Mixed odge Module M of normal crossing type respect to, a normal crossing divisor that contains, then we have the following two quasi-isomorphisms in G X,S : 0 M[ ] L X, X,S S 0 M[ ]. <0M[!] L X, X,S S <0 M[!]. One observation is that, if we assume is smooth, then the previous theorem holds for any Mixed odge module on X. One big advantage of thinking of Mixed odge Modules is due to the fact that they behave well under the direct image functor of projective morphsims. Similarly, the log- -modules we considered above also behave well under direct image functors in the following sense: Theorem. Fix a projective morphism of log-smooth pairs f : X, X Y, Y and a Mixed odge Module M on X. Assume Y is smooth. If X is not smooth, we further assume that M is of normal crossing type respect to a normal crossing divisor that contains X. Then we have i f # 0 X M [ X] = 0 Y i f # <0 X M [! X] = <0 Y i f + M [ Y ]. i f + M [! Y ]. In particular, the funtor f # is strict on X 0 M[ X ] and X <0 M[!X ]. We also show that such log- -modules behave well under the dual functor: Theorem. Let M be a Mixed odge Module on X, be a smooth divisor on X. Let M be the dual Mixed odge Module of M. We have X, 0 M[ ] <0 M [!] X, <0 M[!] 0 M [ ] In particular, we have that both X, 0 M[ ] and X, <0 M[!] are strict. All the results above can be stated without the odge filtration. owever, the existence of the odge filtration and its compatibility with the Kashiwara- Malgrange filtration is essential in the proof of the theorems. By keeping track the odge filtration and the strictness properties that we obtain, we can also get some explicit formulae for the associated graded modules: Corollary 16, Corollary 18, which can be very useful in some geometric applications. We first fix the notations in Section 2, 3, 4. They will be stated in the general setting of log--modules. In Section 5, we prove the comparison theorem. The theorems about the direct image functor and dual functor will be proved in Section 6 and 7 respectively.
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 3 We will consider right log--modules in this paper if not otherwise specified, since the direct image functor is easier to be explicitly written down in the right module setting. The Mixed odge Modules that we are discussing here are all assumed to be algebraic. In particular, they are extendable [Sai90, 4]. We use strict right -module to represent a Mixed odge Module, forgetting the weight filtration. All algebraic varieties that we work with in this paper are smooth and over the complex number field C. Acknowledgment. The author would like to express his gratitude to his adviser Christopher acon for suggesting this topic and useful discussions. The author thanks Linquan Ma, Christian Schnell, Lei Wu and Ziwen Zhu for answering his questions. uring the preparation of this paper, the author was partially supported by MS-1300750 and a grant from the Simons Foundation, Award Number 256202. 2. Preliminaries In this section, we first define the log--module corresponding to a log-smooth pair. Then we recall the notions appeared in [SS16, Appendix A.], but in the log -modules setting. We will also define some basic functors: the Spencer functor, the pushforward functor and the dual functor, on the derived category of log- modules. If we set the boundary divisor = 0, we will get the same functors on the derived category of modules. Given a log-smooth pair X,, i.e., a smooth variety X with a normal crossing divisor on X, with = 1 +... + n being its decomposition of irreducible components. enote dimx = d X. Working locally on X, we can assume that X is just a polydisc dx in C dx. Let x 1,...,x dx be a local analytic coordinate system on X and is the simply normal crossing SNC divisor that defined by y := x 1... x n = 0. Let 1,..., n be the dual basis of dx 1,...,dx dx. We define T X,, the sheaf of the log-tangent bundle on X,, to be the locally free sheaf that is locally generated over O X by {x 1 1,...,x n n, n+1,..., dx }. T X, is naturally a sub-sheaf of T X, the sheaf of the tangent bundle over X. Similarly, we define X,, the sheaf of logarithmic differential operators on X,, to be the sub-o X -algebra of X that is locally generated by {x 1 1,...,x n n, n+1,..., dx }. If = 0, X, = X. There is a natural filtration F on X given by the degree of the differential operators. We denote X = R F X := p F p X, the Rees algebra induced by X,F. For any differential operator t F 1 X, we denote := t z R F X. enote by MG X, the category of graded X modules whose morphisms are graded morphisms of degree zero, and we call it the category of associated Rees modules of the category of filtered X -modules MF X. Note that MG X is an abelian category. See [SS16, efinition A.2.3] for details. ence, we can define the derived category of MG X, and we denote it by G X. We also use
4 CUANAO WEI G X, {,+,b} to denote the bounded above, bounded below, bounded condition respectively on the derived category. Recall that, we say M MG X is a coherent X -module if M is locally finitely presented, i.e. if for any x X, there exists an open neighborhood U x of x and an exact sequence: q X Ux p X Ux M Ux. We denote by MG coh X, the full subcategory MG X with the objects that are coherent. Note that we know MG coh X is also an abelian category. We refer to [SS16, A.9. A.10.] for a discussion on this topic. Fix M MG X, we can write M p, the p-th graded piece as M p = F p M z p, where F p M is an O X -module. If M is a coherent X -module, then F p M is a coherent O X -module. Similarly, we have a natural filtration F on X,, and denote X, = R F X, := p F p X,. We similarly use MG coh X,, G X, to denote the corresponding categories as above. Let ÕX bethesheafgivenbythegradedring O X [z] = R F O X, wherethefiltration F on O X is the trivial one. We denote C X = C X [z] as a graded sub-ring of Õ X. Further, given a coherent O X -module L, we denote L = L[z], the induced graded Õ X -module. From now on, we will omit the word graded as long as it is clear from the context. As in [SS16, A.5.] and [CM99, 3.1], we define the logarithmic Spencer complex on a X, -module M, by Sp X, M := {0 M n TX,... M T X, M 0}, with the C X -linear differential map locally given by k m ξ 1... ξ k 1 i 1 mξ i ξ 1... ˆξ i... ξ k + i=1 i 1 m [ξ i,ξ j ] ξ 1... ˆξ i... ˆξ j... ξ k, i<j whereξ 1,...,ξ dx isalocalbasisoft X, z. Wehavethefollowing[CM99,Theorem 3.1.2.] Theorem 1. Sp X, X, is a resolution of Õ X as left X, -modules. Proof. We only need to show it respect to Gr F Sp X, X,, the complex of its associated graded pieces. owever, it is straight forward to check that it is just K ξ 1,...,ξ dx ;Gr F X,,
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 5 the Koszul complex of Gr F X,, with ξ 1,...,ξ dx Gr F 1 X,. It is obvious that ξ 1,...,ξ dx is a regular sequence and generates Gr F X, over Gr F Õ X. ence Gr F Sp X, X, Gr F Õ X. ence, we can view the logarithmic Spencer complex in the following way. We have the right exact functor: X, Õ X : MG X, MG CX, by M M X, Õ X. We can define its left derived functor Sp X, = L X, Õ X : G X, by Sp X, M = M L X, Õ X. G CX, Note that each term of Sp X, X, is locally free over X,. ence due to Theorem 1, we have Sp X, M M L X, Õ X To relieve the burden of notation, in the rest of the paper, means Õ or O depending on the context if not otherwise specified. As in the X case [Sai88, 2.4.1. 2.4.2] [SS16, Exercise A.3.9], there are two right X, -module structures on M X, which are defined by : { m P triv f = m Pf right triv : m P triv ξ = m Pξ { m P tens f = mf P right tens : m P tens ξ = mξ P m ξp. We call the first one the trivial right X, -module structure, and the second one the right X, -module structure induced by tensor product. Proposition 2. There is a unique involution τ : M X, M X, which exchanges both structures and is the identity on M 1. It is given by m P mp 1 m P. Since it can be checked directly, we omit the proof here. See [Sai88, 2.4.2] for details. Consider the identity functor: ÕX : G X, G X,.
6 CUANAO WEI Replacing ÕX by Sp X, X,, we get that, for any M MG X,, M Sp X, M X, triv {0 M X, n TX,... triv M X, T X, M X, 0}, triv triv which is a resolution of M by right X, -modules where X, acts on M X, triv i TX, using the induced tensor X, -module structure: m P tf =mf P t, m P tq =mq P t m QP t, for any function f and differential operator P. Apply the involution in 2 onto the previous Spencer complex, we get that M Sp X, M 1 X, tens {0 M X, tens n TX,... M X, T X, M X, 0}. tens tens It is also a resolution of M by right X, -modules where X, acts on M X, using the trivial X, -module structure: tens i TX, m P tf = m Pf t, m P tq = m PQ t. Now we start to define the pushforward functor. Given a morphism of logsmooth pairs f : X, X Y, Y, we can similarly define MG f 1 Y,Y, the category of f 1 Y, Y -modules, which is an abelian category. ence we can define the corresponding derived category G f 1 Y, Y. We denote X, X Y, Y = ÕX f 1 Õ Y f 1 Y, Y, which is a left X, X, right f 1 Y, Y -module. We define the relative logarithmic Spencer functor: Sp X,X Y, Y : G X, X G f 1 Y, Y, by Sp X, X Y, M = M Y L X,X X, X Y, Y. The topological direct image gives a functor f : MG f 1 Y,Y MG Y,Y.
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 7 Since MG f 1 Y, Y is an abelian category with enough injectives, we can define the right derived functor of f : Rf : + G f 1 Y, Y G + Y, Y We define the pushforward functor f # : G + X,X G + Y,Y being the composition of the following two functors: G + SpX,X Y, Y X,X + G f 1 Y,Y Rf + G Y,Y Putting them together, we have 2 f # M = Rf M ÕX L X,X f 1ÕY f 1 Y, Y. If M is an object in MG coh X,, then each cohomology of f # M is an object in MG coh X,, [SS16, Theorem A.10.26]. Note that if both X and Y are trivial, we have f # = f +, where f + is the pushforward functor on the derived category of -modules. In [SS16], they use the notation f instead of f +. Proposition 3. Let f : X, X Y, Y and g : Y, Y Z, Z be two morphisms of log-smooth pairs, and assume that f is proper, we have a functorial canonical isomorphism of functors g f # g # f #. Proof. Since X,X is locally free over ÕX, we have X, X Y, Y L f 1 Y,Y =ÕX f 1 Õ f 1 Y,Y Y = ÕX L f 1 Y, f 1ÕY Y ÕX L g f 1ÕZ g f 1 Z,Z = X, X Z, Z. f 1 Y, Y Z, Z L f 1 Y,Y L f 1 Y,Y f 1 Õ Y g 1 Õ Z g 1 Z,Z f 1 Õ Y L g f 1 Z, g f 1ÕZ Z The equalities above are as left X, X, right g f 1 Z, Z bi-modules. Now by the definition 2 and the projection formula, we can conclude the proof. We refer to [SS16, Theorem A.8.11. Remark A.8.12.] for a more detailed explanation on the composition of direct images. Recall that we define the dual functor [Sai88, 2.4.3] X : G X + G X, by X M = Rom X M, ω X [d X ] X, tens where ω X [d X ] means shifting the sheaf ω X to the cohomology degree d X.
8 CUANAO WEI by Similarly, we define the dual functor X, X : G X, X G + X, X, 3 X, X M = Rom X,X M, ω X [d X ] X, X, tens We use the induced tensor right X, X -module structure on ω X X, X when we take the Rom X,X M, functor, and we use the trivial right X, X -module structure on ω X X, X to give the right X, X -module structure on X, X M. 3. Strictness Condition In this section, we recall the definition of the strictness condition. We also recall the associated graded functor in this section. At the end of the section, we explicitly write down the formulae for the induced direct image functor and dual functor on the associated graded module, under certain strictness condition. They can be very useful in some geometric applications. Fix a filtered ring A,F. We alwaysassume the filtration F is exhaustive, which means i F i A = A, and F 0 = 0. enote à := R FA, the associated Rees algebra. We define the strictness condition on MGà by [SS16, efinition A.2.7.]: efinition 1. 1 An object of MGà is said to be strict if it has no C[z]-torsion, i.e., comes from a filtered A-module. 2 A morphism in MGà is said to be strict if its kernel and cokernel are strict. Note that the composition of two strict morphisms need not be strict. 3 A complex M of MGà is said to be strict if each of its cohomology modules isastrictobject ofmgã. ence wecan usethe same definition forthe strictness of M GÃ. We have a right exact functor Gr F : Gr MGà MG F Ã, defined by Gr F M = M C C/z C. It naturally induces a derived functor: Gr F : Gà G Gr F Ã, given by Gr F M = M L C C/z C. In particular, if we assume that M MGà is strict, which means M is the associated Rees module of a filtered A-module, we have Gr F M Gr F M. ence, it is not hard to see that taking the Gr F functor on M is the same as taking the associated graded module.
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 9 Now given M GÃ, we have a spectral sequence E p,q 2 = p Gr F q M p+q Gr F M. IfM is strict, we havethe spectral sequence degeneratesat the E 2 page, which implies the commutativity of taking cohomology and Gr F functor under the strictness condition: Gr F i M = i Gr F M. In the case that à = X,, we have We denote A X, := Gr F X, = SymT X,. T X, = Spec A X,, the space of log-cotangent bundle over X,. ence, we have a canonical functor Gr F : G X, G A X,, We denote by GM, the object in the derived category of O T -modules that X, is induced by Gr F M. Proposition 4 Laumon s formula. Let f : X, X Y, Y be a morphism of log-smooth pairs. Let M MG Y, Y. Assume that both M and f # M are strict. Then we have i f #Gr F M := R i f Gr F M L A X,X f A Y,Y Gr F i f # M. Proof. enote N = M L X,X ÕX f 1ÕY f 1 Y,Y. By the associativity of tensor product and M being strict, we have Gr F N = Gr F M L A X,X f A Y, Y. Further, by the projection formula, we have Now we only need to show that R i f Gr F N i Gr F Rf N. i Gr F Rf N Gr F R i f N. It follows evidently by the commutativity of taking cohomology and Gr F functor under the strictness assumption on f # M Rf N. Proposition5. Fix a M MG X, and assume X, M MG X,, which means it has only one non-trivial cohomology at cohomology degree 0. We further assume that both M and X, M are strict. Then we have Gr F X, M Rom AX, Gr F M,ω X [d X ] OX A X,. Note that the sections of A X,,k := Sym k T X, act on the right-hand with an extra factor of 1 k. It is due to that we are using the induced tenser log- module
10 CUANAO WEI structure on the right-hand side. If we consider both sides as O T -complexes, X, we get G X, M 1 T X, Rom OT X, GM,p Xω X [d X ] O T X, Proof. Since M is strict, we have Rom X, M, ω X [d X ] C/z C X, L C tens Rom AX, M C C/z C,ω X [d X ] OX A X,, which can be checked locally by taking a resolution of M by free X, -modules. Now the statement is clear by the strictness assumption on X, M. 4. Kashiwara-Malgrange filtration on coherent X -modules In this section, we recall some basic properties of R-indexed Kashiwara-Malgrange filtration on a coherent strict -module respect to a smooth divisor from [Sai88, 3.1]. Then we define muilti-indexed rational Kashiwara-Malgrange filtration respect to a simply normal crossing divisor when the -module is of normal crossing type, and recall some properties that we will need in later sections. We say that an R-indexed increasing filtration is indexed by A + Z, where A is a finite subset of [ 1,0, if gr a := a/ a<0 = 0 if and only if a / A + Z. All R-indexed filtrations in this paper are indexed by A + Z for some finite set A [ 1,0 Fix X be a smooth variety and be a smooth divisor on X. Let t be a local function that defines, and t be a local vector field satisfying [ t,t] = 1. efinition 2. Let M be a coherent strict X -module. We say that a rationally indexed increasing filtration on M is a Kashiwara-Malgrange filtration respect to, if 1 a R a M = M, and each filtered piece a M is a coherent X, -module. 2 a M i X a+i M for all a R,i Z, and a M t = a 1 M if a < 0. 3The action t t a over gr a M is nilpotent for any a R. Note that t 1 X, t 1 X, hence condition 3 implies 4t : gr a M gr a 1 M and t : gr a 1 M gr a Mz are bijective for a 0. 5 a+i M = am i X for a 0, i 0. Further, all the conditions above are independent from the choice of t and t. We know that given a Mixed odge Modue M, in particular being strictly R- specializable along, there exists a rational Kashiwara-Malgrange filtration respect to and it is unique. [Sai88, 3.1.2. Lemme], [SS16, 7.3.c]. Further, we know that every gr a M is a strict gr 0 X -module. [SS16, Proposition 7.3.26] Recallthatgivenacoherent X -modulemequippedwithakashiwara-malgrange filtration, we have a free resolution: Proposition 6. Fix M a strict X -module that is equipped with the Kashiwara- Malgrange filtration respect to a smooth divisor on X. Then locally over X.
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 11 we can find a resolution: where each M i, M, M,, is a direct sum of finite copies of X z p, [a], with 1 a 0,p Z. Further, the resolution is strict respect to, which means it is still a resolution after taking any filtered piece of. Further, if we have that the multiplication by t induces an isomorphism t : gr 0 M gr 1M, in particular, when M = M[ ], the above resolution can be achieved with 1 < a 0. If we have that applying x induces an isomorphism x : gr 1M gr 0 M z, in particular, when M = M[!], the above resolution can be achieved with 1 a < 0. Proof. Without the extra assumption, it is just [Sai88, 3.3.9. Lemme.], see also the proof of [Sai88, 3.3.17. Proposition.]. With the extra assumption, it is evident that we can also achieve the corresponding index interval in [Sai88, 3.3.9. Lemme.]. See [SS16, Proposition 9.3.4 and Proposition 9.4.2], for the corresponding isomorphism in the localization and dual localization case. Remark. Actually, the existence of the free resolution as above only depends on the strictness assumption 4 in efinition 2. See also [Sai88, 3.3.6.-3.3.9.]. Now we start to consider the case that = 1 +...+ n is a normal crossing divisor on X with irreducible components i. For any we denote a = [a 1,...,a n ] Q n, a X = i i a i X. For a = 0 := [0,...,0], 0 X is just X,. Weintroducethenotation X z p, [a],todenote X equippedwithshifted filtrations: X z p, [a] := b a X z p. b Given a strict coherent X -module that possesses the Kashiwara-Malgrange filtration i respecttoallcomponents i of,wedefineamulti-indexedkashiwara- Malgrange filtration respect to by a M = i a i M, for any a = [a 1,...,a n ] Q n. It is not hard to see that M is a multi-indexed graded module over X, in the sense that a M b X a+b M, for any a,b Q n. Given a subset S {1,...,n}, we denote S = i S i. For any such S, we denote a S M := i S a i i M.
12 CUANAO WEI Further, we say that b < a if b i < a i for all i. We denote <a M := b<a b M. In general, the n filtrations i do not behave well between each other, in the sense that when we look 1 as a filtration on 2 0 M, it does not have good strictness conditions as in efinition 2. owever, if M is a strict coherent X - module of normal crossing type respect to, then locally we have the following strictness relations [Sai90, 3.11. Proposition.], [SS16, Lemma 11.2.11.] 4 5 where i : gr i x i : a M a 1 im a i < 0 a i 1a i M gr i a i a i M z a i > 0, 1 i :=[0,...,0,1,0,...,0], with the 1 at the i-th position. Further, if we have the isomorphisms x i : i 0 M i 1M, for every i, in particular when M = M[ ], the identity 4 still holds when a i = 0 If we have i 1 M i + i i <0M z = 0 M z, in particular when M = M[!], the identity 5 still holds when a i = 0. Note that, to make the multi-indexed Kashiwara-Malgrange filtration satisfy 4 and 5, instead of requiring that M is of normal crossing type respect to, we only need to assume that M is of normal crossing type respect to, where is a normal crossing divisor that contains. One advantage of such assumtion is that it behaves well when we do induction on the number of components on the boundaries. 5. Comparison Theorem Fix a smooth variety X and a normal crossing divisor = 1 +...+ n on X as in the previous sections. We start with the following vanishing Lemma 7. Given a strict X, -module M. Let S = 1 and x 1 be a local function that defines 1. Assume that Gr F M is torsion free respect to x 1, then we have M i L X, X,S = 0, for i 0. Proof. Take the canonical left resolution of M as in 1: M Sp X, M X, tens ={0 M X, tens n TX,... M X, T X, M X, 0}. tens tens Recall that the X, -module structure on M X, tens i TX, is the trivial one. Since both X, and X,S are locally free over ÕX, we have
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 13 M X, tens TX, M X,S Sp X, M X,S tens TX, tens. L X, X,S Now we only need to show that Gr F Sp X, M X,S cohomology. owever, it is just K x 1 1,...,x n n, n+1,..., dx ;N, the Koszul complex of N := Gr F M X,S, with tens x 1 1,...,x n n, n+1,..., dx tens has no higher as the elements in Gr F 1 X,. More precisely, for any element m P N, we have the relations m Px 1 1 = mx 1 1 P mx 1 1 P, m Px i i = mx i i P m x i i P, for i = 2,...,n m P j = m j P m j P, for j = n+1,...,d X. Note that these elements act on N homogeneously. We know that 6 1,x 2 2,...,x n n, n+1,..., dx Gr F 1 X,S, is a regular sequence on N, which is due to M Sp X,S M X,S tens, or by an easy argument on the natural grading on Gr F X,S. Now we need to show that x 1 1,...,x n n, n+1,..., dx Gr F 1 X, is also a regular sequence on N. Since we have that x 2 2,...,x n n, n+1,..., dx Gr F 1 X,, is a regular sequence on N, it suffices to show that x 1 1 is torsion free on N := N/N x 2 2,...,x n n, n+1,..., dx. Note that due to the relations we have above, it is not hard to see that N Gr F M O X [ 1 ]. owever, by the assumption that x 1 is torsion-free on Gr F M, we can conclude the proof. For the rest of this section, we will consider M as a Mixed odge Module on X. Given a divisor with normal crossing support on X, we denote and M[ ] = i i 1 M, M[!] = i! i 1 M
14 CUANAO WEI being the localization and dual localization of M on X\ [Sai90, 2.11 Proposition], [SS16, Chapter 9]. Note that both M[ ] and M[!] are Mixed odge Modules, in particular, strict coherent X -modules. Theorem 8 Comparison Theorem with normal crossing boundary. Notations as above. Given two subset S I {1,...,n} and a Mixed odge Module M of normal crossing type respect to, a normal crossing divisor that contains, then we have the following two quasi-isomorphisms in G coh X,X : 7 8 I 0 M[ ] L X,I I <0 M[!] L X,I X,S S 0 M[ ]. X,S S <0 M[!]. Proof. We only show 7 here, 8 follows similarly. Since M[ ] = M[ ][ I ], we can assume I = without lose of generality. Further, by induction, we only need to show that case that S = 1, 9 0 M[ ] X, X,S S 0 M[ ]. ue the the strictness condition on the multi-indexed Kashiwara-Malgrange filtration4, x i istorsion-freeon Gr F 0 M[ ]foranylocalfunction x i thatdefines i, hence so is any filtered piece of M[ ]. Now apply the previous lemma, we get 0 M[ ] L X, X,S 0 M[ ] X, X,S Further, by 5, we have S 0 M[ ] 0 M[ ] X,S M[ ]. To show 9, now we only need to show that 10 0 M[ ] X, X,S = 0 M[ ] X,S. Note that we have a natural injection 0 M[ ] 0 M[ ] X, X,S. Similarly to the proof of [SS16, 9.3.4 7], we give an integer-indexed increasing filtration on N := 0 M[ ] X, X,S by: { 11 U 1 k N = 1 k M[ ] 0 M[ ], for k 0 k j=0 0 M[ ] j 1, for k 1. where k Z and 1 is a differential operator satisfying [x 1, 1 ] = 1, [x i, 1 ] = 0, for i 1, where x 1 is a local holomorphic function that defines 1. It is obvious that k U 1 k N = N. We give a similar filtration on S 0 M[ ] by: { U 1 k S 0 M[ ] 1 k M[ ] 0 M[ ], for k 0 = k j=0 0 M[ ] j 1, for k 1. We have that k U 1 k S 0 M[ ] = S 0 M[ ] due to 5. Consider the following commutative diagram,
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 15 Gr U 1 0 N Gr U 1 k k 1 N Gr U 1 0 S 0 M[ ] k 1 Gr U 1 k S 0 M[ ], where Gr U k := U k/u k 1. Since the upper horizontal map is surjective and the lower horizontal map is isomorphic by the construction of the filtrations, we get that all the arrows appearing in the commutative diagram are isomorphisms, which concludes 10, hence 9. Applying the Spencer functor on both sides, we can recover the classical logarithmic comparison theorem: Theorem 9. Let M be a right X -module that represents a Mixed odge Module on a smooth variety X. Let be a normal crossing divisor on X. Assuming that M is of normal crossing type respect to a normal crossing divisor which contains, then we have Sp X, 0 M[ ] Sp X M[ ] Sp X, <0 M[!] Sp X M[!] in G CX, the derived category of graded C X -modules. Proof. By the definition of the Spencer functor, we have Sp X, 0 M[ ] 0 M[ ] L X, Õ X 0 M[ ] L X, X L X Õ X M[ ] L X Õ X Sp X M[ ]. The third identity is due to the comparison theorem, by taking I =, and S = 0. The dual localization case follows similarly. When the boundary divisor := is smooth, the strictness conditions 4 and 5 are satisfied by efinition 2. ence we have Theorem 10 Comparison Theorem with smooth boundary. Given a Mixed odge Module M, we have the following two quasi-isomorphisms in G X : 0 M[ ] L X, X M[ ], <0 M[!] L X, X M[!]. Proof. It follows by a similar argument as we show 9 in the Comparison Theorem with normal crossing boundary.
16 CUANAO WEI Remark. Actually, it can also be proved by using the free resolution in Proposition 6. More precisely, for the M[ ] case, we can get a resolution with each term being a direct sum of finite copies of X z p, [a], with 1 < a 0,p Z. Note that in this case, we have X z p, [a] = X,, 0 hence by taking the 0 piece, we also get a free X, -module resolution. A similar argument will be used in proving Theorem 17. 6. Pushforward functor Fix M, a strict coherent X -module of normal crossing type on a log-smooth pair X,. Let be a divisor with its support contained in, and = r i := r 1 1 +...+r n n be the decomposition of reduced and irreducible components. Let i : X, Y, be the graph embedding given by the local function y = x r1 1...xrn n, where each x i is a local function that defines the divisor i on X. We first build a relation between the Kashiwara-Malgrange filtration on M and on i + M. Let 1,..., n, y Y that are mutually commutative and satisfying [ i,x i ] =[ y,y] = 1, [ i,x j ] =[ i,y] = [ y,x i ] = 0.for i j Let u = x r1 1...xrn n y, and consider x 1,...,x n,u being a local coordinate system on Y. We can get x1,..., xn, u Y that satisfying and we can further require that m δ xi =m xi δ, m δ u =m δ u, [ xi,x i ] =[ u,u] = 1, [ xi,x j ] =[ xi,u] = [ u,x i ] = 0,for i j. We can write i # M = k N M δ k u as a Y = X [u] u -module. By changing of coordinates, we have the following relations: m δ y k = m δ u, k m δ i = m xi δ mr i x r1 1...xrn n /x i δ u, m δy = mx r1 1...xrn n δ, m δõx = mõx δ, and with the usual commutation rules. Lemma 11. Notations as above, we can get the Kashiwara-Malgrange filtration on i + M from on M by: 12 a i + M = a rm δ Y,, for a < 0. For a 0, we can get it inductively by a i + M z = <ai + M z + a 1i + M y.
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 17 If we further have that x i : i 0 M i 1 M is an isomorphism for every component i, in particular when M = M[ ], we have 12 still holds when a = 0. Proof. The proof is similar to the proof in [SS16, Theorem 11.3.1]. owever, since our settings are a little bit different, we spell out the details here. We need to check the filtration we defined above satisfies the Condition 1, 2, 3 in efinition 2. We only check the case with the extra condition here. The other case follows similarly. For Condition 1, if a 0 and if m 1,...,m l generate a rm over X,X, then m 1 δ,...,m l δ generate a i +M over Y, Y, due to the fact that we have the relation mx i xi δ = m δ x i i +r i y y. ence, we can conclude that, for every a, a i + M is coherent over Y,Y. To show that a a i +M = i + M, we only need to show that Note that M δ a a i + M 0 M X = M, which is due to the assumption that M, hence M[ ] is of normal crossing type, hence the strictness condition 5. Then, we can use the relation m xi δ = m δ i +mr i x r1 1...xrn n /x i δ u, to get that a a i +M 0 M δ Y M δ. For Condition 2, we have that, for any a, a i + M y a 1 i +M, which can be checked by using the relation m δy = mx r1 1...xrn n δ. Further, the equality for a < 0 can also be deduced from the corresponding properties on M. We are left to show a 1 i + M y a i + M. When a > 0, it follows by definition. When a 0, for any m δ a i + M, considering the relation [SS16, 11.2.10] and our extra assumption 13, we obtain a M x i = a 1 im for a 0. ence, for any m δ a 1 i +M we can write m δ = mr i x r1 1...xrn n /x i δ, for some m a r 1 M. Now using the relation i m δ i = m xi δ mr i x r1 1...xrn n /x i δ u,
18 CUANAO WEI we obtain m δ y = m xi δ m δ i a i +M, which concludes the proof of the Condition 2. For Condition 3, it is straight forward to check that we have m δy y az = m 1ri 1 x i xi r i az δ x i m δ i. r i Using this relation and assuming a 0, we can get that, if r i M νri a ia x i xi r i az i <r iam, we have that a r M δ y y az νria a r ǫ im δ Y,, where ǫ i := [0,...,0,ǫ,0,...,0], where 0 < ǫ 1 is located at the i-th position. enote µ a = i ν r ia, do the computation above inductively, we obtain a r M δ µa y y az <a r M δ Y,. ence y y az is nilpotent on gr a i + M, for a 0. When a > 0, we can get the same conclusion by induction and using the relation y y y az = y y a 1z y. Lemma 12. Notations as above. Fix a strict X, -module N. Assume Gr F N is torsion-free respect to every x i, then we have i # N 0 i # N. Proof. Let i be the divisors on Y that are defined by the the local functions x i. enote := 1 +... + n. We can decompose i : X, Y, into j : X, Y, + and id : Y, + Y,. By 2, we have j # N j N ÕX L X, f 1 Õ Y f 1 Y, + owever, it is straight forward to check that the natural map is surjective. ence its dual dj : j Ω Y log + Ω X log j : T X, j T Y, has a locally free cokernal and we denote it by N. Then we have Õ X f 1 Õ Y f 1 Y, + X, Sym Ñ z. In particular, we have that ÕX f 1 Õ Y f 1 Y, + is locally free over X,, hence we obtain j # N 0 j # N. For id : Y, + Y part, by 2, we have id # j # N j # N L Y, + Y,.
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 19 owever, by the assumption that Gr F N is torsion-free respect to x i, hence so is Gr F j # N. By applying Lemma 7 indectively, we obtain id # j # N 0 id # j # N. Lemma 13. Notations as in Lemma 11. We have Further, if the morphism i # a r M a i +M for a < 0. 13 x i : i 0 M i 1 M is an isomorphism for every component i, e.g. M = M[ ], the second identity still holds when a = 0 Proof. Since Gr F a rm is x i torsion free for a < 0 resp. a 0 with the extra assumption, due to the strictness condition 4, applying the previous lemma, we get i # a r M 0 i # a r M, for a < 0 resp. a 0. We have and 0 i # a r M i a r M X, ÕX i 1ÕY i 1 Y,, i + M i M X ÕX i 1 Y Y. There is a natural morphism i a r M X, ÕX i 1 Y Y, i M X ÕX i 1 Y Y, which is an injection due the torsion free condition on Gr F a rm, and its image is exactly a r M δ Y,. We can conclude the proof by using Lemma 11. Fix a projective morphism of two smooth varieties f : X Y. Given X, Y two reduced divisors with normal crossing support on X, Y respectively, we say that f : X, X Y, Y is a projective morphism of log-smooth pairs if we further have that f 1 Y X. Proposition 14. Notations as above, assume that f 1 Y = X. Fix a Mixed odge Module M on X. Then we have i f + M [ X] = i f + M [ Y ] i f + M [! X] = i f + M [! Y ]. Proof. By [Sai90, 2.11. Proposition], we only need to check the identities above at the level of perverse sheaves, which is obvious by considering the following commutative diagram:
20 CUANAO WEI X \ X f i X X f i Y Y \ Y Y. Noticing that i X and i Y are affine and f and f are projective, we have Rf Ri X K =Ri Y Rf K Rf Ri X! K =Ri Y! Rf K, for any perverse sheaf K on X \ X. Theorem 15. Fix a projective morphism of log-smooth pairs f : X, X Y, Y and a Mixed odge Module M on X. Assume Y is smooth. If X is not smooth, we further assume that M is of normal crossing type respect to a divisor with normal crossing support that contains X. Then we have i f # 0 X M [ X] = 0 Y i f + M [ Y ]. i f # <0 X M[! X] = <0 Y i f + M [! Y ]. In particular, the direct image funtor f # is strict on X 0 M [ X] and X <0 M [! X]. Proof. We only show the proof of the first identity here. The second one follows similarly. Consider the morphism of log pairs induced by the identity map id : X, X X,f 1 Y, and its induced direct image of X 0 M[ X ]. We can forget the logarithmic structure on those components of X f 1 Y. More precisely, we have id # X 0 M[ X ] X 0 M[ X ] L X,X X,f 1 Y f 1 Y 0 M[ X ]. The second identity is due to the comparison theorem. ence, without loss of generality, we can further assume that X = f 1 Y. Consider the following commutative diagram X, X i S,S f Y, Y j T, T, g where i : X, X S, S is the graph embedding given by a local function f y, where y is a local function on Y that defines Y, and is the graph embedding given by y. j : Y, Y T, T, g : S, S T, T
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 21 is the naturally induced morphism, and it is straight forward to check that we have g T = S. By the previous lemma, we have [ i # 0 X M X ] 0 S i + M[ S ]. Now by [SS16, 7.8.5], [Sai88, 3.3.17] we have i [ g # 0 S i + M S ] 0 T i g i + M [ T]. ence we obtain 14 g i # X 0 M [ X ] T 0 i g i + M [ T]. Note that since both Y and T are smooth and j T = Y, it is evident that the natural map dj : j Ω T log T Ω Y log Y is surjective. ence as in Lemma 12, we have that Y,Y T, T is a flat left Y, Y -module. ence j # i f # 0 X M [ X] 0 j # i f # 0 X M [ X]. By the degeneration of Leray Spectral sequence and 14, we obtain j # i f # 0 X M [ X] 0 T i j f + M [ T]. Apply the previous lemma again, we have j # i f # 0 X M [ X] j # 0 Y i f + M [ Y]. By the construction of j, it is not hard to see that j # is a fully faithful functor, see also the remark below, hence we can conclude that i f # 0 X M [ X] 0 Y i f + M [ Y]. In particular, it is a strict Y, Y -module. Remark. Note that to deduce j # being fully faithfully, we need to use the assumption that Y is smooth, hence by a change of coordinates, we have Y,Y T, T Y,Y ÕY[ u ]. Now the fully faithfulness is evident by a local computation. If Y is not smooth, Y,Y T, T is not flat as a left Y,Y -module in general. See also the computation in Lemma 12. Consider the associated graded complex, by Proposition 4, we obtain Corollary 16. Notaions as in Theorem 15 and Proposition 4, we have i f # Gr F 0 X M [ X] Gr F 0 Y i f + M [ Y ], Gr F <0 X M [! X] Gr F <0 Y i f + M [! Y ]. i f #
22 CUANAO WEI 7. ual Functor Theorem 17. Let M be a Mixed odge Module on X, be a smooth divisor on X. Let M be the dual Mixed odge Module of M: M := Rom X M, ω[d X ] X. We have X, 0 M[ ] <0M [!] X, <0 M[!] 0 M [ ] In particular, we have that both X, 0 M[ ] and X, <0 M[!] are strict. Proof. We only prove the first equation here, the second one follows similarly. Apply the resolution in Proposition 6 on M[ ], locally we have M, M[ ],, is a direct sum of finite copies of X z p, [a], with where each M i, 1 < a 0,p Z. Now we only need to show that 15 Rom X, 0 M, = <0Rom X M,,ωX [d X ] X,,ωX [d X ] X owever, for 1 < a 0, we have X z p, [a] = X, z p. ence Rom X, 0 0 X z p, [a],ω X [d X ] X, = ω X [d X ] X, z p. On the other hand, to get the Kashiwara-Malgrange filtration on M, we set Rom X X z p, [a],ω X [d X ] X = ω X [d X ] X z p, [ 1 a], which gives a filtration on the complex Rom X M,,ωX [d X ] X. This filtration is actually strict and induces the Kashiwara-Malgrange filtration on M by [Sai88, 5.1.13. Lemme.]. Note that Saito only show the Pure odge Module case there, but the proof still works in the Mixed odge Module case by induction on the weights. ence for 1 < a 0, we have <0 Rom X X z p, [a],ω X [d X ] X = ω X [d X ] X, z p. Now 15 is clear. Pass to the associated graded pieces, by Proposition 5, we obtain
LOG-COMPARISON WIT SMOOT BOUNARY IISOR IN MM 23 Corollary 18. Notations as in the previous theorem, we have Gr F 0 M [ ] Rom AX, Gr F <0M[!],ω X [d X ] OX A X,, Gr F <0M [!] Rom AX, Gr F 0 M[ ],ω X [d X ] OX A X,. If we consider both sides as graded O T -complexes on T X, X,, we get G 0 M [ ] 1 T Rom OT G X, X, <0M[!],p Xω X [d X ] O T X, G <0M [!] 1 T X, Rom OT X, G 0 M[ ],p Xω X [d X ] O T X, References [CM99] Francisco J. Calderón-Moreno. Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor. Ann. Sci. École Norm. Sup. 4, 325:701 714, 1999. [PS14] Mihnea Popa and Christian Schnell. Kodaira dimension and zeros of holomorphic oneforms. Ann. of Math. 2, 1793:1109 1120, 2014. [Sai88] Morihiko Saito. Modules de odge polarisables. Publ. Res. Inst. Math. Sci., 246:849 995 1989, 1988. [Sai90] Morihiko Saito. Mixed odge modules. Publ. Res. Inst. Math. Sci., 262:221 333, 1990. [SS16] Claude Sabbah and Christian Schnell. The MM project. 2016. [Wei17] Chuanhao Wei. Logarithmic kodaira dimension and zeros of holomorphic log-one-forms. preprint, 2017. [Wu16] Lei Wu. Multi-indexed deligne extensions and multiplier subsheaves. preprint, 2016.,.