Perverse Leray filtration and specialization with applications to the Hitchin morphism
aa r X i v : . [ m a t h . AG ] F e b PERVERSE LERAY FILTRATION AND SPECIALIZATIONWITH APPLICATIONS TO THE HITCHIN MORPHISM
MARK ANDREA A. DE CATALDO
Abstract.
We initiate and develop a framework to handle the specialization morphismas a filtered morphism for the perverse, and for the perverse Leray filtration, on thecohomology with constructible coefficients of varieties and morphisms parameterized bya curve. As an application, we use this framework to carry out a detailed study offiltered specialization for the Hitchin morphisms associated with the compactification ofDolbeault moduli spaces in [de-2018].
Contents
1. Introduction, notation and preliminaries 21.1. Motivation and outline of the results 21.2. More precise outline of the contents of the paper 41.3. Notation 51.4. The vanishing and nearby cycles formalism 81.5. Nearby cycle functor and nearby points 102. The morphisms of type δ δ δ and vanishing/nearby cycles 152.3. Relative hard Lefschetz and morphisms of type δ G m -quotients 384.2. Compactification of Dolbeault moduli spaces via G m -actions 414.3. Special properties of Q X and IC X for the compactification 414.4. The long exact sequence of the triple ( Z, X, X o ) 44References 47 Introduction, notation and preliminaries
Let v : Y → S be a morphism into a connected nonsingular curve, let s ∈ S be a point,let G ∈ D bc ( Y ) be a bounded constructible complex, let t ∈ S be a suitably general point(in a more algebraic set-up, the geometric generic point of the curve, or of an Henseliantrait), let Y s and Y t be the corresponding fibers.It is natural, and of fundamental importance, to compare the cohomology groups H j ( Y s , G | Y s ) and H j ( Y t , G | Y t ) . One classical example, is the study of Lefschetz pencils.Similarly, if X → Y is an S -morphism, then it is equally important to compare, for F ∈ D bc ( X ), the cohomology groups H j ( X s , F | X s ) → H j ( X t , F | X t ).When it is defined, the specialization morphism H j ( Y s , G | Y s ) → H j ( Y t , G | Y t ) , and sim-ilarly for X s and X t , is a key tool for this comparison.The paper [de-Ma-2018] studies the Hitchin S -morphism f : X → Y for a smoothand projective family X /S and establishes that the resulting specialization morphisms inintersection cohomology exist, and that they are filtered isomorphisms for the respectiveperverse Leray filtrations. The two authors realized that there seems to be no availablediscussion in the literature of the specialization morphism as a filtered morphism, so thatthey developed a criterion for having such a filtered isomorphism that worked in thecontext of the Hitchin morphism associated with a family of smooth varieties over a base.In this paper, we initiate and develop a general framework to study the specializationmorphism as a filtered morphism for the perverse (Leray) filtration. We then apply sucha framework to a more detailed study of the Hitchin morphism.1.1. Motivation and outline of the results.
Let us consider the toy model situation in Remark 3.5.7 based on the Set-up 3.4.1,where we start with a smooth morphism f : X o → Y o over a a smooth curve S , and wecompactify f by adding divisors Z and W , to get f : X o ∪ Z = X → Y = Y o ∪ W , so thatall resulting morphisms are smooth and proper, except for X o and Y o over S . Then themorphisms of long exact sequences (69) for the relative singular cohomology of the pairs( X t,s , X ot,s ) and ( Y t,s , Y ot,s ) relating any two point s, t ∈ S , is a filtered isomorphisms forthe Leray spectral sequences for the morphisms f t : X t → Y t and f s : X s → Y s .What happens when the morphisms are not proper, the varieties are singular, we takecoefficients in an arbitrary bounded constructible complex of sheaves in D bc ( X ), and weconsider the perverse Leray filtration? This is the question addressed in this paper.First of all, why the perverse Leray filtration? The middle perversity t -structure is moresuitable to study singularities, so we focus on the perverse (Leray) filtration, instead ofthe Grothendieck (Leray) filtration. In principle the Leray filtration can be studied withthe methods of this paper, but the results I can think of are much weaker, essentiallydue to the fact that the vanishing/nearby cycle functors have several important perverse t -exactness properties, without having a counterpart for the usual standard t -structure.Let us focus on an arbitrary situation X/Y /S with K ∈ D bc ( X ), with s ∈ S a pointand t ∈ S a suitably general point (the geometric generic point of the curve/Henseliantrait, if the reader prefers that language). In this general context, the specializationmorphism (“from a special point to a general point”) H ∗ ( X s , K | X s ) → H ∗ ( X t , K | X t ) is ERVERSE LERAY FILTRATION AND SPECIALIZATION 3 not even defined. This is due to the failure of the relevant base-change morphisms tobe isomorphisms. Even when the specialization morphism is defined, its perverse filteredcounterpart may fail to be well-defined.The purpose of this paper is to develop a framework where these questions can be studiedsystematically. What follows is a list of some of the outcomes of this study, presented in aweaker and less complete form with respect to what is found in the body of the paper, sothat the reader may get an idea of the techniques introduced and of the results that areproved.Let us start with some of the preparatory results in §
2. In the set up of morphisms X → Y → S and of the specialization to a point s on the curve S , the fibers X s and Y s overthe special point are Cartier divisor. A key point is to study, in the more general set-up of aCartier divisor T ′ inside a variety T , the failure of commutativity of the perverse truncationfunctors p τ ≤ k with the restriction i ∗ to T ′ , when applied to a complex G ∈ D ( T ). Thisis codified in the failure to be isomorphisms of a certain morphisms that we introduceand denote by δ (22). Proposition 2.1.5 says that the δ are isomorphisms when G has noconstituents (e.g. no non-trivial direct summands) supported on the divisor T ′ . Lemma2.2.1, which is placed in the context of specialization, gives a criterion for the δ to beisomorphisms in terms of the vanishing φG = 0 of the vanishing cycles of the complex G . Proposition 2.3.2 is another criterion for the δ being isomorphisms for a complex f ∗ F , when we are in a projective morphism f : X → Y situation (not necessarily over acurve S ), where a Cartier divisor on X , pulled back from Y , has suitable transversalityconditions with respect to the complex F on X : essentially, one requires the complex F on X to be semisimple and to stay semisimple after restriction to the Cartier divisor.Corollary 2.4.2 shows that this kind of transversality can be achieved in a simple normalcrossing singularities situation, where the restriction does not stay semisimple.As it may be clear by now, the main goal of this paper is to identify conditions thatensure that specialization morphisms are defined, and, when they are defined, that theyare isomorphisms, and, in the filtered context, that they are filtered isomorphisms.Following the preparation in §
2, concerning the morphisms δ , we zero in on the problemby defining the perverse filtered version of the specialization morphism (when it exists),and by offering some criteria in the main section § v isnot proper, without additional constraints on the situation, the specialization morphismmay fail to be well-defined. We are in the situation f : X → Y, v : Y → S , F ∈ D bc ( X ). Inorder to try to convey the flavor of the theorem, we offer two different sets of conditions,each of which is a sufficient set of conditions: f and v are proper, and φ ( F ) = 0; f proper, F semisimple and φ ( F ) = 0.We apply these results to a compactification of the morphism f as above, i.e. as in Set-up3.4.1. In this case, we prove Theorem 3.5.4, i.e that, under suitable sets of hypotheses, thesituation of the toy model discussed at the beginning of this section can be reproducedin its entirety: the specialization morphisms for X s , X t , X os , X ot and Z s , Z t are filteredisomorphisms compatibly with the restriction and Gysin morphisms (which one needs toshow are well-defined) stemming from the inclusions. MARK ANDREA A. DE CATALDO
Finally, Theorem 4.4.2 is our application of the methods of this paper, and especiallyof Theorem 3.5.4, to the compactification of Dolbeault moduli spaces constructed in[de-2018], thus showing that the various criteria developed in this paper can actuallybe implemented in a highly non-trivial and geometrically interesting situation. Perhapssurprisingly, this is true whether we consider intersection cohomology, or singular coho-mology.1.2.
More precise outline of the contents of the paper. § § § § § § δ ; theseare key to this paper since their being isomorphisms is necessary to the specializationmorphisms being filtered isomorphisms for the perverse (Leray) filtration. The heart ofthis paper, i.e. the discussion of the specialization morphism as a filtered morphism for theperverse Leray filtration, is §
3. In order to carry out that discussion we need to measure thefailure of the restriction-to-the-special-fiber functor to commute with perverse truncation.This failure is measured by the morphisms of type δ , which are defined in Lemma 2.1.2 andRemark 2.1.3, in the more general context of effective Cartier divisors (the special fiberbeing one such). Proposition 2.1.5 gives a sufficient condition for the morphisms of type δ to be isomorphisms. Lemma 2.2.1 establishes the key facts we need when dealing withthe morphisms of type δ together with the vanishing/nearby cycle functors; in particular,the vanishing of the vanishing cycle functor gives a sufficient condition for the morphismsof type δ being isomorphisms. Proposition 2.3.2 provides another such criterion under theassumption that certain relative hard Lefschetz symmetries are in place; Corollary 2.4.2ensures that if we are in a simple normal crossing divisor situation, then such symmetriesare in place.The aforementioned framework is developed in §
3. We refer to the beginning of thatsection for a more detailed account of its contents. Here we simply list the main points.The set-up is the one of an S -morphism X → Y , where S is a nonsingular connected curve,and s ∈ S is a point, the “special” point. The definition of the specialization morphismas a filtered morphism for the perverse Leray filtration is contained in Definition 3.3.3.The main Theorem of this paper is Theorem 3.3.7, which establishes various criteria forthis morphisms to exist and to be a filtered isomorphism. On of the main themes here isto work with non proper structural morphisms X → S and Y → S , for in this case, thebase change morphisms are not isomorphisms in general. Compactifying the situation isone traditional way to circumvent this issue; this introduces additional base change issues,and Proposition 3.4.2 provides criteria to resolve them. Once a compactification is inplace, one has the long exact sequence of the resulting triple (boundary, compactification,original space), and Theorem 3.5.4 provides three criteria ensuring that the resulting threespecialization morphisms gives rise to an isomorphism of filtered long exact sequencesassociated with the special and general triples. ERVERSE LERAY FILTRATION AND SPECIALIZATION 5 § § G m -quotients, which could be of general independentinterest, are established along the way. Remark 1.2.1. ( Other algebraically closed fields, generic geometric points vsgeneral points ) We have chosen to work with vanishing/nearby cycles wrt a morphisminto a nonsingular curve, over the field of complex numbers, with the classical topology,and with finite algebraic Whitney stratifications (which play a role only in the background,by merely existing and having the usual properties). While this is mostly a matter ofexpository style, we also work in a more global (not over a curve/Henselian trait) contextin parts of §
2. By complementing the references given in § [Il-1994, § , whichalso deals with vanishing/nearby cycles and the perverse t-structure, the exposition andthe results of this paper remain valid, mutatis mutandis, for varieties over algebraicallyclosed fields of characteristic zero and for the ´etale cohomology with coefficients in Q ℓ , forany prime ℓ ; for example, the role played in this paper by a suitably general point t on anonsingular curve S is now played by the geometric generic point of S , or by the geometricgeneric point of an Henselian trait. Similarly, except for §
4, where in the application to thecompactification of the Hitchin morphism we need to consider quotients by finite groupson which we have little control, the results remain valid over an algebraically closed fieldof positive characteristic and Q ℓ -coefficients, with ℓ not dividing the characteristic of thefield. Acknowledgments.
The author thanks: Michel Brion, Victor Ginzburg, Jochen Hein-loth, Luca Migliorini, J¨org Sch¨urmann and Geordie Williamson for useful suggestions.Special thanks to Davesh Maulik, with whom the author has discussed various prelimi-nary versions of Theorem 4.4.2. The author, who is partially supported by N.S.F. D.M.S.Grants n. 1600515 and 1901975, would like to thank the Freiburg Research Institute forAdvanced Studies for the perfect working conditions; the research leading to these resultshas received funding from the People Programme (Marie Curie Actions) of the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n.[609305]. Finally, we thank very warmly the referee for pointing out some inaccuracies inthe first submission.1.3.
Notation.
By variety, we mean a separated scheme of finite type over the field of complex numbers C . By point, we mean a closed point. See [de-Mi-2009] for a quick introduction and forstandard references concerning the constructible derived category and other concepts inthis subsection.Given a variety Y, we denote by D bc ( Y ) the constructible bounded derived categoryof sheaves of rational vector spaces on Y, and by DF ( Y ) its filtered variant [Il-1971,Be-Be-De-1982]. We endow D bc ( Y ) with the middle perversity t -structure. A functor that MARK ANDREA A. DE CATALDO is exact with respect to this t -structure is said to be t -exact. The full subcategory ofperverse sheaves is denoted by P ( Y ). We employ the following standard notation forthe objects associated with this t -structure: the full subcategories p D ≤ j ( Y ) and p D ≥ j ( Y ), ∀ j ∈ Z , and p D [ j,k ] ( Y ) := p D ≥ j ( Y ) ∩ p D ≤ k ( Y ) , ∀ j ≤ k ∈ Z , of D bc ( Y ); the truncationfunctors p τ ≤ j : D bc ( Y ) → p D ≤ j ( Y ) and p τ ≥ j : D bc ( Y ) → p D ≥ j ( Y ); the perverse cohomologyfunctors p H j : D bc ( Y ) → P ( Y ). We denote derived functors using the un-derived notation,e.g. if f : X → Y is a morphism of varieties, then the derived direct image (push-forward)functor Rf ∗ : D bc ( X ) → D bc ( Y ) is denoted by f ∗ , etc. Distinguished triangles in D bc ( Y ) aredenoted G ′ → G → G ′′ . At times, we drop the space variable Y from the notation.The following operations preserve constructibility of complexes: ordinary and extraor-dinary push-forward and pull-backs, hom and tensor product, Verdier duality, nearby andvanishing cycles.When we write a Cartesian diagram of morphisms of varieties: X ′ g / / f (cid:15) (cid:15) X f (cid:15) (cid:15) Y ′ g / / Y, (1)the ambiguities arising by having denoted different arrows by the same symbol, are au-tomatically resolved by the context in which they are used; e.g. when we write the basechange morphism of functors, the expression g ∗ f ∗ → f ∗ g ∗ is unambiguous.The k -th (hyper)cohomology groups of Y with coefficients in G ∈ D bc ( Y ) are denoted by H k ( Y, G ). The complex computing this cohomology is denoted by R Γ( Y, G ) and it lives inthe bounded derived category D bc ( pt ) whose objects are complexes of vector spaces withcohomology given by finite dimensional rational vector spaces.The filtrations we consider are finite and increasing. A sequence of morphism G • :=( . . . → G n → G n +1 → . . . ) in D bc ( Y ) subject to G i = 0 ∀ i ≪
0, and to G i ∼ → G i +1 ∀ i ≫
0, gives rise to objects ( G, F ) ∈ DF ( Y ) and ( R Γ( Y, G ) , F ) ∈ D bc ( pt ) . We call sucha sequence of morphisms a system in D bc ( Y ) . There is the evident notion of morphismof systems G • → G ′• and, for each such, the resulting morphisms ( G, F ) → ( G ′ , F ) in DF ( Y ) , and ( R Γ( Y, G ) , F ) → ( R Γ( Y, G ′ ) , F ) in DF ( pt ). Given G ∈ D bc ( Y ) we havethe system of perverse truncation morphisms: . . . → p τ ≤ n G → p τ ≤ n +1 G → . . . → G. Amorphism G → G ′ in D bc ( Y ) gives rise to a morphism of systems p τ ≤• G → p τ ≤• G ′ , whichgives rise to a morphisms of filtered objects:( R Γ( Y, G ) , P )) / / ( R Γ( Y, G ′ ) , P )) in DF ( pt ) . (2)The filtration P is called the perverse filtration. There is also the evident notion of systemsof functors; e.g. the truncation functors.Let f : X → Y be a morphism of varieties, let F ∈ D bc ( X ) and let G = f ∗ F . Then wehave the filtered objects ( R Γ( X, F ) , P ) = ( R Γ( X, F ) , P f ) := ( R Γ( Y, f ∗ F ) , P ) . The filtra-tion P f is called the perverse Leray filtration associated with f. Given a morphism F → F ′ in D bc ( X ) , the analogue of (2) holds. We have the corresponding objects P fj H k ( X, F ),Gr P f j H k ( X, F ). ERVERSE LERAY FILTRATION AND SPECIALIZATION 7
If a statement is valid for every value of an index, e.g. the degree of a cohomologygroup, or the step of a filtration, then we denote such an index by a bullet-point symbol,or by a star symbol, e.g. H • ( X, F ), P • , p τ ≤• , P ⋆ H • ( X, F ), Gr P f ⋆ H • ( X, F ).We employ the following convention for shifts of filtered increasing filtrations: F ( j ) • := F •− j . (3)We have the evident and equivalent relations between truncations and shifts:[ ⋆ ] ◦ p τ ≤• = p τ ≤•− ⋆ ◦ [ ⋆ ] , p τ ≤• ◦ [ ⋆ ] = [ ⋆ ] ◦ p τ ≤• + ⋆ , (4)which are valid for p τ ≥• and p H • as well.The category P ( Y ) of perverse sheaves on Y is Abelian and Artinian, so that the Jordan-Holder theorem holds in it. The constituents of a non-zero perverse sheaf G ∈ P ( Y ) are theisomorphisms classes of the perverse sheaves appearing in the unique and finite collectionof non-zero simple perverse sheaves appearing as the quotients in a Jordan-Holder filtrationof G. The constituents of a non-zero complex G ∈ D bc ( Y ) are defined to be the constituentsof all of its non-zero perverse cohomology sheaves.In general, we drop decorations (indices, parentheses, space variables, etc.) if it seemsharmless in the context.Given a morphism of varieties X → Y and a point y ∈ Y , we denote by X y the fiberover y in X .We are going to use the nearby/vanishing cycle functors. See § Theorem 1.3.1. ( Decomposition and Relative Hard Lefschetz theorems ) Let f : X → Y be a proper morphism of varieties and let F ∈ P ( X ) be semisimple.Then f ∗ F is semisimple.If f is projective, then the Relative Hard Lefschetz holds: the choice of an f -ample linebundle induces isomorphisms p H −• ( f ∗ F ) ≃ → p H • ( f ∗ ( F ) . We define the intersection complex IC T of a variety T as the direct sum of the inter-section complexes of its irreducible components T j : IC T := ⊕ j IC T j . Then, as it is shownin [de-2012]: IC T ∈ P ( T ) is a semisimple perverse sheaf; it is the intermediate extensionfrom the smooth locus of T of the direct sum of the constant sheaves on each componentshifted by its dimension (see [Wu-2019] for a topological characterization); the Decompo-sition Theorem, Relative Hard Lefschetz Theorem, and allied Hodge-Theoretic facts holdfor it and its cohomology. The complex IC T underlies a polarizable Hodge module. Thesimple objects in P ( Y ) are the intersection complexes IC T ( L ), where T ⊆ Y is closedand irreducible and L is an irreducible local systems on some Zariski dense open subset T o ⊆ T reg .We define the topologist’s intersection complex as follows: IC T := ⊕ j IC T j [dim T j ].We define the intersection cohomology groups by setting IH • ( T ) := H • ( T, IC T ); theystart in degree − dim T . We define the topologist’s intersection cohomology groups bysetting IH • ( T ) := H • ( T, IC T ); they start in degree zero. E.g. for T = o ` Λ the disjoint
MARK ANDREA A. DE CATALDO union of a point and a line, we have: Q T = Q o ⊕ Q Λ ; IC T = Q o ⊕ Q Λ [1]; IC T = Q o ⊕ Q Λ ; IH • ( T ) = H • ( Q T ) = H • ( o ) ⊕ H • (Λ); IH • ( T ) = H • ( o ) ⊕ H • +1 ( T ).In order to avoid repeating naturality-type statements, we employ systematically thefollowing notation. The symbol “=” denotes either equality, or a canonical isomorphism.The symbol ≃ → and denotes the fact that a canonical arrow is, in the context where itappears, an isomorphism. The symbol ∼ = denotes an isomorphism; if the direction ofthe arrow is relevant, we write ≃ → . E.g.: the base change canonical isomorphism reads: g ! f ∗ = f ∗ g ! ; the proper base change isomorphism for f proper reads: g ∗ f ∗ ≃ → f ∗ g ∗ .1.4. The vanishing and nearby cycles formalism.
This section is an expanded version of [de-Ma-2018, § v = v Y : Y → S be a morphism of varieties, where S is a nonsingular curve. Let s ∈ S be a point, usually called the special point.We have the exact functors of triangulated categories: i ∗ , i ! , ψ = ψ v , φ = φ v : D bc ( Y ) → D ( Y s ) , (5)where, i : Y s → Y is the closed embedding of the (special) fiber of v over s , φ is thevanishing cycle functor and ψ is the nearby cycle functor. The functors φ and ψ dependon v ; e.g. [El-Lˆe-Mi-2010, Example 2.0.5]; this is not an issue in this paper.We have the two, Verdier dual, canonical distinguished triangle of functors: (we oftenwrite σ instead of σ ∗ and σ ! ) i ∗ [ − σ ∗ / / ψ [ − can / / φ / / /o/o/o , φ var / / ψ [ − σ ! / / i ! [1] / / /o/o/o . (6) Fact 1.4.1. ( t -exactness for ψ [ − and φ ) The functors ψ [ − and φ are t -exact andcommute with Verdier duality. We thus have the following canonical identifications: p τ ≤• φ = φ p τ ≤• ; p τ ≤• ψ [ −
1] = ψ [ − p τ ≤• ; ditto for p τ ≥• and p H • . (7)The following is a key property of the vanishing cycle functor. Fact 1.4.2. ( Smooth morphisms and vanishing of φ ) If v : Y → S is smooth overa neighborhood of s , and G ∈ D bc ( Y ) has locally constant cohomology sheaves near s, then φ G = 0 ∈ D ( Y s ) . See [De-1972, XIII, 2.1.5] . In the special case where v = id S , thisimplies that a complex G ∈ D ( S ) has constant cohomology sheaves near s if and only if φG = 0 . We recall, for later use, the following simple
Lemma 1.4.3. ( Vanishing of φ implies no constituents ) Let G ∈ D bc ( Y ) . If φG = 0 , then no constituent of G is supported on Y s . Proof.
See [de-Ma-2018, Lemma 3.1.5] (cid:3)
Remark 1.4.4.
The converse to Lemma 1.4.3 is false; see [Ka-Sh-1990, Ex. VIII.14] (Lefschetz degeneration to an ordinary double point).
ERVERSE LERAY FILTRATION AND SPECIALIZATION 9
Fact 1.4.5.
The composition i ∗ [ − → ψ [ − → i ! [1] yields a natural morphism of functors D bc ( Y ) → D ( Y s ) : ν : i ∗ [ − / / i ! [1] . (8) The morphism ν coincides with the morphism obtained via Verdier’s specialization functor,(cf. [Sc-2003] , for example), so that it depends only on the closed embedding Y s → Y, i.e.it is independent of v. If φG = 0 , then σ ∗ ( G ) , σ ! ( G ) and ν ( G ) are isomorphisms. Fact 1.4.6. ( Base change diagrams for ψ and φ ) Let f : X → Y be an S -morphismand let v X = v Y ◦ f : X → S be the structural morphism. The various base change naturaltransformation associated with i and f give rise to functorial morphisms of distinguishedtriangles as follows. See [De-1972, XIII, 2.1.7] . (1) The one in D ( X s ) stemming from the morphism of functors i ∗ f ! → f ! i ∗ , and fromthe isomorphism of functors f ∗ i ∗ = → i ∗ f ∗ , respectively: (we write φ Y instead of φ v Y , etc.) i ∗ [ − f ! σ ◦ f ! / / (cid:15) (cid:15) ψ X [ − f ! / / (cid:15) (cid:15) φ X f ! (cid:15) (cid:15) / / /o/o/o i ∗ [ − f ∗ σ ◦ f ∗ / / ψ X [ − f ∗ / / φ X f ∗ / / /o/o/o f ! i ∗ [ − σ ◦ f ! / / f ! ψ Y [ − / / f ! φ Y / / /o/o/o , f ∗ i ∗ [ − σ ◦ f ! / / = O O f ∗ ψ Y [ − / / O O f ∗ φ Y O O / / /o/o/o . (9) When f is ´etale, so that f ∗ = f ! , the two resulting morphisms of triangles areisomorphisms, inverse to each other. (2) The one in D ( Y s ) stemming from the base change morphism of functors i ∗ f ∗ → f ∗ i ∗ , and from the base change isomorphism f ! i ∗ = → i ∗ f ! : i ∗ [ − f ∗ σ ◦ f ∗ / / (cid:15) (cid:15) ψ Y [ − f ∗ / / (cid:15) (cid:15) φ Y f ∗ (cid:15) (cid:15) / / /o/o/o i ∗ [ − f ! σ ◦ f ! / / ψ Y [ − f ! / / φ Y f ! / / /o/o/o f ∗ i ∗ [ − f ∗ ◦ σ / / f ∗ ψ X [ − / / f ∗ φ X / / /o/o/o , f ! i ∗ [ − f ! ◦ σ / / = O O f ! ψ X [ − / / O O f ! φ X O O / / /o/o/o . (10) When f is proper, so that f ! ≃ → f ∗ , the two resulting morphisms of triangles areisomorphisms, inverse to each other. (3) We dualize the four diagrams associated in (9) and (10), and obtain the fouranalogous diagrams: φ X f ∗ / / ψ X [ − f ∗ σ ◦ f ∗ / / i ! [1] f ∗ / / /o/o/o φ X f ! / / (cid:15) (cid:15) ψ X [ − f ! σ ◦ f ! / / (cid:15) (cid:15) i ! [1] f != (cid:15) (cid:15) / / /o/o/o f ∗ φ Y / / O O f ∗ ψ Y [ − f ∗ ◦ σ / / O O f ∗ i ! [1] / / /o/o/o O O , f ! φ Y / / f ! ψ Y [ − f ! ◦ σ / / f ! i ! [1] / / /o/o/o , (11) which are isomorphisms inverse to each other when f is ´etale, and φ Y f ! / / ψ Y [ − f ! σ ◦ f ! / / i ! [1] f ! / / /o/o/o φ Y f ∗ / / (cid:15) (cid:15) ψ Y [ − f ∗ σ ◦ f ∗ / / (cid:15) (cid:15) i ! [1] f ∗ = (cid:15) (cid:15) / / /o/o/o f ! φ X / / O O f ! ψ X [ − f ! ◦ σ / / O O f ! i ! [1] O O / / /o/o/o , f ∗ φ X / / f ∗ ψ Y [ − f ∗ ◦ σ / / f ∗ i ! [1] / / /o/o/o , (12) which are isomorphisms inverse to each other when f is proper. (4) by combining the l.h.s. square commutative diagram in (10) with the r.h.s. squarecommutative diagram in (12), we obtain the following commutative diagram ofmorphisms of functors: i ∗ [ − f ∗ (cid:15) (cid:15) / / ψ [ − f ∗ / / (cid:15) (cid:15) i ! [1] f ∗ = (cid:15) (cid:15) f ∗ i ∗ [ − / / f ∗ ψ [ − / / f ∗ i ! [1] , (13) where the compositions of the horizontal arrows are given by the correspondingmorphims (8), suitably pre/post-composed with f ∗ . Nearby cycle functor and nearby points.
Even if logically not necessary, it maybe helpful to the intuition to clarify the use of theword “nearby.”Let t ∈ S be any point. By abuse of notation, denote by t : t → S , t : X t → X and t : Y t → Y the resulting closed embeddings. There is the morphism (8) of functors D ( − ) → D ( − t ) : t ∗ [ − −→ t ! [1] . (14)Let us start with the following Fact 1.5.1.
For what follows, see [de-Ma-2018, Fact 2.2.7] . Given a finite collection G ∈ D bc ( Y ) of complexes, there is a Zariski-dense open subset S o ( G ) ⊆ S such that:the direct images v ∗ p τ ≤• G have locally constant cohomology sheaves, and their formationcommutes with arbitrary base change; the p τ ≤• G have no constituent supported on any ofthe fibers of the morphisms v over S o ( G ) ; the strata on Y, of a stratification with respectto which G -hence the p τ ≤• G - are constructible, are smooth over S o ( G ) . Definition 1.5.2.
We say that t ∈ S is general for G if t ∈ S o ( G ) as in Fact 1.5.1. Fact 1.5.3.
Let G ∈ D bc ( Y ) . Let t ∈ S o ( G ) be a general point for G. Then: (1)
The natural morphism t ∗ [ − G → t ! [1] G is an isomorphism in D ( Y t ) . See [de-Ma-2018,Fact 2.2.5] . (2) We have the identifications of [de-Ma-2018, Fact 2.2.6] : t ∗ [ − p τ ≤• G = p τ ≤• t ∗ [ − G, t ! [1] p τ ≤• G = p τ ≤• t ! [1] G, ditto for p τ ≥• and p H • . (15) Remark 1.5.4.
In the special case where v Y : Y → S is the identity on S, we have that i ∗ , i ! , ψ, φ : D ( S ) → D ( s ) and that t ∗ , t ! : D ( S ) → D ( t ) . In general, we have canonical ERVERSE LERAY FILTRATION AND SPECIALIZATION 11 identifications D ( s ) = D ( pt ) = D ( t ) , where pt is just a point, so that all three categories arenaturally equivalent to the bounded derived category of finite dimensional rational vectorspaces. Similarly, in the filtered case: DF ( s ) = DF ( pt ) = DF ( t ) . We use the catch-allnotation D bc ( pt ) and DF ( pt ) . Given G ∈ D bc ( Y ) and s ∈ S our special point, we want to think: of the points t ∈ S o ( G )as the points nearby s ; of the nearby vanishing cycle as capturing the complex G at points t nearby s. The following important fact makes this precise.
Fact 1.5.5. (Fundamental isomorphism)
For what follows, see the fundamental iden-tity [De-1972, XIV, 1.3.3.1] . Let v : Y → S be the identity, let G ∈ D bc ( Y ) , let t ∈ S o ( G ) be a general point for G. Choose a disk ∆ ⊆ S centered at s and passing through t, suchthat ∆ ∗ := ∆ \ { s } ⊆ S o ( G ) . Choose an universal covering of ∆ ∗ and a point e t on it over t. This choice of data gives rise to isomorphisms: t ∗ [ − G ∼ / / ψ [ − G ∼ / / t ! [1] G, in D ( pt ) , (16) where the composition is (14), hence it is independent of the choices made above, but theindicated morphisms depend on the choice of data made above. Note that (16) is Verdierself-dual. Changing the choice of data changes the two individual arrows by the monodromyautomorphism induced by an appropriate element in π ( S o ( G ) , t ) . Let v : Y → S be any morphism and let G ∈ D bc ( Y ) . Then: (1)
By taking v ∗ G ∈ D ( S ) in (16), we get collections of isomorphisms: ψv ∗ G ∼ / / t ∗ v ∗ G = v ∗ t ∗ G = R Γ( Y t , t ∗ G ) , in D ( pt ) , (17) where any two such identifications ∼ → differ by the action induced by an elementof π ( S o ( G ) , t ) on the r.h.s. The second = sign is clear. The first can be seen byusing the base change properties relative to the general point t expounded in § t general for G, made in Definition 1.5.2, allows us to:replace ψ with t ∗ (cf. (16)); use the identification t ∗ v ∗ = v ∗ t ∗ . One cannot select apoint t ∈ S that is general for every G ∈ D bc ( Y ) . Because of this it is preferable towork with ψ : D ( S ) → D ( s ) instead of with a general point t ∈ S. When workingwith finitely many complexes in D bc ( Y ) , or with a family of complexes constructiblewith respect to an arbitrary fixed stratification, we can always choose a point t ∈ S that is general for all of them. (2) By taking v ∗ p τ ≤• G ∈ D ( S ) and v ∗ p τ ≤• f ∗ F ∈ D ( S ) in (16), we reach conclusionsanalogous to part (1), namely, we obtain isomorphisms: ( ψv ∗ G, P ) ∼ / / ( R Γ( Y t , t ∗ G ) , P ) , in DF ( pt ); (18)( ψv ∗ f ∗ F, P ) ∼ / / ( R Γ( X t , t ∗ F ) , P ) , in DF ( pt ) . (19)2. The morphisms of type δ This section is a necessary technical interlude on the way to §
3, where we study thebehavior of the perverse filtration relative to the specialization morphism.
The distinguished triangles (6) and the t -exactness of the functors φ and ψ [ −
1] suggestthe need to quantify the failure of t -exactness for the functors i ∗ [ −
1] and i ! [1].In this section we quantify this failure by introducing the morphisms of functors of type δ (27) which play an important role starting with §
3, diagram (49). When the morphismsof type δ are evaluated against a complex G ∈ D bc ( Y ), they are isomorphisms if and onlyif perverse truncations and perverse cohomology commute with the appropriately shiftedpull-backs.The special fiber Y s in Y is an effective Cartier divisor. We introduce the morphismsof type δ in the more general context of embedded effective Cartier divisors in (22). Foreasier bookkeeping, we use the shifted version (24).Proposition 2.1.5 is a criterion, in the general context of Cartier divisors (i.e. when weare not necessarily working over a curve S ), for the morphisms of type δ to be isomor-phisms, hence it is a criterion for the aforementioned commutativity of perverse trunca-tions/cohomology with shifted pullbacks. It states that if a complex has no constituentssupported on an effective Cartier divisor, then the resulting morphisms of type δ areisomorphisms.Lemma 2.2.1 yields, in the context of the vanishing/nearby cycle functors (here we areworking over a curve S , and the Cartier divisor in question is the fiber over the specialpoint s ) the important diagram (28) of morphisms of distinguished triangles of functorsinvolving the morphisms of type δ needed in this paper. In particular, Lemma 2.2.1.(3)combines Lemma 1.4.3 (if φG = 0, then G has no constituents on Y s ) with Proposition2.1.5 (“if no constituents, then the δ ’s are isomorphisms).Proposition 2.3.2 is another general (i.e not necessarily related to a situation over acurve) criterion of a different flavor for the δ ’s to be isomorphisms in the context of acomplex G = f ∗ F being a direct image under a projective morphism f . It replaces theassumptions on constituents and/or on the vanishing of the vanishing cycle functor, withthe assumptions that both F and i ∗ F [ −
1] are semisimple; the proof uses the RelativeHard Lefschetz Theorem.2.1.
Pull-back to Cartier divisors and truncations: the morphisms δ . In the context of the vanishing cycle functor, i.e. v : Y → S and s ∈ S , the special fiber Y s is a Cartier divisor in the total space Y. The vanishing cycle functor φ and the shiftednearby cycle functor ψ [ −
1] are t -exact. It is important to study the defect of t -exactnessof the shifted restriction to the special fiber functors i ∗ [ −
1] and i ! [1] . In this section, which amplifies [de-Ma-2018, § δ in the context of embeddings of Cartier divisors in Lemma 2.1.2, and we provea criterion for when these morphisms δ are isomorphisms in Proposition 2.1.5.Let us emphasize that in this section we work with varieties and not with S -varieties.We start by listing the needed general t -exactness properties related to embeddings ofeffective Cartier divisors on varieties. Lemma 2.1.1. ( Inequalities for embeddings of Cartier divisors ) Let ι : T ′ → T bea closed embedding of varieties such that the open embedding T \ T ′ is an affine morphism ERVERSE LERAY FILTRATION AND SPECIALIZATION 13 (e.g. T ′ is an effective Weil divisor supporting an effective Cartier divisor). Then: (weomit the space variables) (1) The functor ι ∗ is is right t -exact and the functor ι ! is left t -exact: ι ∗ : p D ≤• → p D ≤• , ι ! : p D ≥• → p D ≥• . (20)(2) The functor ι ∗ [ − is left t -exact and the functor ι ! [1] is right t -exact: ι ∗ : p D ≥• → p D ≥•− , ι ! : p D ≤• → p D ≤• +1 . (21) In particular, we have that: ι ∗ P ( T ) ⊆ p D [ − , ( T ′ ) and ι ! P ( T ) ⊆ p D [0 , ( T ′ ) .Proof. See [de-Ma-2018, Lemma 3.1.1]. (cid:3)
The following lemma extends [de-Ma-2018, Lemma 3.1.2] in the direction needed in thispaper.We denote by γ ≤• : p τ ≤•− → p τ ≤• and γ ≥• : p τ ≤• → p τ ≤• +1 the natural morphisms.They are Verdier dual to each other. We set: γ ∗≤• := γ ≤• ι ∗ , γ ! ≤• := γ ≤• ι ! , γ ∗≥• := γ ≥• ι ∗ , γ ! ≥• := γ ≥• ι ! . The entries of each of the two pairs ( γ ∗≤• , γ ! ≥• ) and ( γ ∗≥• , γ ! ≤• ) are exchanged by Verdierduality. Lemma 2.1.2. ( The morphisms δ ) There are natural morphisms of distinguished tri-angles of functors: p τ ≤•− ι ∗ γ ∗≤• / / δ ∗≤• (cid:15) (cid:15) ι ∗ = (cid:15) (cid:15) / / p τ ≥• ι ∗ δ ∗≥• (cid:15) (cid:15) γ ∗≥• { { / / /o/o/o/o p τ ≤•− ι ! γ ! ≤• " " / / ǫ ! ≤• (cid:15) (cid:15) ι != (cid:15) (cid:15) / / p τ ≥• ι ! ǫ ! ≥• (cid:15) (cid:15) γ ! ≥• | | / / /o/o/o/o ι ∗ p τ ≤• / / ǫ ∗≤• (cid:15) (cid:15) ι ∗ O O = (cid:15) (cid:15) / / ι ∗ p τ ≥• +1 ǫ ∗≥• (cid:15) (cid:15) / / /o/o/o ι ! p τ ≤•− / / δ ! ≤• (cid:15) (cid:15) ι ! O O = (cid:15) (cid:15) / / ι ! p τ ≥• δ ! ≥• (cid:15) (cid:15) / / /o/o/o/o p τ ≤• ι ∗ / / ι ∗ / / O O p τ ≥• +1 ι ∗ / / /o/o/o , p τ ≤• ι ! / / ι ! / / O O p τ ≥• +1 ι ! / / /o/o/o , (22) which are exchanged by Verdier duality.By iteration, the morphisms in (22) induce natural morphisms: δ ∗• : p H •− [ − • +1] ι ∗ / / ι ∗ p H • [ −• ] , δ ! • : ι ! p H • [ −• ] / / p H • +1 [ − • − ι ! , (23) which are exchanged by Verdier duality.Proof. We prove the lemma for the l.h.s. of (22). The r.h.s. follows by Verdier duality.Consider the l.h.s. diagram of (22), but with the arrows of type δ and ǫ removed. Byapplying [Be-Be-De-1982, Prop. 1.1.9, p.23], whose hypotheses are met by virtue of theinequalities (20) and (21), we see that we can fill in the diagram and make it commutativewith unique arrows. The compositions ǫδ of the vertical arrows give the desired arrows oftype γ because in any t -structure the structural morphism p τ ≤•− → Id factors uniquelythrough the structural morphism p τ ≤• → Id via γ ≤• : p τ ≤•− → p τ ≤• . The morphism on the l.h.s. of (23) arises by composing truncation applied to δ ∗ , with δ ∗ applied to the truncation: p τ ≥•− p τ ≤•− ι ∗ p τ ≥•− δ ∗≤• / / p τ ≥•− ι ∗ p τ ≤• δ ∗≥•− p τ ≤• / / p τ ≥•− ι ∗ p τ ≤• . The morphism on the r.h.s. is obtained in a similar fashion and the verification that thetwo arrows in (23) are Verdier dual is left to the reader. (cid:3)
Remark 2.1.3. (1)
By virtue of the identity (4) concerning truncations and shifts, the morphisms oftype δ in (22) and (23) may be re-written as follows: δ ∗≤• : p τ ≤• ι ∗ [ − / / ι ∗ [ − p τ ≤• , δ ! ≤• : ι ! [1] p τ ≤• / / p τ ≤• ι ! [1] ,δ ∗≥• : p τ ≥• ι ∗ [ − / / ι ∗ [ − p τ ≥• , δ ! ≥• : ι ! [1] p τ ≥• / / p τ ≥• ι ! [1] δ ∗• : p H •− ι ∗ / / ι ∗ [ − p H • , δ ! • : ι ! [1] p H • / / p H • +1 ι ! . (24) where Verdier duality exchanges the morphisms along the two diagonals of the firsttwo rows, and exchanges the two terms in the third row. (2) By looking at (22), and by virtue of Verdier duality, we deduce that for a given G ∈ D ( T ) we have that: δ ∗≤• ( G ) is an isomorphism if and only if δ ∗≥• ( G ) is anisomorphism if and only if δ ! ≤• ( G ∨ ) is an isomorphism if and only if δ ! ≥• ( G ∨ ) isan isomorphism. (3) The same example in Remark 2.2.2.(2) shows that for a given G ∈ D ( T ) , having δ ∗≤• ( G ) an isomorphism does not imply that δ ! ≤• ( G ) is an isomorphism. (4) Proposition 2.1.5 gives a criterion for all six morphisms of type δ to be isomor-phisms. When the morphisms (24) are isomorphisms, we may say that in that case“perverse truncation/cohomology commutes up to a suitable shift with restrictionto the Cartier divisor”. While standard truncation commutes with such an un-shifted restriction -in fact with any pull-back-, in general perverse truncation doesnot. Remark 2.1.4.
In view of the l.h.s. inequality in (21), the r.h.s. vertex of the distin-guished triangle in the middle row of the l.h.s. of (22) satisfies the inequality ι ∗ p τ ≥• +1 : D → p D ≥• . By taking the long exact sequence of perverse cohomology sheaves of saiddistinguished triangle, the aforementioned inequality yields the natural isomorphism offunctors: p τ ≤•− ι ∗ p τ ≤• ≃ / / p τ ≤•− ι ∗ . (25) Proposition 2.1.5. ( If no constituents, then the δ ’s are isomorphisms ) Let ι : T ′ → T be as in Lemma . If G ∈ D ( T ) has no constituent supported on T ′ , then themorphisms of type δ in (24) are isomorphisms. ERVERSE LERAY FILTRATION AND SPECIALIZATION 15
Proof.
This is [de-Ma-2018, Proposition 3.1.4]. For the reader’s convenience, we includeit here.We prove the conclusion for δ ∗≤• ( G ). We have that p τ ≤• G ∈ p D ≤• , so that, by (20), wehave that ι ∗ p τ ≤• G ∈ p D ≤• , and then, clearly, we have that: ι ∗ p τ ≤• G = p τ ≤• ι ∗ p τ ≤• G. (26)In view of (26) and of (25), and by considering the truncation triangle p τ ≤•− → p τ ≤• → p H • [ −• ] applied to ι ∗ p τ ≤• G , in order to prove the desired conclusion, it is necessaryand sufficient to show that p H • ( ι ∗ p τ ≤• G ) = 0.This can be argued as follows. By taking the long exact sequence of perverse cohomologyof the distinguished triangle ι ∗ p τ ≤•− G → ι ∗ p τ ≤• G → ι ∗ p H • G [ −• ] , we see that it isnecessary and sufficient to show that ι ∗ p H • G [ −• ] ∈ p D ≤•− , or, equivalently, that ι ∗ p H • G ∈ p D ≤− . By [Be-Be-De-1982, 4.1.10.ii], we have the distinguished triangle p H − ( ι ∗ p H • G )[1] → ι ∗ p H • G → p H ( ι ∗ p H • G ) and an epimorphism p H • G → p H ( ι ∗ p H • G ). Since G is as-sumed to not have constituents supported at Y s , we must have p H ( ι ∗ p H • G ) = 0, so that ι ∗ p H • G [ −• ] ∈ p D ≤•− , as requested.We have proved that if G has no constituents supported on T ′ , then δ ∗≤• ( G ) is anisomorphism. By Remark 2.1.3, we have that δ ∗≥• ( G ) is an isomorphism as well.Note that G has no constituents supported on T ′ if and only if the same is true for p τ ≤• G . We thus have that δ ∗≤• ( p τ ≤• G ) and δ ∗≥• ( p τ ≤• G ) are isomorphisms.Now, δ ∗• is the composition of two isomorphisms, namely p τ ≥• ( δ ∗≤• ( G )), followed by δ ∗≥• ( p τ ≤• G ).We have proved that the morphisms of type δ ∗ are isomorphisms.Note that G has no constituents supported on T ′ if and only if the same is true for G ∨ .By Remark 2.1.3.(2), it follows that the morphisms of type δ ∗ for G ∨ are isomorphisms,so that so are the morphisms of type δ ! for G .The proposition is proved.Alternatively, one could also retrace the proof given in detail above for the morphismsof type δ ∗ and give a proof for the morphisms of type δ ! . (cid:3) Proposition 2.1.5 can be informally summarized as follows: under special circumstances,the perverse truncation and restriction to a “non-special” Cartier divisor commute witheach other. One can also say that we can slide one functor across the other.2.2.
The morphisms δ and vanishing/nearby cycles. Lemma 2.1.2 is about a Cartier divisor on a variety. Let us specialize to the case whenthe Cartier divisor is the special fiber in the context of the vanishing and nearby cyclefunctors as in § S and we fixa point s ∈ S . Lemma 2.2.1. ( Morphisms of type δ for ψ and φ ) (1) We have natural morphisms of distinguished triangles: p τ ≤• ψ [ − γ ψ ≤• / / δ ψ ≤• ≃ (cid:15) (cid:15) ψ [ − = (cid:15) (cid:15) / / p τ ≥• +1 ψ [ − δ ψ ≥• ≃ (cid:15) (cid:15) γ ψ ≥• { { / / /o/o/o p τ ≤• φ γ φ ≤• / / δ φ ≤• ≃ (cid:15) (cid:15) φ = (cid:15) (cid:15) / / p τ ≥• φ δ φ ≥• ≃ (cid:15) (cid:15) γ φ ≥• { { / / /o/o/o/o ψ [ − p τ ≤• / / ǫ ψ ≤• (cid:15) (cid:15) ψ [ − O O = (cid:15) (cid:15) / / ψ [ − p τ ≥• +1 ǫ ψ ≥• (cid:15) (cid:15) / / /o/o/o φ p τ ≤• / / ǫ φ ≤• (cid:15) (cid:15) φ O O = (cid:15) (cid:15) / / φ p τ ≥• ǫ φ ≥• (cid:15) (cid:15) / / /o/o/o/o p τ ≤• +1 ψ [ − / / ψ [ − / / O O p τ ≥• +2 ψ [ − / / /o/o/o , p τ ≤• +1 φ / / φ / / O O p τ ≥• +2 φ / / /o/o/o , (27) where each diagram is self-dual. (2) If we denote by i ∗ [ − and i ! [1] the two diagrams (22) (after having replaced i ∗ with i ∗ [ − and i ! with i ! [1] ; see Remark 2.1.3.(1)), and if we denote by ψ [ − and by φ the two diagrams in (27), then there are morphism of diagrams i ∗ [ − → ψ [ − → i ! [1] , in the sense that all the corresponding square diagrams are commutative. Inparticular, we have the commutative diagram: i ∗ [ − p τ ≤• / / ψ [ − p τ ≤• δ ψ ≤• ≃ (cid:15) (cid:15) / / i ! [1] p τ ≤• δ ! ≤• (cid:15) (cid:15) p τ ≤• i ∗ [ − / / δ ∗≤• O O p τ ≤• ψ [ − / / O O p τ ≤• i ! [1] . (28) Similarly, for p τ ≥• and p H • . The resulting diagrams are exchanged -in the case oftruncations- or preserved -in the case of perverse cohomology- by Verdier Duality. (3) Let G ∈ D bc ( Y ) be such that φG = 0 . Then all the vertical arrows in (28), and in itsvariants involving p τ ≥• and p H • , evaluated at G, are isomorphisms. In particular,all the arrows of type δ evaluated at G in (24) are isomorphisms. (4) Let G ∈ D bc ( Y ) be such that G has no constituents supported on Y s . Then all thevertical arrows in (28), and in its variants involving p τ ≥• and p H • , evaluated at G are isomorphisms. In particular, all the arrows of type δ evaluated at G in (24)are isomorphisms.Proof. (1) The proof that there are the natural diagrams (27) is the same as the one ofLemma 2.1.2. The fact that the arrows of type δ ψ and δ φ in (27) are isomorphisms simplytranslates the t -exactness of ψ [ −
1] and φ .(2) Once we note that the dual to i ∗ is i ! , and that ψ [ −
1] and φ are self-dual, hence theself-duality of (27), there is nothing left to prove.(3) That the horizontal arrows in (28) are isomorphisms is an immediate consequenceof the distinguished triangles (6) and the hypothesis φG = 0. Once the horizontal arrowsare isomorphisms, since one of the vertical arrows is an isomorphism, so are the remainingvertical ones.(4) Follows from Proposition 2.1.5. (cid:3) Remark 2.2.2.
ERVERSE LERAY FILTRATION AND SPECIALIZATION 17 (1)
We have the following two implications: a) if φ ( G ) = 0 , then G has no constituentson the special fiber (Lemma 1.4.3); b) if G has no constituents on the special fiber,then the morphisms of type δ in (28) are isomorphisms. These two implicationsare combined in Lemma 2.2.1.(3), which is an amplified version of [de-Ma-2018,Proposition 3.1.5] . (2) Both the implication a) and b) directly above, as well the implication in Lemma2.2.1.(3), cannot be reversed. For a), see the counterexample in Remark 1.4.4. For2), as well as for Lemma 2.2.1.(3), consider the following counterexample: take ι to be the embedding of the origin s into a disk ∆ , and G to be j ! Q [1] ∈ P (∆) .Then G admits Q s as a constituent and φG = 0 . On the other hand i ∗ j ! = 0 , sothat the morphisms of type δ ∗ are trivially isomorphisms of zero objects. (3) The fact that G has no constituents supported on Y s does not prevent the followingphenomenon: the number of constituents of t ∗ G may be strictly smaller than thenumber of constituents of i ∗ G . Consider a family X → Y → S , where X/S is afamily of K3 surfaces fibered onto curves, where the general member has irreduciblefibers, while a special member has some reducible fibers. Then G := f ∗ Q will exhibitthis kind of behaviour. I thank Giulia Sacc`a, Antonio Rapagnetta and JunliangShen to help me sort this out. Relative hard Lefschetz and morphisms of type δ . The goal of this section is to prove Proposition 2.3.2 which is a slight generalizationof [Mi-Sh-2018, Corollary 3.8], and yields another criterion, this time involving the Rel-ative Hard Lefschetz Theorem (RHL), for the morphisms of type δ to be isomorphisms.Proposition 2.3.2 is used in the proofs of Theorems 3.3.7 and 3.5.4.Proposition 2.3.2.(2) is the special case of Proposition 2.3.2.(1) when the Cartier divisoris the special fiber in the context of the vanishing/nearby cycle functors. We thus startwith a variety X and a complex F ∈ P ( X ) perverse and semisimple such that it staysperverse and semisimple after restriction and shift by [1] to the pre-image X ′ on X of aCartier divisor Y ′ on Y. This condition on the restriction should be thought as some kindof weakened transversality condition of the divisor Y ′ on Y, with respect to the morphism,and to the semisimple perverse sheaf F. Remark 2.3.1.
The hypotheses of the upcoming Proposition 2.3.2 are met, for example,when the morphism f is projective, the varieties X and X ′ are irreducible orbifolds and F = Q X [dim X ]. Note also that in this section, we do not assume that X, Y , etc., arevarieties over S . Proposition 2.3.2. ( RHL criterion for the morphisms δ being isomorphisms) (1) Let f : X → Y be a projective morphism of varieties. Let ι : Y ′ → Y be aclosed embedding of pure codimension one such that the resulting open embedding Y \ Y ′ → Y is affine and let: X ′ f (cid:15) (cid:15) ι / / X f (cid:15) (cid:15) Y ′ ι / / Y (29) be the resulting Cartesian diagram. Let F ∈ D bc ( X ) be perverse semisimple andsuch that ι ! F [1] ∈ D ( X ′ ) (resp. ι ∗ F [ − ) is perverse semisimple.Then we have the following identities: p τ ≤• (cid:16) f ∗ ι ! F [1] (cid:17) = (cid:16) ι ! p τ ≤• f ∗ F (cid:17) [1] , ( resp. p τ ≤• ( f ∗ ι ∗ F [ − ι ∗ p τ ≤• f ∗ F ) [ − , (30) and similarly if p τ ≤• is replaced by p τ ≥• , or by p H • .The complex f ∗ F has no constituent supported on Y ′ .The morphisms of type δ in (24) associated with f ∗ F are isomorphisms. (2) Assume, in addition, that f is an S -morphism of S -varieties, where S is a non-singular curve, that s ∈ S is a point and that Y ′ = Y s .Then the morphisms of type δ in (24), (27) and (28) associated with f ∗ F areisomorphisms.The complex f ∗ F has no constituent supported on the special fiber Y s .Proof. Since part (2) is a special case of part (1), it is enough to prove part (1). We do soin the case of ι ! F [1]; the case of ι ∗ F [ −
1] can be proved in the same way.We prove the identities (30) for ι ! . By the Decomposition and Relative Hard LefschetzTheorems 1.3.1, we have isomorphisms: f ∗ F ∼ = / / L • p H • ( f ∗ F )[ −• ] , f ∗ ι ! F [1] ∼ = / / L • p H • ( f ∗ ι ! F [1])[ −• ] , (31) p H −• f ∗ F ∼ = p H • f ∗ F, p H −• ( f ∗ ι ! F [1]) ∼ = p H • ( f ∗ ι ! F [1]) . (32)By the base change identity f ∗ ι ! F [1] = ι ! f ∗ F [1], and by the splitting assumptions (31),we get an isomorphism: M k ι ! (cid:16) p H k ( f ∗ F X ) [1] (cid:17) [ − k ] ∼ = M k (cid:16) p H k (cid:16) f ∗ ι ! F [1] (cid:17)(cid:17) [ − k ] . (33)At this juncture, it does not seem a priori clear that the desired conclusion follows bytaking cohomology sheaves on both sides, the reason being that the unshifted summandson the lhs are not immediately seen to be perverse. What it is a priori clear is that, dueto the shift [1], they live in p D [ − , ( Y ′ ) (cf. [BBD, 4.1.10.ii, p.106]). CLAIM : the complex ι ! p H k ( f ∗ F X )[1] is a perverse sheaf, ∀ k . Proof of the CLAIM . We need to show that p H − ( ι ! p H k ( f ∗ F X )[1]) = 0, ∀ k . To simplifythe notation, we set: P kX ′ := p H k ( f ∗ ι ! F [1]) , Q kX := ι ! p H k ( f ∗ F X )[1] ∈ p D [ − , ( W ) , Q k,lX = p H l ( Q kX ) , l = − , . (34)We need to show that Q k, − X = 0, ∀ k .We know that p H − k ∼ = p H k , ∀ k , both for f ∗ F and for f ∗ ι ! F [1], so that we have that: P − kX ′ = P kX ′ , Q − k,lX = Q k,lX , ∀ k, ∀ l = − , . (35)We argue by contradiction and assume that the desired conclusion fails for some k , i.e.that there is k such that Q k, − X = 0.By taking perverse cohomology in (33), we get an isomorphism P kX ′ ∼ = Q k, X ⊕ Q k +1 , − X . (36) ERVERSE LERAY FILTRATION AND SPECIALIZATION 19
Let − k be the smallest index − k such that Q − k, − = 0. By symmetry, k ≥
0. We thushave P − k − X ′ = Q − k − , X ⊕ Q − k , − X = Q − k − , X . By symmetry, P k +1 X ′ = Q k +1 , X On the other hand, we have that P k +1 X ′ = Q k +1 , X ⊕ Q k +2 , − X = Q k +1 , X , because Q k +2 , − X = Q − k − , − X = 0, by the minimality property of − k . This contradiction establishes the CLAIM .At this point, (33), joint with the CLAIM, implies, by taking perverse cohomologysheaves, that (30) holds with “ = ” replaced by an “ ∼ = ”. In order to establish (30) with= (canonical isomorphism), we simply observe that now that we know that the lhs of(33) is a direct sum of shifted perverse sheaves, we reach the desired conclusion by takingperverse cohomology in the isomorphism ι ! f ∗ F [1] = f ∗ ι ! F [1].We have thus proved identities (30). At this point, the two final assertions of (1) canbe seen to be equivalent; we show that they hold as follows.The assertion on the lack of constituents supported on the Cartier divisor Y ′ followsfrom the fact that any such constituent would have to show up as a direct summandsupported on Y ′ and, as such, its ι ! = ι ∗ would show up as a non-zero Q k, − in the proofabove, a contradiction.The last assertion of part (1) follows from Proposition 2.1.5. (cid:3) Remark 2.3.3.
As the proof shows, one may replace the assumption that ι ! F [1] is per-verse semisimple, with the assumption that f ∗ ι ! F [1] satisfies the conclusion of the RHL.Similarly, if we replace ι ! F [1] with ι ∗ F [ − . This assumption is met if the boundary is asimple normal crossing divisor and we take F to be the intersection complex, which, bythe very definition of snc divisor, is necessarily the shifted constant sheaf near the divisor;see Corollary 2.4.2. See also the proof of part II of Theorem 3.5.4, and Remark 4.4.3. RHL and boundary with normal crossing divisors.
Let us place ourselves in the situation of Proposition 2.3.2, except that we do not assumethat ι ∗ F [ −
1] is perverse semisimple on the Cartier divisor on X ′ . The goal of this sectionis to prove Lemma 2.4.1, to the effect that if we are in a simple normal crossing divisorssituation on a nonsingular variety X of dimension n , so that ι ∗ Q X [ n −
1] is not perversesemisimple on X ′ , then we still have the useful RHL symmetry (cf. Remark 2.3.3). Foran application, with details left to the reader, see Remark 4.4.3.Let X be an irreducible variety, let n be its dimension, and let Z ⊆ X be a simplenormal crossing divisor; in particular, X is nonsingular near Z . Let I be a finite setindexing the irreducible components Z i of Z , let I = { i , . . . , i k } ⊆ I be any subset, let Z I := ∩ i ∈ I Z i . We have the standard long exact sequence of sheaves on X :0 / / Q Z / / L | I | =1 Q Z I / / . . . / / L | I | = n Q Z I / / . (37) By splicing (37), and by shifting in an appropriate manner, we get the system of shortexact sequences of perverse sheaves in P ( X ):0 / / / / K n [0] ≃ / / L | I | = n +1 Q Z I [0] / / , / / K n [0] / / K n − [1] / / L | I | = n Q Z I [1] / / / / ,. . . , / / K [ d Z − / / K [ d Z − / / L | I | =2 Q Z I [ d Z − / / / / , / / K [ d Z − / / Q Z [ d Z ] / / L | I | =1 Q Z I [ d Z ] / / / / . (38)Note that Q Z [ d Z ] is not semisimple in general, even when X is nonsingular. Neverthe-less, we have: Lemma 2.4.1.
Let f : X → Y be a proper morphism with X nonsingular and irreducible.Then f ∗ Q Z [ d Z ] has the RHL symmetry, i.e. cupping with the first Chern class L of an f -ample line bundle induces isomorphisms L • : p H −• f ∗ Q Z [ d Z ] ≃ / / p H • f ∗ Q Z [ d Z ] , (39) so that the complex f ∗ Q Z [ d Z ] is t -split (splits as the direct sum of its shifted perversecohomology sheaves). Moreover, we have that: p H • ( ι ∗ p H ⋆ ( f ∗ Q Z [ d Z ])) = 0 , ∀• 6 = − , ∀ ⋆ . (40) Proof.
For the RHL symmetry:We use descending induction on n : start from the bottom of (38); go up one step at thetime; at each step use the five lemma applied to:0 / / p H −• A / / L • (cid:15) (cid:15) p H −• B / / L • (cid:15) (cid:15) p H −• C / / L • (cid:15) (cid:15) / / p H • A / / p H • B / / p H • C / / (cid:3) Corollary 2.4.2.
Let f : X → Y be a projective morphism of varieties. Consider Carte-sian diagram like (29) and assume that X ′ is supported on a simple normal crossingdivisor. Let F = IC X .Then the conclusions of Proposition 2.3.2 hold. ERVERSE LERAY FILTRATION AND SPECIALIZATION 21
Proof.
We simply observe that X is assumed to be nonsingular near X ′ , so that, near X ′ , IC X is the constant sheaf, up to shifts. The shifts are locally constant integers, andthis does not affect the arguments. Now, F is semisimple on X . While i ∗ F [ −
1] is notsemisimple, Lemma 2.4.1 applies, so that the RHL holds, and we can repeat verbatim theproof of Proposition 2.3.2. (cid:3) The specialization morphism as a filtered morphism
In this section, we initiate a systematic discussion of when the specialization morphismis defined and when it is a filtered isomorphism for the perverse Leray filtrations associatedwith a morphism. The case of the perverse filtration is the special case when said morphismis the identity. § sp ∗ ( − ) and frames it in a formalismsuited to handle it together with perverse truncation in the later sections. The special-ization morphism and its filtered counterparts are not defined for an arbitrary complex.The main goal of this paper is to identify conditions that ensure that these morphismsare defined, and that when they are defined they are isomorphisms and, in the filteredcontext, filtered isomorphisms. § P -filtered specialization morphism sp ∗ P ( − ), where P standsfor the perverse filtration. This can be viewed as a special case of the perverse Lerayspecialization morphism seen later in § § P f -filtered spe-cialization morphism sp ∗ P f ( − ). In order to streamline the discussion, we introduce thecommutative diagram (55). We then prove Theorem 3.3.7 which lists criteria for the spe-cialization morphism to be a filtered isomorphism for the perverse Leray filtration; theearlier Propositions 3.1.9 and 3.2.7 are the analogous criteria for the specialization and(perverse) P -filtered specialization morphisms sp ∗ P ( − ) and sp ∗ P f ( − ) . § § § s and at the general point t. We discuss the relation between the various criteria in Remark 3.3.10.We apply these criteria to the Hitchin morphism in § The specialization morphism revisited.
Let things be as in § v : Y → S, a point s ∈ S ,a complex G ∈ D bc ( Y ) . By combining the base change properties in (10) and (12), and by considering themorphisms of type ν in (8), we obtain the commutative diagram of morphisms of functors D bc ( Y ) → D bc ( pt ) (cf. Remark 1.5.4): i ∗ [ − v ∗ σ ∗ / / bc i ∗ v ∗ (cid:15) (cid:15) ψ [ − v ∗ σ ! / / bc ψv ∗ (cid:15) (cid:15) sp ! ( ( PPPPPPPPPPPPP i ! [1] v ∗ = bc i ! v ∗ (cid:15) (cid:15) ( φv ∗ ) v ∗ i ∗ [ − σ ∗ / / ? sp ∗ ? ♠ ♠ ♠ ♠ ♠ ν = ν v ∗ ψ [ − σ ! / / v ∗ i ! [1] O O ( v ∗ φ ) , (42)where: there is a ν for each of the two rows, i.e. ν and ν , but we have indicated only theone for the second row; the terms φv ∗ and v ∗ φ in parentheses are, up to shift, the conesof the morphisms of type σ (cf. Remark 3.1.1).We have the functors ν, sp ! : D bc ( Y ) → D bc ( pt ). Due to the directions of the arrows onthe l.h.s. of (42), the broken arrow “? sp ∗ ?” is not defined as a morphism of functors, but,as it is discussed below, it can happen to be defined for some G ∈ D bc ( Y ). Remark 3.1.1. ( Cones in (42) ) By (6), the cones of the morphisms σ ∗ and σ ! on thetop row of (42) are φv ∗ and φv ∗ [1] , respectively. Then, for G ∈ D bc ( Y ) , the morphism σ ∗ ( v ∗ G ) is an isomorphism, if and only the morphism σ ! ( v ∗ G ) is an isomorphism, if andonly if φv ∗ G = 0 . What above remains true for the bottom row, with v ∗ φ replacing φv ∗ , etc. Remark 3.1.2. ( Interrelations between vanishings ) In general, the two conditions v ∗ φG = 0 and φv ∗ G = 0 are independent. We have the following implications: ( φG = 0) = ⇒ ( v ∗ φG = 0); v proper = ⇒ (( v ∗ φG = 0) ⇐⇒ ( φv ∗ G = 0)) . Since in general the base change morphism bc i ∗ v ∗ is not an isomorphism, the sought-aftermorphisms ? sp ∗ ? of functors D bc ( Y ) → D bc ( pt ) is not defined. Definition 3.1.3. ( The specialization morphism sp ∗ ( G )) Let G ∈ D bc ( Y ) . We saythat the specialization morphism sp ∗ ( G ) is defined for G if the base change morphism bc i ∗ v ∗ ( G ) is an isomorphism. In this case, we define the morphism in D bc ( pt ) : sp ∗ ( G ) := σ ∗ ( bc i ∗ v ∗ ) − : R Γ( Y s , i ∗ G ) = v ∗ i ∗ G / / ψv ∗ G = R Γ( s, ψv ∗ G ) . (43)When sp ∗ ( G ) is defined we have that: ν ( G ) = sp ! ( G ) ◦ sp ∗ ( G ) . (44) Remark 3.1.4.
The part of diagram (42) that is needed to define specialization morphismsis: v ∗ i ∗ [ − i ∗ [ − v ∗ σ ∗ / / bc i ∗ v ∗ o o ψ [ − v ∗ . (45) The harmless shift by [ − is convenient later on, when dealing with the morphisms of type δ ; see diagram (49). The reason for having embedded (45) in (42) is explained in Remark3.1.10. Remark 3.1.5.
ERVERSE LERAY FILTRATION AND SPECIALIZATION 23 (1)
By using (17), we may re-write (43) as follows: sp ∗ ( G ) : R Γ( Y s , i ∗ G ) / / ψv ∗ G ∼ / / R Γ( Y t , t ∗ G ) . (46) The ambiguity involved in this re-writing (see Fact 1.5.5, especially (17)) plays norole in this paper: firstly, the image lands in the monodromy invariants; secondly,we have limited ourselves to criteria for when the specialization morphism is: de-fined; an isomorphism; filtered for the perverse filtrations; a filtered isomorphismfor the perverse filtrations. All of these issues can be settled at a point t generalfor G, independently of the monodromy action at t . (2) Even when sp ∗ ( G ) is defined, I do not see a reason for the base change morphism bc ψv ∗ : ( R Γ( Y t , t ∗ G ) ≃ ) ψv ∗ G / / v ∗ ψG = R Γ( Y s , ψG ) (47) to be necessarily an isomorphism. (3) In view of the isomorphism ψv ∗ G ≃ R Γ( Y t , t ∗ G ) , typically, and this paper is noexception, we are interested in ψv ∗ G , and not as much in v ∗ ψG . This is importantto keep in mind when, later we consider the filtered version of the specializationmorphisms see (49), (50) and (53) for the perverse version, and see (55), (56)and (59) for the perverse Leray version. Of course, the base change morphism bc ψv ∗ : ψv ∗ → v ∗ ψ is an isomorphism when v is proper. Remark 3.1.6. ( v proper ) If v : Y → S is proper, then, by the proper base changetheorem, we get the functor: sp ∗ : D bc ( Y ) → D ( pt ) . Remark 3.1.7. ( The morphism of functors sp ! : ψ [ − v ∗ → v ∗ i ! [1]) Regardless ofwhether v is proper or not, the functor sp ! : D bc ( Y ) → D bc ( pt ) is defined. In analogy with(46), and taking into account the definition of cohomology with supports on the closed set Y s , for a given G ∈ D bc ( Y ) , it takes the form of a morphism: sp ! ( G ) : R Γ( Y t , t ∗ G ) ≃ / / ψv ∗ G / / R Γ( Y s , i ! [2] G ) = R Γ Y s ( Y, G [2]) . (48) Most of the results proved in this paper for the morphism sp ∗ ( G ) (when defined), hold forthe morphism sp ! . In fact, the proofs are simpler in this case, since we do not need toestablish the existence of sp ! along the way, the way we must do for a sp ∗ ( G ) . On the otherhand, in general, the r.h.s. of (48) is not easily relatable to, say, R Γ( Y s , i ∗ G ) . For thisreason, we have not included in this paper the sp ! -versions of the results. We simply observethat when, for some necessarily special G , the natural morphism ν : i ∗ [ − G ) → i ! [1]( G ) is an isomorphism, then, we do have that sp ! ( G ) : R Γ( Y t , t ∗ G ) → R Γ( Y s , i ∗ G ) and thiscan be a very useful tool, especially, when sp ∗ ( G ) fails to be defined. This proved to becrucial in the paper [de-Ma-Sh-2020] (where due to the particular set-up there, we optedfor a more direct construction of the morphism sp ! ). Remark 3.1.8.
In what follows, “( ⇐ ( φG = 0) )” is short for “(which is implied by ( φG = 0) )”. Proposition 3.1.9. ( Criteria for the existence of sp ∗ ( G ) and for it being anisomorphism ) Let G ∈ D bc ( Y ) . Recall Remark 3.1.8. (1) If v is proper, then sp ∗ : D bc ( Y ) → D ( pt ) is defined as a functor.If in addition φv ∗ G = 0 ( ⇐ ( φG = 0) ), then sp ∗ ( G ) is an isomorphism. (2) If φv ∗ G = 0 and φG = 0 , then sp ∗ ( G ) is defined and an isomorphism.Proof. (1). For the first assertion, see Remark 3.1.6. For the second one, we are left withshowing that the morphism σ ∗ ( v ∗ G ) on the top row of (42) is an isomorphism. This followsfrom the fact that its cone φv ∗ G = 0.(2). The hypotheses imply that the cones of the four horizontal arrows in (42) are zero.These four arrows are then all isomorphisms. Since one of the three vertical arrows in (42)is an isomorphism, so are the remaining two, and we are done. (cid:3) Remark 3.1.10. ( Vanishing of φ and base change ) In order to define the specializa-tion morphism, we only need the l.h.s. of diagram (42). The proof of Proposition 3.1.9.(2)shows that the two conditions φG = 0 and φv ∗ G = 0 combined, imply that the base changemorphism bc i ∗ v ∗ ( G ) for G is an isomorphism. This is the reason why we have introduceddiagram (42): if, after having plugged some G ∈ D bc ( Y ) , the four horizontal arrows in(42) are isomorphisms, then, because of the presence of base change isomorphism bc i ! v ∗ ,the three vertical arrows evaluated at G are also isomorphism, i.e. the base change mor-phism bc i ∗ v ∗ is also an isomorphism, and the specialization morphism sp ∗ ( G ) is defined. The specialization morphism and the perverse filtration.
Let things be as in § v : Y → S, a point s ∈ S ,a complex G ∈ D bc ( Y ) . By analogy to (42), and by using the naturality properties of the morphisms of type δ in §
2, we obtain the commutative diagram of morphism of systems of functors: i ∗ [ − v ∗ p τ ≤• σ ∗≤• / / bc i ∗ v ∗≤• (cid:15) (cid:15) ψ [ − v ∗ p τ ≤• σ ! ≤• / / bc ψv ∗ (cid:15) (cid:15) sp ! ≤• ●●●●●●●●●●●●●●●●●●●●●●● i ! [1] v ∗ p τ ≤• = bc i ! v ∗≤• (cid:15) (cid:15) ( φv ∗ p τ ≤• ) v ∗ i ∗ [ − p τ ≤• σ ∗≤• / / v ∗ ψ [ − p τ ≤• σ ! ≤• / / δ ψ ≤• = (cid:15) (cid:15) v ∗ i ! [1] p τ ≤• δ ! ≤• (cid:15) (cid:15) O O ( v ∗ φ p τ ≤• ) v ∗ p τ ≤• i ∗ [ − σ ∗≤• / / δ ∗≤• O O ? sp ∗≤• ? : : ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ν ≤• = ν ≤• v ∗ p τ ≤• ψ [ − σ ! ≤• / / O O v ∗ p τ ≤• i ! [1] , ( φ ) . (49) Remark 3.2.1. ( Cones/not cones in (49) ) In the top two rows of diagram (49), wehave indicated in parentheses the cones -up to shift- of the morphisms of type σ : apply(42) and Remark 3.1.1 to p τ ≤• . Since truncation is not an exact functor, the cones of themorphisms of type σ in the third row are not shifts of v ∗ p τ ≤• φ ; the term in parentheseson the third bottom row is a term whose vanishing when evaluated at G ensures that thearrows of type σ ( G ) on the third row are isomorphisms. The reason why (49) has three rows, instead of the two in (cf. 42), is that p τ ≤• doesnot commute with i ∗ [ −
1] -nor with any other shift-; the morphisms of type δ measure thefailure of this commutativity. ERVERSE LERAY FILTRATION AND SPECIALIZATION 25
Remark 3.2.2. ( Interrelations between vanishings ) The conditions φv ∗ p τ ≤• G = 0 and v ∗ φ p τ ≤• G = 0 are independent; if v is proper, then they are equivalent. By the t -exactness of φ and the additivity of v ∗ , we have that: v ∗ φ p τ ≤• = v ∗ p τ ≤• φ , and that: ( φG = 0) ⇐⇒ ( φ p τ ≤• G = 0) = ⇒ v ∗ φ p τ ≤• G = 0 . Since in general the base change morphism bc i ∗ v ∗ ≤• are not isomorphisms, the sought-aftermorphism of systems of functors ? sp ∗≤• ? is not defined. On the other hand, there can bea corresponding arrow for special G ∈ D bc ( Y ). Recall the notion of “systems” (cf. § Definition 3.2.3. ( The P -filtered specialization morphism sp ∗ P ( G )) Let G ∈ D bc ( Y ) .We say that the P -filtered specialization morphism sp ∗ P ( G ) is defined for G if the basechange morphisms bc i ∗ v ∗ ≤• ( G ) are isomorphisms, in which case, since the inverse isomor-phisms form an isomorphism of systems, we can define the arrow in DF ( pt ) : ( bc i ∗ v ∗ P denotes the evident resulting filtered isomorphism, and δ P the evident resulting filteredmorphism) sp ∗ P ( G ) := σ ∗ P ( bc i ∗ v ∗ P ) − δ ∗ P : v ∗ i ∗ G = ( R Γ( Y s , i ∗ G ) , P ) / / ( ψv ∗ G, P ) , (50) where the last term is the one associated with the system ψv ∗ p τ ≤• G . We have functors ν P , sp ! P : D bc ( Y ) → DF ( pt ). When sp ∗ P ( G ) is defined, we have that: ν P ( G ) = sp ! P ( G ) ◦ sp ∗ P ( G ) . (51) Remark 3.2.4.
Remark 3.1.4 holds essentially verbatim in the context of diagram (49).In particular, we have: v ∗ p τ ≤• i ∗ [ − δ ∗≤• / / v ∗ i ∗ [ − p τ ≤• i ∗ [ − v ∗ p τ ≤• σ ∗ / / bc i ∗ v ∗ o o ψ [ − v ∗ p τ ≤• . (52) Remark 3.2.5. (1)
By using (18), the filtered analogue of (43) takes the following form: sp ∗ P ( G ) : ( R Γ( Y s , i ∗ G ) , P ) / / ( ψv ∗ G, P ) ∼ / / ( R Γ( Y t , t ∗ G ) , P ) . (53) The analogue of Remark 3.1.5 on the ambiguities due to monodromy holds in thiscontext. (2)
Even when sp ∗ ( G ) is defined, the base change morphism: bc ψv ∗ : ( R Γ( Y t , t ∗ G ) ≃ ) ψv ∗ G / / v ∗ ψG = R Γ( Y s , ψG ) (54) is not necessarily an isomorphism. In particular, the filtration P in ( ψv ∗ G, P ) isnot the perverse filtration for a complex on Y s . Remark 3.2.6. ( v proper ) The evident analogue of Remark 3.1.6 holds in this perversefiltered context: if v : Y → S is proper, then, by the proper base change theorem, we getthe functor: sp ∗ P : D bc ( Y ) → DF ( pt ) . Proposition 3.2.7 below is the P -filtered counterpart to the un-filtered Proposition 3.1.9. Proposition 3.2.7. ( Criteria for the existence of sp ∗ P ( G ) and for it being a filteredisomorphism )(1) If v is proper, then sp ∗ P : D bc ( Y ) → DF ( pt ) is defined as a functor.If in addition φG = 0 , then sp ∗ P ( G ) is a filtered isomorphism. (2) If φv ∗ p τ ≤• G = 0 and v ∗ φ p τ ≤• G = 0 , then sp ∗ P ( G ) is defined.If in addition φG = 0 , then sp ∗ P ( G ) is a filtered isomorphism. (3) If G is semisimple, φv ∗ G = 0 and v ∗ φG = 0 , then sp ∗ P ( G ) is defined.If in addition φG = 0 , then sp ∗ P ( G ) is a filtered isomorphism.Proof. (1). For the first assertion, see Remark 3.2.6. If we assume that φG = 0, then, inview of Remarks 3.2.1 and 3.2.2, all horizontal arrows in the commutative diagram (49)are isomorphisms. Since each row of vertical arrows contains an isomorphisms, all arrowsin (49) are isomorphisms, so that sp ∗ P is a filtered isomorphism.(2). The two vanishing assumptions imply that all the four horizontal arrows in thefirst two rows of (49) are isomorphisms. As seen above, this implies that the vertical basechange arrows are isomorphisms as well, so that sp ∗ P is defined. If we we assume that φG = 0, then we conclude as above that sp ∗ P is a filtered isomorphism.(3). Since G is semisimple, we have that: φv ∗ G = 0 is equivalent to φv ∗ p τ ≤• G = 0; v ∗ φG = 0 is equivalent to v ∗ φ p τ ≤• G = 0, Now (3) follows from (2). (cid:3) Remark 3.2.8. ( Vanishing of φ and base change ) The analogue of Remark 3.1.10,on the vanishing of φ assumptions implying base change isomorphisms, holds in the contextof the proof of Proposition 3.2.7. Question 3.2.9.
I do not know of an example where: v is proper -so that sp ∗ and sp ∗ P are defined-, where φv ∗ G = 0 -so that sp ( G ) is an isomorphism-, but where sp P ( G ) is notan isomorphism (i.e. in the filtered sense). The issue is that while δ ∗≤• ( G ) in (49) is anisomorphism for • ≫ , it is not clear to me what happens for other values of • . We knowthat if in addition φG = 0 , then sp ∗ P is an isomorphism. The specialization morphism and the perverse Leray filtration.
Let things be as in § f : X → Y and v : Y → S, a point s ∈ S , a complex G := f ∗ F ∈ D bc ( Y ) for F ∈ D bc ( X ). ERVERSE LERAY FILTRATION AND SPECIALIZATION 27
By analogy to (42) and (49), and by using the naturality properties of the morphismsof type δ in §
2, we obtain the commutative diagram of morphisms of systems of functors: i ∗ [ − v ∗ p τ ≤• f ∗ σ ∗≤• / / bc i ∗ v ∗≤• (cid:15) (cid:15) ψ [ − v ∗ p τ ≤• f ∗ (cid:15) (cid:15) σ ! ≤• / / sp ! Pf , ≤• (cid:27) (cid:27) ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ i ! [1] v ∗ p τ ≤• f ∗ = bc i ! v ∗≤• (cid:15) (cid:15) (( φv ∗ p τ ≤• f ∗ F = 0) ⇔ σ isos) v ∗ i ∗ [ − p τ ≤• f ∗ σ ∗≤• / / v ∗ ψ [ − p τ ≤• f ∗ σ ! ≤• / / = (cid:15) (cid:15) v ∗ i ! [1] p τ ≤• f ∗ δ ! ≤• (cid:15) (cid:15) O O (( v ∗ φ p τ ≤• f ∗ F = 0) ⇔ σ isos) v ∗ p τ ≤• i ∗ [ − f ∗ σ ∗≤• / / δ ∗≤• O O bc i ∗ f ∗≤• (cid:15) (cid:15) v ∗ p τ ≤• ψ [ − f ∗ σ ! ≤• / / = O O ′ (cid:15) (cid:15) v ∗ p τ ≤• i ! [1] f ∗ bc i ! f ∗≤• = (cid:15) (cid:15) (( φf ∗ F = 0) ⇒ σ isos) v ∗ p τ ≤• f ∗ i ∗ [ − σ ∗≤• / / ? sp ∗ Pf , ≤• ? C C ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ν ≤• = ν ≤• v ∗ p τ ≤• f ∗ ψ [ − σ ! ≤• / / v ∗ p τ ≤• f ∗ i ! [1] O O (( f ∗ φF = 0) ⇒ σ isos) . (55) Remark 3.3.1. ( Cones/not cones in (55) ) This remark is analogous to Remark 3.2.1.The cones of the morphisms of type σ appearing on the first two rows are indicated inparentheses on the r.h.s. The terms in parentheses on the third and fourth row are not thecones of the morphisms of type σ (truncation is not an exact functor), but their vanishingimplies that the morphisms of type σ are isomorphisms. Remark 3.3.2. ( Interrelations between vanishings ) The conditions φF = 0 and φf ∗ F = 0 are independent; if f is proper, then φF = 0 implies φf ∗ F = f ∗ φF = 0 . Notethat φf ∗ F = 0 is equivalent to φ p τ ≤• f ∗ F = 0 , and it implies v ∗ φ p τ ≤• f ∗ F = 0 .The conditions v ∗ φ p τ ≤• f ∗ F = 0 and φv ∗ p τ ≤• f ∗ F = 0 are independent; if v is proper,then they are equivalent. Definition 3.3.3. ( The P f -filtered specialization morphism sp ∗ P f ( F )) We say thatthe P f -filtered specialization morphism sp ∗ P f ( F ) is defined for F ∈ D bc ( X ) if the basechange morphisms bc i ∗ v ∗ ≤• ( F ) and bc i ∗ f ∗ ≤• ( F ) are isomorphisms. In this case, since the in-verse isomorphisms form a morphism of systems, we can define the arrow in DF ( pt ) : sp ∗ P f ( F ) := σ ∗ P ( bc i ∗ v ∗ P ) − δ ∗ P f ( bc i ∗ f ∗ P ) − : v ∗ i ∗ G = ( R Γ( X s , i ∗ F ) , P f ) / / ( ψv ∗ f ∗ F, P ) , (56) where the last term is the one associated with the system ψv ∗ p τ ≤• f ∗ F. We have functors ν P f , sp ! P f : D bc ( X ) → DF ( pt ). When sp ∗ P f ( F ) is defined, we havethat: ν P f ( F ) = sp ! P f ( F ) ◦ sp ∗ P f ( F ) . (57) Remark 3.3.4.
Remark 3.1.4 holds essentially verbatim in the context of diagram (55).In particular, we have: v p τ ≤• f ∗ i ∗ [ − v ∗ p τ ≤• i ∗ [ − f ∗ δ ∗≤• / / bc i ∗ f ∗≤• o o v ∗ i ∗ [ − p τ ≤• f ∗ i ∗ [ − v ∗ p τ ≤• f ∗ σ ∗ (cid:15) (cid:15) bc i ∗ v ∗ o o ψ [ − v ∗ p τ ≤• f ∗ . (58) Remark 3.3.5.
By using (19), we may re-write (56) as follows: sp ∗ P f ( F ) : ( R Γ( X s , i ∗ F ) , P f s ) / / ( ψv ∗ f ∗ F, P ) ∼ / / ( R Γ( X t , t ∗ F ) , P f t ) . (59) The analogue of Remark 3.2.5, on the ambiguities due to monodromy and on the meaningof P in ( ψv ∗ f ∗ F, P ) , holds in this context. Remark 3.3.6. ( v proper ) The evident analogue of Remarks 3.1.6 and 3.2.6 holdsin this perverse Leray filtered context: if f and v are proper, then we get the functor: sp ∗ P f : D bc ( X ) → DF ( pt ) . The following theorem is a perverse Leray analogue of Proposition 3.2.7. Recall thenotational Remark 3.1.8.
Theorem 3.3.7. ( Criteria for the existence of sp ∗ P f ( F ) and for it being a filteredisomorphism )(1) If v and f are proper, then sp ∗ P f : D bc ( X ) → DF ( pt ) is a functor.If in addition φf ∗ F = 0 ( ⇐ ( φF = 0) ) then sp ∗ P f ( F ) is defined and a filteredisomorphism. (2) If v proper, φf ∗ F = 0 , and f ∗ φF = 0 ( ⇐ ( φF = 0) ), then sp ∗ P f ( F ) is defined anda filtered isomorphism. (3) If f is proper, φv ∗ p τ ≤• f ∗ F = 0 , and v ∗ φ p τ ≤• f ∗ F = 0 ( ⇐ ( φf ∗ F = 0) ⇐ ( φF = 0) ),then sp ∗ P f ( F ) is defined.If in addition φf ∗ F = 0 ( ⇐ ( φF = 0) ), then sp ∗ P f ( F ) is defined and a filteredisomorphism. (4) If φv ∗ p τ ≤• f ∗ F = 0 , φf ∗ F = 0 , and f ∗ φF = 0 ( ⇐ ( φF = 0) ), then sp ∗ P f ( F ) isdefined and a filtered isomorphism. (5) If f is proper and F is semisimple, φv ∗ f ∗ F = 0 and v ∗ φf ∗ F = 0 ( ⇐ ( φf ∗ F =0) ⇐ ( φF = 0) ), then sp ∗ P f ( F ) is defined.If in addition, φf ∗ F = 0 ( ⇐ ( φF = 0) ), then sp ∗ P f ( F ) is a filtered isomorphism. (6) If v is proper, f is projective and F and i ∗ F [ − are semisimple, and φv ∗ f ∗ F = 0 ( ⇐ ( φf ∗ F = 0) ⇐ ( φF = 0) ), then sp ∗ P f ( F ) is defined and a filtered isomorphism. (7) If f is projective and F and i ∗ F [ − are semisimple, φv ∗ f ∗ F = 0 and v ∗ φf ∗ F = 0 ( ⇐ ( φf ∗ F = 0) ⇐ ( φF = 0) , then sp ∗ P f ( F ) is defined and a filtered isomorphism.Proof. (1). For the first assertion in (1) , see Remark 3.3.6. We now prove the secondassertion in (1). Keep in mind that the base change morphisms are all isomorphisms.Since f is proper, φf ∗ F = 0 implies that -it is equivalent to- f ∗ φF = 0, as well as v ∗ φ p τ ≤• f ∗ F = 0. It follows that the four horizontal arrows on row two and three in (55) ERVERSE LERAY FILTRATION AND SPECIALIZATION 29 are isomorphisms. This implies that so are the vertical arrows of type δ . It remainsto show that the morphisms σ ∗≤• are isomorphisms, which follows from φv ∗ p τ ≤• f ∗ F = v ∗ φ p τ ≤• f ∗ F = 0.(2). Since v is proper, rows one and two are identified. The vanishing assumptionsimply that all seven arrows on rows three and four are isomorphisms. In particular,the six base change morphisms in (55) are isomorphisms, so that sp ∗ P f is defined. Theassumption φf ∗ F = 0 implies that the cones of the four horizontal arrows in rows one andtwo are isomorphisms, so that all seven arrows on these two rows are isomorphisms. Thesame argument shows that the seven arrows in rows two and three are isomorphisms. Inparticular, the morphisms of type δ and the morphisms σ ∗≤• are isomorphisms, and theconclusion follows.(3). We prove the first assertion in (3). The base change morphisms for f are isomor-phisms. The vanishing assumptions imply that we can identify the first two rows, so that sp ∗ P f is defined. We prove the second assertion in (3). The hypothesis φf ∗ F = 0 impliesthat the morphisms of type δ are isomorphisms, and we are done.(4). The vanishing assumptions imply that all horizontal arrows in (55) are isomor-phisms. It follows that all arrows in (55) are isomorphisms and the conclusion follows.(5). Since f is proper and F is semisimple, f ∗ F is semisimple by the DecompositionTheorem 1.3.1, so that p τ ≤• f ∗ F is a direct summand of f ∗ F . It follows that: φv ∗ f ∗ F = 0is equivalent to φv ∗ p τ ≤• f ∗ F = 0; v ∗ φf ∗ F = 0 is equivalent to v ∗ φ p τ ≤• f ∗ F = 0. This showsthat (5) follows from (3).(6). Since v and f are proper, sp ∗ P f is defined. The semisimplicity assumptions on F andon i ∗ F [ −
1] and the projectivity of f imply, via Proposition 2.3.2, that the morphisms oftype δ are isomorphisms. It remains to observe that the assumption φv ∗ f ∗ F = 0, coupledwith the semisimplicity of f ∗ F , implies, as seen in the proof of (5), that φv ∗ p τ ≤• f ∗ F = 0.It follows that σ ∗ is an isomorphisms and we are done.(7). Since f is proper, the corresponding base change morphisms are isomorphisms. Asseen in the proof of (6), the morphisms of type δ are isomorphisms. As seen in the proofof (5), the vanishing assumptions imply that φv ∗ p τ ≤• f ∗ F = 0 and v ∗ φ p τ ≤• f ∗ F = 0. As inthe proof of (3), the base change morphisms for v are isomorphisms and, as in the proofof (1), the morphisms σ ∗≤• are isomorphisms. The proof of (7) is complete. (cid:3) Remark 3.3.8. ( Vanishing of φ and base change ) The analogue of Remark 3.1.10holds in the context of the proof of Theorem 3.3.7.
Remark 3.3.9.
Recall that a condition of type φv ∗ ( − ) = 0 means that v ∗ ( − ) has locallyconstant cohomology sheaves R • v ∗ ( − ) on S near s . Remark 3.3.10. (1) [de-Ma-2018, Theorem 3.2.1 parts (i,ii,ii)] are implied by the slightly more preciseTheorem 3.3.7 parts (1,3,5), respectively. (2)
Proposition 3.2.7 parts (1,2,3) are also the special cases of Theorem 3.3.7 parts(1,3,5) when one takes f : X → Y to be the identity. The specialization morphism via a compactification.
Consider Theorem 3.3.7: parts (3,4,5,7) do not assume that v = v X : X → S isproper, but make some local constancy assumptions for certain direct images to S , namely φv ∗ ( − ) = 0.In this subsection we provide, in the context of a proper S -morphism f : X → Y andof a non proper morphisms v X : X → S , sufficient conditions ensuring that we obtainwell-defined sp ∗ P ( G ) or sp ∗ P f ( F ), but that do not require local constancy assumptions. Onthe other hand, they require the existence of good compactification. As we shall see, here“good” is made precise by conditions relating vanishing cycles at the boundary. In general,it may be difficult to achieve such a vanishing. See Remark 3.4.5 for one situation in whichthis is possible. They are also achieved in the compactification of Dolbeault moduli spacesconstructed in [de-2018]; see § S -morphism f : X o → Y o , and we do not assume that v = v Y o : Y o → S is proper.In order to circumvent base change issues, it is natural to first compactify the picture.We start with f : X o → Y o and we get, by Nagata’s completion theorems: Zariski openand dense embeddings X o ⊆ X and Y o ⊆ Y ; a proper morphism f : X → Y extending f : X o → Y o ; a proper morphism v Y : Y → S extending v Y o : Y o → S. By blowing-up Y, if necessary, we may assume that W := Y \ Y o supports an effective Cartier divisor.It follows that we can place ourselves in the following: Set-up 3.4.1.
Let S be a variety. Consider a Cartesian diagram of S -morphisms, with ( a : W → Y ← Y o : b ) closed/open complementary immersions: Z a / / f (cid:15) (cid:15) X f (cid:15) (cid:15) X ob o o f (cid:15) (cid:15) W a / / Y Y ob o o (60) such that: all morphisms f are proper; the varieties Y and W proper over S ; the boundary W on Y supports an effective Cartier divisor. In particular, the functors b ! , b ∗ : Y o → Y are t -exact. Many of us are used to denote the pair of closed/open embeddings ( a, b ) by ( i, j ).However, in this paper, and in large part of the vanishing cycle literature, the closedembedding of the special fiber is systematically denoted by i .We denote by ad the distinguished triangle of endofunctors of D bc ( Y ):ad := a ∗ a ! a / / Id b / / b ∗ b ∗ / / /o/o/o ; (61)by plugging G ∈ D bc ( Y ) in (61), we obtain the distinguished triangle ad( G ) in D bc ( Y ) , functorial in G .Proposition 3.4.2 below is an “ f -proper, but v X -non-proper” analogue to Theorem3.3.7.(1), where it was assumed that f and v are proper and that φF = 0. Recall Remark3.3.5, which allows us to use a general point t to express the target of specializationmorphisms. ERVERSE LERAY FILTRATION AND SPECIALIZATION 31
Note that the assumptions (B) in Proposition 3.4.2 imply the assumptions of Theorem3.3.7.(5) which lead to sp ∗ P f being an isomorphism. We have included (B) for completenessin the context of this subsection. Recall the notational Remark 3.1.8. Proposition 3.4.2.
Assume we are in the Set-up . Assume that S is a nonsingularand connected curve and let s ∈ S be a point. Let F ∈ D bc ( X ) .Consider the following two sets of conditions: (A) φb ∗ f ∗ b ∗ F = 0 ( ⇐ ( φb ∗ b ∗ F = 0) ), and φf ∗ b ∗ F = 0 ( ⇐ ( φb ∗ F = 0) ⇐ ( φb ∗ F =0) ⇐ ( φF = 0) . (B) F semisimple, φv ∗ b ∗ f ∗ b ∗ F = 0 ( ⇐ ( φf ∗ b ∗ b ∗ F = 0) ⇐ ( φb ∗ b ∗ F = 0) ), and φf ∗ b ∗ F = 0 ( ⇐ ( φb ∗ F = 0) ⇐ ( φF = 0) ).If either conditions (A) or (B) are met, then the P f -filtered specialization morphism: sp ∗ P f ( F ) : ( R Γ( X os , i ∗ b ∗ F ) , P f s ) / / ( ψv ∗ f ∗ i ∗ F, P ) ≃ ( R Γ( X ot , t ∗ b ∗ F ) , P f t ) , (62) where t ∈ S is general for F with respect to X/S and for f ∗ F with respect to Y /S , isdefined and it is a filtered isomorphism for the perverse Leray filtrations.If f : X → Y is the identity, then the same conclusion holds with sp ∗ P f ( F ) = sp ∗ P ( F ) . Proof.
Since the proof is analogous to the proof of Theorem 3.3.7.(1), we only indicate themain line of the arguments.In complete analogy with the formation of the commutative diagram (55), we form thecommutative diagram: (with many decorations omitted) i ∗ [ − v ∗ b ∗ p τ ≤• f ∗ b ∗ σ ∗ / / bc (cid:15) (cid:15) ψ [ − v ∗ b ∗ p τ ≤• f ∗ b ∗ bc (cid:15) (cid:15) σ ! / / i ! [1] v ∗ b ∗ p τ ≤• f ∗ b ∗ = bc (cid:15) (cid:15) ( φv ∗ b ∗ p τ ≤• f ∗ b ∗ F ) v ∗ i ∗ [ − b ∗ p τ ≤• f ∗ b ∗ σ ∗ / / bc i ∗ b ∗ (cid:15) (cid:15) v ∗ ψ [ − b ∗ p τ ≤• f ∗ b ∗ σ ! / / bc (cid:15) (cid:15) v ∗ i ! [1] b ∗ p τ ≤• f ∗ b ∗ bc = (cid:15) (cid:15) O O ( v ∗ φb ∗ p τ ≤• f ∗ b ∗ F ) v ∗ b ∗ i ∗ [ − p τ ≤• f ∗ b ∗ σ ∗ / / v ∗ b ∗ ψ [ − p τ ≤• f ∗ b ∗ σ ! / / = (cid:15) (cid:15) v ∗ b ∗ i ! [1] p τ ≤• f ∗ b ∗ δ ! ≤• (cid:15) (cid:15) O O ( φf ∗ b ∗ F ) v ∗ b ∗ p τ ≤• i ∗ [ − f ∗ b ∗ σ ∗ / / δ ∗≤• O O bc (cid:15) (cid:15) v ∗ b ∗ p τ ≤• ψ [ − f ∗ b ∗ σ ! / / bc (cid:15) (cid:15) O O v ∗ b ∗ p τ ≤• i ! [1] f ∗ b ∗ bc = (cid:15) (cid:15) ( φf ∗ b ∗ F ) ,v ∗ b ∗ p τ ≤• f ∗ i ∗ [ − b ∗ σ ∗ / / D D ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ v ∗ b ∗ p τ ≤• f ∗ ψ [ − b ∗ σ ! / / v ∗ b ∗ p τ ≤• f ∗ i ! [1] b ∗ O O ( φb ∗ F ) , (63)the properties of which are similar to the ones of (55). The second row from the top in(63) does not have a counterpart in (55), and this is because here we need to considerthe base change morphisms for i ∗ b ∗ . The terms in parentheses for the first three rows are,up to shift, the cones of the arrows of type σ . After having plugged an F ∈ D bc ( X ) in(63), the vanishing of the term in parentheses implies that the corresponding morphismsof type σ are isomorphisms. Since v : X → S is proper, the first two rows get identified by proper base change.Since f is proper, the same is true for the last two.Since every row contains a vertical arrow which is an isomorphism, we only need toshow that all horizontal arrows in (63) are isomorphisms. It is enough to show that thecorresponding terms in parentheses vanish. By the identifications above, we need to do soonly for rows two, three and four.The assumption φb ∗ F = 0, which is common to (A) and (B), together with the proper-ness of f , implies the vanishing of the terms in parentheses for the bottom three rows (forthe third row, one uses that φ commutes with perverse truncation).We are left with proving that the morphisms of type σ on row two are isomorphisms.We prove that (A) implies the desired conclusion. The assumption φb ∗ b ∗ F = 0 impliesthe vanishing of the terms in parentheses for the second row because of what follows:0 = f ∗ φb ∗ b ∗ F = φf ∗ b ∗ b ∗ F = φb ∗ f ∗ b ∗ F = φb ∗ b ∗ f ∗ F, so that: 0 = p τ ≤• φb ∗ b ∗ f ∗ F = φ p τ ≤• b ∗ b ∗ f ∗ F = φb ∗ p τ ≤• b ∗ f ∗ F = φb ∗ p τ ≤• f ∗ b ∗ F, where in the identities above we have used proper base change for f , the fact that f b = bf in the r.h.s. of (60), the t -exactness of b ∗ ( b is affine and quasi-finite) and of φ .(B) has already been proved in Theorem 3.3.7.(5). (cid:3) Remark 3.4.3.
The proof of Proposition 3.4.2 shows that under its hypotheses, (A) or(B), the base change morphism bc i ∗ b ∗ : v ∗ i ∗ b ∗ p τ ≤• f ∗ b ∗ F → v ∗ b ∗ i ∗ p τ ≤• f ∗ b ∗ F in (65) is anisomorphism. Remark 3.4.4.
Remark 3.1.4, on the part of (55) that is needed to define sp ∗ P f ( F ) , holdsessentially verbatim in the context of diagram (63). Remark 3.4.5.
The conditions φb ∗ b ∗ F = 0 = φb ∗ F in Proposition 3.4.2 are met if, forexample, X/S is smooth, X \ X o is a simple normal crossing divisor over S and F haslocally constant cohomology sheaves. See [De-1972, XIII, Lemme 2.1.11 (dualized)] . Wedo not know of a weaker, but similar set of conditions that leads to sp ∗ P f being defined, butnot necessarily an isomorphisms. Corollary 3.4.6.
Let things be as in Proposition 3.4.2. Then we have the followingcommutative diagram, with horizontal arrows given by restriction, and with vertical arrowsgiven by the well-defined P f -filtered specialization morphisms: ( R Γ( X t , t ∗ F ) , P f t ) / / ( R Γ( X ot , t ∗ b ∗ F ) , P f t )( R Γ( X s , i ∗ F ) , P f s ) / / sp ∗ Pf O O ( R Γ( X os , i ∗ b ∗ F ) , P f s ) . sp ∗ Pf ∼ = O O (64) Proof.
We artificially add to digram (55) one row identical to the second row from thetop and place it between the second and third row; there are now five rows; we connectthe second row to the new third row by the identity; we connect the new third row to thenew fourth row the same way as rows two and three in (55) are connected. The reader
ERVERSE LERAY FILTRATION AND SPECIALIZATION 33 can verify that: if we apply the adjunction Id → b ∗ b ∗ to this new diagram, call it (55 ′ ),then we obtain what can be called the adjunction morphism between (55 ′ ) and (63). Theconclusion follows. (cid:3) Specialization morphisms and the long exact sequence of a triple.
In the context of the compactification Set-up 3.4.1, with S a nonsingular curve and s ∈ S a point, it is natural to ask about the relation between specialization morphismsand the long exact sequence in cohomology associated with the triple ( Z, X, X o ) . Becauseof base change issues, and because of the possible failure of the morphisms of type δ for theclosed embedding a to be isomorphisms, the relation is not always the expected one, i.e.the filtered cones of the horizontal morphisms in (64) are not necessarily the correspondingobjects for i ∗ a ! F and t ∗ a ! F .Recall Remarks 3.1.4, 3.2.4, 3.3.4 and 3.4.4 on the parts of the diagrams involved in thedefinitions of the specialization morphisms sp ∗ , sp ∗ P , sp ∗ P f . We discuss the case of sp ∗ P f . The case sp ∗ P is then the special case when f : X → Y isthe identity. The case sp ∗ is the special case, when we remove p τ ≤• from the picture. Lemma 3.5.1.
Let things be as in Set-up 3.4.1, except that we do not assume that v and f are proper, nor that W is Cartier. Assume that S is a nonsingular curve and s ∈ S isa point. The following diagram is commutative: ψ [ − v ∗ a ! a ! p τ ≤• f ∗ / / ψ [ − v ∗ p τ ≤• f ∗ / / ψ [ − v ∗ b ∗ b ∗ p τ ≤• f ∗ / / /o/o/o i ∗ [ − v ∗ a ! a ! p τ ≤• f ∗ / / σ ∗ O O bc i ∗ v ∗ (cid:15) (cid:15) i ∗ [ − v ∗ p τ ≤• f ∗ / / σ ∗ O O bc i ∗ v ∗ (cid:15) (cid:15) i ∗ [ − v ∗ b ∗ b ∗ p τ ≤• f ∗ / / /o/o/o σ ∗ O O bc i ∗ v ∗ (cid:15) (cid:15) v ∗ i ∗ [ − a ! a ! p τ ≤• f ∗ / / i ∗ a ! a ! → a ! a ! i ∗ (cid:15) (cid:15) v ∗ i ∗ [ − p τ ≤• f ∗ / / (cid:15) (cid:15) v ∗ i ∗ [ − b ∗ b ∗ p τ ≤• f ∗ / / /o/o/o (cid:15) (cid:15) i ∗ b ∗ b ∗ → b ∗ b ∗ i ∗ bc i ∗ b ∗ (cid:15) (cid:15) v ∗ a ! a ! i ∗ [ − p τ ≤• f ∗ / / v ∗ i ∗ [ − p τ ≤• f ∗ / / = O O v ∗ b ∗ b ∗ i ∗ [ − p τ ≤• f ∗ / / /o/o/o v ∗ a ! a ! p τ ≤• i ∗ [ − f ∗ / / δ i ∗ O O bc i ∗ f ∗ (cid:15) (cid:15) v ∗ p τ ≤• i ∗ [ − f ∗ / / δ i ∗ O O bc i ∗ f ∗ (cid:15) (cid:15) v ∗ b ∗ b ∗ p τ ≤• i ∗ [ − f ∗ / / /o/o/o δ i ∗ O O bc i ∗ f ∗ (cid:15) (cid:15) v ∗ a ! a ! p τ ≤• f ∗ i ∗ [ − / / v ∗ p τ ≤• f ∗ i ∗ [ − / / v ∗ b ∗ b ∗ p τ ≤• f ∗ i ∗ [ − / / /o/o/o (65) Proof.
This follows at once from the following list of formal properties.Let things be as in Set-up 3.4.1. Recall (61). Let i : T → S be a morphism of varieties.The verification of the following facts is formal and is left to the reader.(1) σ ∗ is a morphism of functors.(2) The attaching triangles are compatible with base change. More precisely, the basechange morphisms i ∗ v ∗ → v ∗ i ∗ applied to the distinguished triangle of functors (61) (recall that it contains the morphisms a , b ) yield morphisms of the correspondingdistinguished triangles.(3) Let µ : i ∗ a ! a ! → a ! a ! i ∗ be the composition of the base change isomorphism i ∗ a ! a ! → a ! i ∗ a ! with the natural morphism a ! i ∗ a ! → a ! a ! i ∗ [Ka-Sh-1990, Proposition 3.1.9.iii].Then i ∗ a = ai ∗ ◦ µ. (3 ′ ) Let µ ′ : i ∗ b ∗ b ∗ → b ∗ b ∗ i ∗ be the composition of the base change isomorphism i ∗ b ∗ b ∗ → b ∗ i ∗ b ∗ with the natural isomorphism b ∗ i ∗ b ∗ → b ∗ b ∗ i ∗ . Then µ ′ ◦ i ∗ b = bi ∗ . (4) The attaching triangles are compatible with the morphisms of type δ. More pre-cisely, the morphisms of type δ applied to the distinguished triangle of functors(61) yield morphisms of the corresponding distinguished triangles.(5) Part (2) holds for the base change morphisms i ∗ f ∗ → f ∗ i ∗ . (cid:3) We need the following remark and lemma in the proof of Proposition 3.5.4.
Remark 3.5.2.
The base change morphism bc i ∗ b ∗ in (65) and (63) coincide. Lemma 3.5.3.
Let things be as in the compactification Set-up 3.4.1 and assume that S isa nonsingular connected curve and s ∈ S is a point. Let G ∈ D bc ( Y ) be such that φG = 0 and φa ! G = 0 . Then the natural morphisms i ∗ a ! p τ ≤• G → a ! i ∗ p τ ≤• G are isomorphisms.Proof. The assumptions on vanishing are equivalent to assuming that φ p τ ≤• G = 0 and φ p τ ≤• a ! G = 0 , so that it is enough to prove the conclusion without truncations. Wehave i ∗ [ − G ∼ = i ! [1] G and i ∗ [ − a ! G ∼ = i ! [1] a ∗ G. The conclusion follows from the naturalidentification i ! a ! = a ! i ! as follows: i ∗ [ − a ! G = i ! [1] a ! G = a ! i ! [1] G = a ! i ∗ [ − G. (cid:3) Recall Remark 3.3.5, which allows us to use a general point t to express the targetof specialization morphisms. Recall the notational Remark 3.1.8, and convention (3) onshifts of filtrations. Theorem 3.5.4. (Specialization and long exact sequence of a triple)
Let thingsbe as in the compactification Set-up 3.4.1 and assume that S is a nonsingular connectedcurve and s ∈ S is a point. Let F ∈ D bc ( X ) .Consider the following three sets of conditions: (I) (a) f ∗ F has no constituents supported on W. (b) f ∗ i ∗ F has no constituents supported on W s . (c) φf ∗ F = 0 ( ⇐ ( φF = 0) ). (d) φf ∗ a ! F = 0 ( ⇐ ( φa ! F = 0) ). (II) (a) f is projective. (b) F, a ! F [1] , i ∗ F [ − a ! i ∗ F are perverse semisimple. (c) φf ∗ F = 0 ( ⇐ ( φF = 0) ). (d) φf ∗ a ! F = 0 ( ⇐ ( φa ! F = 0) ). (III) (a) f is projective. (b) F, a ! F [1] , i ∗ F [ − and a ! i ∗ F are perverse semisimple. (c) φv ∗ b ∗ f ∗ b ∗ F = 0 (cf. Remark 3.3.9) ( ⇐ ( φf ∗ b ∗ b ∗ F = 0) ⇐ ( φb ∗ b ∗ F = 0) ). (d) φf ∗ b ∗ F = 0 . (e) φf ∗ a ! F = 0 . ERVERSE LERAY FILTRATION AND SPECIALIZATION 35
Assume that either (I), (II), or (III) holds. Then, for t ∈ S general, we have the isomor-phism of distinguished triangles in DF ( pt ) : (cid:0) R Γ( Z t , t ∗ a ! F ) , P f t ( − (cid:1) / / (cid:0) R Γ( X t , t ∗ F ) , P f t (cid:1) / / (cid:0) R Γ( X ot , t ∗ b ∗ F ) , P f t (cid:1) / / /o/o/o (cid:0) R Γ( Z s , i ∗ a ! F ) , P f s ( − (cid:1) / / sp ∗ Pf ( a ! F ) ∼ = O O (cid:0) R Γ( X s , i ∗ F ) , P f s (cid:1) / / sp ∗ Pf ( F ) ∼ = O O (cid:0) R Γ( X os , i ∗ b ∗ F ) , P f s (cid:1) / / /o/o/o sp ∗ Pf ( b ∗ F ) ∼ = O O . (66) Proof.
We plug F ∈ D bc ( X ) in diagram (65). We still denote the resulting diagram by(65) in what follows. Then (65) consists of three columns, C , C and C , and six rows R , . . . , R . Each row is a distinguished triangle. The vertical arrows yield morphisms ofdistinguished triangles.The goal is to prove that, under the assumptions of the theorem: all the vertical arrowsin (65) are isomorphisms. In fact, then, by considering the compositum of the arrowsfrom R to R , and in view of (19), we obtain the following system of isomorphisms ofdistinguished triangles: v ∗ a ! p τ ≤• +1 f ∗ t ∗ [ − a ! F / / v ∗ p τ ≤• f ∗ t ∗ [ − F / / v ∗ b ∗ p τ ≤• f ∗ t ∗ [ − b ∗ F / / /o/o/o ψ [ − v ∗ a ! p τ ≤• +1 f ∗ a ! F / / ≃ ( ) O O ψ [ − v ∗ p τ ≤• f ∗ F / / ≃ ( ) O O ψ [ − v ∗ b ∗ p τ ≤• f ∗ b ∗ F / / /o/o/o ≃ ( ) O O v ∗ a ! p τ ≤• +1 f ∗ i ∗ [ − a ! F / / ≃ sp ∗ ( p τ ≤• +1 a ! F ) O O v ∗ p τ ≤• f ∗ i ∗ [ − F / / ≃ sp ∗ ( p τ ≤• F ) O O v ∗ b ∗ p τ ≤• f ∗ i ∗ [ − b ∗ F / / /o/o/o ≃ sp ∗ ( p τ ≤• b ∗ F ) O O , (67)which yields the desired conclusion (66).We first prove that (I) implies that desired goal that all the vertical arrows are isomor-phisms.The plan of the proof is as follows. Prove that columns C , C and C coincide with di-agrams (58) involved in the definitions of sp ∗ P f ( a ! F ) , sp ∗ P f ( F ) and sp ∗ P f ( b ∗ F ), respectively.Prove that all arrows in said columns are isomorphisms, so that all three morphisms sp ∗ P f are defined and isomorphisms, and the composition of the arrows from R to R yields(66). Step 1: we carry out the plan for C . In this case, P f ( −
1) will enter the picturenaturally.Let us identify C with (58) for a ! F . In order to accomplish this, we need to show that:(i) the top of C , i.e. ψ [ − v ∗ a ! a ! p τ ≤• f ∗ F, coincides with ψ [ − v ∗ a ! p τ ≤• +1 f ∗ a ! F ;(ii) the morphism v ∗ i ∗ [ − a ! a ! p τ ≤• f ∗ F → v ∗ a ! a ! i ∗ [ − p τ ≤• f ∗ F is an isomorphism;(iii) the bottom of C , i.e. v ∗ a ! a ! p τ ≤• f ∗ i ∗ [ − F, coincides with v ! a ! p τ ≤• +1 f ∗ i ∗ [ − a ! F. To prove (i) it is enough to prove that a ! p τ ≤• f ∗ F coincides with p τ ≤• f ∗ a ! F. This followsfrom the fact that the natural morphism (22) of type δ a ! for the closed embedding a is anisomorphism in view of our assumptions that f ∗ F has no constituents supported on W, sothat Proposition 2.1.5 applies. For (ii), we argue as follows. Since i ∗ a ! = a ! i ∗ , is is enough to prove that i ∗ a ! p τ ≤• f ∗ F coincides with a ! i ∗ p τ ≤• f ∗ F. This follows from our vanishing hypotheses and from Lemma3.5.3, in view of the assumptions φf ∗ F = 0 and φf ∗ a ! F = 0.For (iii), we argue as we have done for (i), by using the assumption that f ∗ i ∗ F has nosupports on W s . We have shown that C agrees with (58) for a ! F . Since f and v are proper, we havethat sp ∗ P f ( a ! F ) is defined.Let us now prove that all arrows in C are isomorphisms.The base change maps are isomorphisms. The assumption that φF = 0 implies φf ∗ F =0, so that δ i ∗ is an isomorphism by Lemma 2.2.1.(3). Finally, we claim that σ ∗ on the topof C is an isomorphism. This follows from the fact that its cone φv ∗ a ! a ! p τ ≤• f ∗ F = 0: infact, by using the same kind of argument employed in (i) above, this is implied by theassumption φf ∗ a ! F = 0.This completes the proof of Step 1: all the vertical arrows in C are isomorphisms andgive rise to sp ∗ P f ( a ! F ), which is an isomorphism for the shifted P f ( − Step 2.
Clearly, C is on the nose the collection of morphisms in (58) involved in thedefinition of sp ∗ P f ( F ). Moreover, in view of our hypotheses, Theorem 3.3.7.(1) appliesand sp ∗ P f ( F ) is defined and an isomorphism. In particular, all the arrows in C areisomorphisms. Step 3: we carry out the plan for C by following the template of Step 1.Since b ∗ is ´etale, b ∗ is t -exact and it commutes with f ∗ . Clearly, b ∗ commutes with i ∗ .We thus have the identities: ψ [ − v ∗ b ∗ b ∗ p τ ≤• f ∗ = ψ [ − v ∗ b ∗ p τ ≤• f ∗ b ∗ , v ∗ b ∗ b ∗ p τ ≤• f ∗ i ∗ [ −
1] = v ∗ b ∗ p τ ≤• f ∗ i ∗ [ − b ∗ . (68)This implies that the top and bottom of C give rise to the domain and target of theputative sp ∗ P f ( b ∗ F ) as in Step 1.Since the morphisms of rows are morphisms of distinguished triangles and the verticalarrows in C and in C are isomorphisms, then, by the Five Lemma, so are the ones in C .This concludes Step 3 and we have shown that (I) implies the desired goal that all thevertical arrows are isomorphisms.We prove that (II) implies the same desired goal.To do so, it is enough to prove that (II) implies (I). Since (c) and (d) are common to(I) and (II), we need to show that (II) implies (Ia) and (Ib). By Proposition 2.3.2, theassumptions f projective, F and a ! F [1] perverse semisimple imply (Ia). For the samereason, the assumptions f projective, i ∗ F [ −
1] and a ! i ∗ F perverse semisimple imply (Ib).The desired goal is thus met.Note that instead of assuming that a ! F [1] (resp. a ! i ∗ F ) is perverse semisimple, it isenough to assume that f ∗ a ! F [1] (resp. f ∗ a ! i ∗ F ) satisfies the conclusion of the relativeHard Lefschetz Theorem; see Remark 2.3.3.We prove that (III) implies the desired goal that all the vertical arrows are isomorphisms.We go back to (65). Again, the goal is to prove that the bottom and top rows areas in the proof that (I) implies the desired conclusion, and that all vertical arrows areisomorphisms. ERVERSE LERAY FILTRATION AND SPECIALIZATION 37
The top and bottom of C are already in the desired form.The top and bottom of C are already in the desired form; see (68), which uses onlythat b ∗ is t -exact and that it commutes with f ∗ .By Remarks 3.5.2 and 3.4.3, the base change morphism bc i ∗ b ∗ in (65) is an isomor-phism. As in the case of (II), all the morphisms of type δ , connecting R to R areisomorphisms. The base change morphisms for i ∗ f ∗ and i ∗ v ∗ are also isomorphisms in allcolumns, including C .All the arrows in C are thus isomorphisms as soon as σ ∗ is. This follows from thehypothesis that φv ∗ b ∗ f ∗ b ∗ F = 0 as follows. Since F is semisimple, so is b ∗ F . The Decom-position Theorem implies that φv ∗ b ∗ p τ ≤• f ∗ b ∗ F is a direct summand of φv ∗ b ∗ f ∗ b ∗ F = 0,so that φv ∗ b ∗ b ∗ p τ ≤• f ∗ F = φv ∗ b ∗ p τ ≤• f ∗ b ∗ F = 0, where we have used the t -exactness of b ∗ and proper base change. The cone of σ ∗ in C is zero and σ ∗ in C is an isomorphism.We have proved that C has the desired form and that all of the arrows in C areisomorphisms, so that sp P f ( b ∗ G ) is an isomorphism.Since the base change arrow bc i ∗ v ∗ is an isomorphism in C it follows that all arrowsconnecting R to R are isomorphisms. In particular, the arrow denoted by i ∗ a ! a ! → a ! a ! i ∗ is an isomorphism.The arrow σ ∗ in C is an isomorphism if and only if φv ∗ a ! a ! p τ ≤• f ∗ F = 0, which wenow prove. As in the case of (II), we have that f ∗ F has no constituents supported on W ,so that φv ∗ a ! a ! p τ ≤• f ∗ F = φv ∗ a ! p τ ≤• +1 f ∗ a ! F = v ∗ a ! p τ ≤• +1 f ∗ φa ! F = 0, by the assumption0 = φf ∗ a ! F = f ∗ φa ! F ( f is proper).What above also implies that the top of C is ψ [ − v ∗ a ! p τ ≤• +1 f ∗ a ! F , i.e. the systemgiving rise to the target of the specialization map. As to the bottom of C , since allarrows are isomorphisms, it is identified with v ∗ i ∗ [ − a ! a ! p τ ≤• f ∗ F , which, by what above, isidentified with v ∗ a ! i ∗ [ − p τ ≤• +1 f ∗ a ! F , which –using the fact that f ∗ a ! F has no constituentssupported on W s by virtue of the hypotheses that f is projective, a ! F [1] and i ∗ a ! F areperverse semisimple, coupled with Proposition 2.3.2– equals v ∗ a ! p τ ≤• +1 f ∗ i ∗ [ − a ! F , i.e.the system giving rise to the source of sp ∗ P f ( a ! F ).We have proved that C has the desired form and that all arrows in C are isomorphisms.We have already proved that the same holds for C . As seen earlier by using the FiveLemma, it follows that the same holds for C . The conclusion follows. (cid:3) Remark 3.5.5.
Condition (I) in Proposition 3.5.4 is met if: f is projective, F is perversesemisimple on X and a ∗ [ − F is perverse semisimple on Z . Condition (II) in Proposition3.5.4 is met if: f is projective, i ∗ F [ − is perverse semisimple on X s and a ∗ i ∗ [ − F isperverse semisimple on Z s . Both cases follow from Proposition 2.3.2. Remark 3.5.6.
Proposition 3.5.4 yields a different proof, under different hypotheses, ofthe conclusions of both Proposition 3.4.2 and Corollary 3.4.6.
Remark 3.5.7.
The toy-model for Theorem 3.5.4 is when we assume that v X , v Y , v Z , v W and f are proper and smooth. We do not assume that v Y o is proper. In this case,we leave to the reader to verify that if we set F = Q X , and we consider the usual Lerayfiltration on the cohomology of X s and X t ( Z s , Z t , resp.), with respect to f s : X s → Y s and f t : X t → Y t , ( f s : Z s → W s and f t : Z t → W t , resp.), then we end up with: thespecialization morphisms are defined, they are filtered isomorphisms, and they fit in the long exact Gysin sequences: (in what follows ⋆ is arbitrary, but fixed) . . . / / L ⋆ − H •− ( Z t , Q ) Gysin / / L ⋆ H • ( X t , Q ) / / L ⋆ H • ( X ot , Q ) +1 / / . . .. . . / / L ⋆ − H •− ( Z s , Q ) Gysin / / sp Z O O L ⋆ H • ( X s , Q ) / / sp X O O L ⋆ H • ( X os , Q ) +1 / / sp Xo O O . . . (69)where L stands for the increasing classical Leray filtration (which classically starts at zero).4. Applications to the Hitchin morphism
In this section we apply the results of §
3, especially equation (66) in Theorem 3.5.4to the triples arising from the compactification of Dolbeault moduli spaces in families[de-2018]. The main result is Theorem 4.4.2, to the effect that the triple given by theDolbeault moduli spaces over a curve, its compactification and the boundary gives riseto a specialization morphism of long exact sequences for the (intersection) cohomology ofthe special and general points that is a filtered isomorphisms of triples for the perverseLeray filtrations. This is what one would obtain if we were in the oversimplified and idealsituation of a fiber bundle with boundaries, and we were to take the Leray filtrations.The compactification [de-2018] is obtained by taking a G m -quotient; this is quicklyreviewed in § § § § Descending along G m -quotients. In this section, we assume we have two Cartesian diagrams of morphisms over a variety S as in the compactification Set-up 3.4.1: Z f (cid:15) (cid:15) a / / X f (cid:15) (cid:15) X of (cid:15) (cid:15) b o o Z f (cid:15) (cid:15) a / / X f (cid:15) (cid:15) X of (cid:15) (cid:15) b o o W a / / Y Y ob o o W a / / Y Y o , b o o (70)such that: G m -acts equivariantly on the l.h.s. with finite stabilizers; the actions cover thetrivial action over S ; the r.h.s. is obtained by taking the quotient under the G m -action;the quotient morphisms π are geometric quotients. Lemma 4.1.1.
Each quotient morphism π above factors as π = qp, where p is a quotient S -morphism by a finite subgroup C of G m , and q is a smooth quotient S -morphism by a G m -action, so that, for example, we have: π : X p / / X ′ := X /C q / / X = X / G m = X ′ / G ′ m (:= G m /C ) . (71) ERVERSE LERAY FILTRATION AND SPECIALIZATION 39
Proof.
Let C ⊆ G m be any finite subgroup containing all the stabilizers (recall that we arein a finite type situation). The variety X ′ := X /C over S is endowed with the induced G ′ m := G m /C ≃ G m -action, and we have diagram (71). The quotients morphism p isfinite. By Luna ´Etale Slice Theorem, the quotient morphism q is smooth: the G ′ m -actionhas trivial stabilizers, so that q is ´etale locally a projection map with scheme-theoreticfiber G m . (cid:3) We record the following remark for use in the proof of Lemma 4.3.3.
Remark 4.1.2.
The following commutative diagrams are Cartesian in the category oftopological spaces: Z a / / p (cid:15) (cid:15) X p (cid:15) (cid:15) X ob o o p (cid:15) (cid:15) Z ′ a / / q (cid:15) (cid:15) X ′ q (cid:15) (cid:15) X ′ ob o o q (cid:15) (cid:15) Z a / / X X o . b o o (72) The same is true if we replace ( Z, X, X o ) etc. with ( W, Y, Y o ) etc. Lemma 4.1.3. ( Descending the lack of constituents on the boundary to a quo-tient by G m ) Let G ∈ D ( Y ) and G ∈ D bc ( Y ) be such that q ∗ G is a direct summand of p ∗ G . If G has no constituents supported on the boundary W , then G has no constituentssupported on the boundary W .Proof. CLAIM: p ∗ G has no constituents supported on W ′ . Since p is finite, p ∗ is t -exact, sothat p ∗ and p H • commute. It follows that we may assume that G is perverse. Let J • G bea Jordan-Holder filtration for G , so that the non trivial graded objects Gr J • G are simple,i.e. they are intersection complexes supported at irreducible closed subvarieties of Y andwith simple coefficients. By the t -exactness of p ∗ , we have that p ∗ J • G is a filtration and Gr p ∗ J • p ∗ G = p ∗ Gr J • G . Since p is finite, hence small, we have that p ∗ sends the intersectioncomplex of a closed subvariety T of Y with twisted coefficients, to the intersection complexof the image p ( T ) with appropriately twisted coefficients. By construction, we have that W = p − ( W ′ ) . The subvarieties T appearing as the supports of the simple Gr J • G aboveare not contained in W by assumption, so that their p ( T )’s are not contained in W ′ . Theclaim follows.By construction, we have that W ′ = q − ( W ) . Since q ∗ G is assumed to be a directsummand of p ∗ G , to prove the lemma we need to show that the constituents of q ∗ G are q ∗ [1] of the constituents of G . But this follows from the fact that, since q is smooth ofrelative dimension one with connected fibers, q ∗ [1] is t -exact and it preserves simple objects(cf. [Be-Be-De-1982, p.106, bottom]). (cid:3) We need the following general result in the proof of Lemma 4.1.5.
Lemma 4.1.4.
Let C be a finite group acting on a variety X , let p : X → X ′ be thequotient morphism. Then: (i) p ∗ Q X admits Q X ′ as a direct summand; (ii) p ∗ IC X admits IC X ′ as a direct summand. Proof.
We prove (i). By [Gr-1956, (5.1.1), p.108], we have that Q X ′ = ( p ∗ ( Q X )) C . Wesplit the natural adjunction morphism Q X ′ → p ∗ Q X by sending a section s ∈ Q X ( p − ( U ))to (1 / | C | ) P c ∈ C c ∗ · s .We prove (ii). One can argue as above; we leave this to the reader. We prove a strongerstatement, which may be of independent interest, namely that the conclusion remainsvalid if we assume that p is a small morphism that has the property that every irreduciblecomponent of X maps onto an irreducible component of X ′ . Since IC X is the directsum of the intersection complexes of the irreducible components of X and similarly for IC X ′ , we may assume that X and X ′ are irreducible, and that p is small and surjective.By the conditions of support/co-support characterizing intersection complexes, the factthat p is small and surjective implies that p ∗ IC X is an intersection complex with sometwisted coefficients L . Since p is small and surjective, there is a Zariski dense open subset j : U ′ ⊆ X ′ that is nonsingular, and over which p is finite and ´etale. Then L can be takento be the local system on U associated with the topological covering, i.e. the direct imageof the constant sheaf. Note that this splits off Q U by a standard trace argument. We nowshow that this splitting extends uniquely to a splitting of the corresponding intersectioncomplexes, which proves our contention. Let I ′ , J ′ ∈ P ( X ′ ) be two intersection complexeswith support precisely X ′ . We have the standard identity: Hom ( I ′ , J ′ ) = Hom ( j ∗ I ′ , j ∗ J ′ ):we have j ∗ j ∗ J ′ ∈ p D ≥ ( X ); so Hom ( I ′ , j ∗ j ∗ J ′ ) = Hom ( I ′ , p H ( j ∗ j ∗ J ′ )); we have the shortexact sequence 0 → J ′ → p H ( j ∗ j ∗ J ′ ) → Q →
0, where Q ∈ P ( X ′ ) has support strictlycontained in X ′ ; we thus have Hom ( I ′ , J ′ ) = Hom ( I ′ , p H ( j ∗ j ∗ J ′ )) because Hom ( I ′ , Q ) =0 by reasons of support, and Hom ( I ′ , Q [ − t -structure. We applythe standard identity above to I ′ = p ∗ IC X and to J ′ = IC X ′ , and we lift the splitting offof Q U from L to the splitting off of IC X from p ∗ IC X . (cid:3) Lemma 4.1.5. ( Descending φ = 0 to a quotient by G m ) Let X /S be as in (70).Assume S is a nonsingular curve, and let s ∈ S be a point. Then we have the followingimplications: (1) If φ Q X = 0 , then φ Q X = 0 . (2) If φIC X = 0 , then φIC X = 0 . (3) Let F ∈ D ( X ) , F ∈ D bc ( X ) be such that q ∗ F is a direct summand of p ∗ F , then:if φ F = 0 , then φF = 0 .Proof. We prove (3). Since p is proper, we have that φp ∗ F = p ∗ φ F = 0. It follows that,since φ is additive, we have that φq ∗ F = 0 . Since q is smooth, we have q ∗ φF = φq ∗ F = 0 . Since q is surjective, we have that q ∗ φF = 0 implies the desired conclusion (3). Now,(1) and (2) follow from (3) coupled with Lemma 4.1.4, and the facts q ∗ Q X = Q X ′ and q ∗ IC X = IC X ′ [ −
1] ( q is smooth of relative dimension one; the shifts are innocuous). (cid:3) Lemma 4.1.6. ( Descending other identities )(1) If a ! Q X = Q Z [ − , then a ! Q X = Q Z [ − . (2) If a ! IC X = IC Z [ − , then a ! IC X = IC Z [ − .Proof. We prove (1).We apply p ∗ and take invariants: ( p ∗ a ! Q X ) C = ( a ! p ∗ Q X ) C = a ! ( p ∗ Q X ) C = a ! Q X ′ ;( p ∗ Q Z ) C = Q Z ′ . We thus have that: a ! Q X ′ = Q Z ′ [ − ERVERSE LERAY FILTRATION AND SPECIALIZATION 41
Since q is smooth of relative dimension 1 , we have that q ∗ [2] = q ! , so that q ∗ and a ! commute. It follows that we also have that: q ∗ a ! Q X = a ! Q X ′ = Q Z ′ [ −
2] = q ∗ Q Z [ − q ∗ a ! Q X [2] = q ∗ Q Z . Since q is surjective, we have that a ! Q X [2] is a sheaf. Since q is smoothand surjective, we have that a ! Q X [2] is locally constant of rank one. Since the fibers of q are connected, we deduce that a ! Q X [2] is constant and this proves (1).We prove (2).By repeating the first part of the proof of part (1), we see that: a ! IC X ′ = IC Z ′ [ − q ∗ a ! IC X = q ∗ IC Z [ − q ∗ [1] a ! IC X [1] = q ∗ [1] IC Z .By [Be-Be-De-1982, Corollaire 4.1.12], a ! IC X [1] is perverse. By [Be-Be-De-1982, Propo-sition 4.2.5], q ∗ [1] : P ( Z ) → P ( Z ′ ) is fully faithful, hence conservative. This implies thedesired conclusion (2). (cid:3) Compactification of Dolbeault moduli spaces via G m -actions. Let us summarize the main features of the compactification in [de-2018] that we needin this section. The notation we use here is adapted to the present needs, and differs fromthe one in [de-2018].Let X o /S be the Dolbeault moduli space associated with a reductive group G anda smooth projective family over S. Let f : X o → Y o be the corresponding Hitchin S -morphism; it is projective.The main construction in [de-2018] yields two diagrams as in (70), so that, in particular, X/S is a projective completion over S of X o /S, with boundary the S -relative Cartierdivisor Z, which is proper over S. Similarly for (
W, Y, Y o ). Let us give some more details.Let o S → Y o be the canonical section: each Y os has a canonical distinguished point; e.g. if G = GL n , then this point corresponds to the characteristic polynomial t n , which in turncorresponds to nilpotent Higgs fields. We have the closed S -subvariety f − ( o S ) ⊆ X o . Wehave G m -equivariant open inclusions and equalities as follows: X o × ( A \ { } ) = X o ( X o ∪ ( X o \ f − ( o S )) × { } = X ( X o × A , (73) Y o × ( A \ { } ) = Y o ( Y o ∪ ( Y o \ o S ) × { } = Y ( Y o × A , (74) Z = ( X o \ f − ( o S )) × { } ( X o × { } , W = ( Y o \ o S ) × { } ( Y o × { } . (75) Remark 4.2.1.
Note that we are not ruling out that Z = ∅ . In this case, our goal, i.e.Theorem 4.4.2, still holds and it is not trivial, for it states that the P f -filtered specializationmorphism is an isomorphism, whether we take singular or intersection cohomology groupswith rational coefficients. If Z is not empty, it could still happen that there are points s ∈ S such that Z s is not dense in every irreducible component of X os × { } ; this is notan issue in what follows. Special properties of Q X and IC X for the compactification.Lemma 4.3.1. a ! Q X = Q Z [ − , a ! IC X = IC Z [ − . (76) Proof.
By virtue of Lemma 4.1.6, it is enough to show that: (1) a ! Q X = Q Z [ − a ! IC X = IC Z [ − In view of the open immersions Z = ( X o \ f − ( o S )) × { } ) ⊆ X o × { } , and X ⊆ X o × A , it is enough to prove (1,2) with a : Z ⊆ → X replaced by a : X o × { } ⊆ → X o × A . Oneis easily reduced to the case a : { } ⊆ → A , where the desired conclusions are trivial. (cid:3) The following lemma makes precise the relation between the irreducible components ofthe varieties in the triple (
Z, X, X o ). Let J (resp. K , resp. J o ) be the set of irreduciblecomponents of X (resp. Z , resp. X o ). Lemma 4.3.2. (1)
The function J → J o , X j X j ∩ X o =: X oj is well-defined and a bijection. Wehave that X j ∩ Z =: Z j is either empty, or an irreducible component of Z ; if j, j ′ ∈ J are such that Z j = Z j ′ = ∅ , then j = j ′ ; in particular, this defines aninjection K → J , where Z k = X j ( k ) ∩ Z . (2) The natural adjunction disitinguished triangle (61) a ! a ! IC X → IC X → b ∗ b ∗ IC X reads as follows: L K a ! IC Z k [ − / / L J IC X j / / L J b ∗ IC X oj / / /o/o/o , (77) where the arrows are direct sum arrows (i.e. the components of each arrow withpairs of distinct indices are zero).Proof. Since Z is open in X o = X o × { } , we have that the set of irreducible componentof Z injects in the evident fashion into the one of X o . Since X is open and dense in X o × A , we have that the set of irreducible component of X is naturally identified withthe one of X o × A , which in turn, since A is irreducible, is naturally identified withthe one of X o . It follows that the desired statement holds if we replace ( Z, X, X o ) with( Z , X , X o ). The first triple is obtained from the second by dividing by the action ofthe connected group G m ; the quotient morphism has the orbits as fibers (in fact, it is ageometric quotient). It follows that the sets of irreducible components and their mutualrelations are preserved by passing to the quotient, and (1) is proved.We have the Cartesian diagram: b Z := ` K Z kν (cid:15) (cid:15) a / / b X := ` J X jν (cid:15) (cid:15) c X o := ` J X ojν (cid:15) (cid:15) b o o Z = ∪ K Z k a / / X = ∪ J X j X o = ∪ J X oj , b o o (78)where: ν are the evident morphisms induced by the closed embeddings of the irreduciblecomponents Z j → Z and X j → X ; a (resp. b ) are the evident closed (resp. open)embeddings. We have IC X = ν ∗ IC ˆ X and IC Z = ν ∗ IC ˆ Z . By base change, by the additivityof adjunction morphisms, and by the identity a ! IC X k = IC Z k [ −
1] –which is due to Lemma4.3.1, which remains valid for the irreducible components, by virtue of the just-proved part(1)–, we have that the distinguished triangle (61) applied to IC X reads as (77), and (2) isproved. (cid:3) ERVERSE LERAY FILTRATION AND SPECIALIZATION 43
Lemma 4.3.3.
Let S be a nonsingular curve and let s ∈ S be a point. Let F ∈ D bc ( X ) beeither Q X , or IC X . Then: φa ! F = 0 , φF = 0 , φb ∗ b ∗ F = 0 . (79) In particular, we have: φv ∗ a ! f ∗ a ! F = 0 , φv ∗ f ∗ F = 0 , φv ∗ b ∗ f ∗ b ∗ F = 0 . (80) Proof.
We refer to [de-Ma-2018] and to [de-2018] for references.We have the evident morphisms X → X o × A → X o . Let F o := b ∗ F = F | X o . Let F ∈ D ( X ) be the pull back of F o to X ; if F o = Q X o , then F = Q X ; if F o = IC X o , then F = IC X [ − . By the Non Abelian Hodge Theorem, X o is topologically locally trivialover S, so that φF o = 0: this is clear when F = Q X , so that F o = Q X o ; when F = IC X ,so that F o = IC X o , we invoke the topological invariance of the intersection complex in thenot necessarily irreducible context [de-Ma-2018]. Since pr X o : X o × A → X o is smooth,we have that φpr ∗ X o F o = pr ∗ X o φF o = 0. Since X is open in X o × A , we have that φ F = 0 . By Lemma 4.1.5.(1).(2), applied to π = qp : X → X ′ → X and F and F , wesee that φF = 0 . Since v and f are proper, this implies that φv ∗ f ∗ F = v ∗ f ∗ φF = 0 . We now wish to apply Lemma 4.1.5 to π = qp : Z → Z ′ → Z , a ! F and a ! F . By thebase change identities associated with (72), we have that p ∗ a ! F = a ! p ∗ F (always valid)and that q ∗ a ! F = a ! q ∗ F (because q is smooth). From what above, we know that p ∗ F admits q ∗ F as a direct summand, so that p ∗ a ! F admits q ∗ a ! F as a direct summand. Weclaim that φa ! F = 0 . Since Z is open in X o × { } , it is enough to show that φ e a ! F = 0,where e a : X o × { } → X o × A . Since pr X o is smooth of relative dimension one, so that pr ! X o = pr ∗ X o [ − e a ! F = F o [ − φF o = 0 , we see that φ e a ! F = 0, as desired. By Lemma 4.1.5, we have that φa ! F = 0. Since v , a and f areproper, this implies that φv ∗ a ! f ∗ a ! F = v ∗ a ! f ∗ φa ! F = 0 . The remaining statements follow from what we have proved and the distinguished tri-angle (61): we get φb ∗ b ∗ F = 0, and then ve apply again v ∗ f ∗ to φb ∗ b ∗ F to conclude. (cid:3) Lemma 4.3.4.
Let F ∈ D bc ( X ) . Then: (1) f ∗ F has no constituents supported on W ; (2) f ∗ i ∗ F has no constituents supported on W s . Proof.
We have Cartesian diagrams: X (cid:15) (cid:15) / / X o × A (cid:15) (cid:15) / / X o (cid:15) (cid:15) X s (cid:15) (cid:15) / / X os × A (cid:15) (cid:15) / / X os (cid:15) (cid:15) Y / / Y o × A / / Y o Y s / / Y os × A / / Y os . (81)Let F o := b ∗ F = F X o . Denote by F the pul-back of F o to X . By proper base change, f ∗ pr ∗ X o F o = pr ∗ X o f ∗ F has no constituent on Y o × { } . In view of (75), the same is true for f ∗ F . We apply Lemma 4.1.3, so that f ∗ F has no constituents supported on W. The proofof the second assertion is identycal in view of the fact that: if s ∈ S is a point, then wehave the − s -version of (70) and it enjoys similar properties; in particular, Lemma 4.1.3applies to it. (cid:3) The long exact sequence of the triple ( Z, X, X o ) . In this section, we start with a smooth family of projective manifolds over a nonsingularconnected curve S , so that, according to § X o /S is the Dolbeault moduli space for the family. We fix a point s ∈ S (the special point) andwe let t ∈ S be a general point.[de-Ma-2018, Theorem 1.1.2] proves that the specialization morphism sp ∗ P f in inter-section cohomology for the Dolbeault moduli spaces of a smooth family of projectivemanifolds is an isomorphism: sp ∗ P f : (cid:0) IH • ( X os , Q ) , P f s (cid:1) ∼ / / (cid:0) IH • ( X ot , Q ) , P f t (cid:1) . (82)This is accomplished by applying [de-Ma-2018, Theorem 3.2.1, part (ii), or part (iii)].In fact, part (ii) implies (iii); this latter seems like a more natural statement, at least whendealing with semisimple coefficients F ∈ D bc ( X ). Since the semisimplicity is essential inwhat above, the methods of [de-Ma-2018] do not seem to afford the same kind of resultsfor the non-semisimple F = Q X o , i.e. for singular cohomology.As it is noted in Remark 3.3.10, Theorem 3.3.7 is an amplification of [de-Ma-2018,Theorem 3.2.1] and it can thus be used to prove (82)): in fact we can use any of theparts (3,4,5,7) of Theorem 3.3.7 to deduce (82). However, none of these approaches yieldsa genuine new proof of (82) since, as it turns out, in verifying the assumptions, we endup verifying the assumptions of part (5), and thus end up using [de-Ma-2018, Theorem3.2.1.(iii)].In this section, we show that:(1) By using a good compactification of Dolbeault moduli spaces, Proposition 3.4.2yields a new proof of (82)); see Proposition 4.4.1. In fact, (82) is made moreprecise in (83). The proof in Proposition 4.4.1 uses Proposition 3.4.2.(A); thereader can verify that, in the case of (topologist’s) intersection cohomology, onecan use Proposition 3.4.2.(B) as well.(2) Proposition 3.4.2 also leads to the proof of a new fact -not seemingly affordable bythe methods of [de-Ma-2018], nor of Theorem 3.3.7-, namely that (83) holds forordinary singular cohomology as well.(3) The good compactification of Dolbeault moduli spaces gives rise to the main newfact proved in this section that the specialization morphisms sp ∗ P f for the triple( Z, X, X o ), in (the topologist’s) intersection cohomology as well as in rationalsingular cohomology, give rise to the isomorphism (84) of long exact sequences ofTheorem 4.4.2Recall that P f on H • ( X, F ) is defined by considering the system p τ ≤• f ∗ F .We shall make implicit use of (4) and (3).In the remainder of this section: when dealing with intersection cohomology groups,all morphisms are direct sum morphisms with respect to the canonical decomposition ac-cording to irreducible components; in Proposition 4.4.1 and Theorem 4.4.2, the horizontalarrows are well-defined without ambiguities, the vertical ones are well-defined modulo theambiguities introduced by the action of monodromy exchanging the irreducible compo-nents. These ambiguities are harmless for the purposes of this paper; see Remark 3.1.5. ERVERSE LERAY FILTRATION AND SPECIALIZATION 45
Moreover, they are removed if instead of using a general point t , we use the nearby cyclefunctor ψ .We state the following proposition using (hyper)cohomology; the reader should haveno difficulty re-writing the stronger version of (83) in DF ( pt ) that uses ( R Γ( − , − ) , P f ).Even ignoring the l.h.s. of (83), as pointed out above, this result, i.e. that the perversefiltered specialization morphisms exists and is an isomorphism for the singular cohomologyof Dolbeault moduli spaces, is new. Proposition 4.4.1.
Let H ( − ) denote either the singular cohomology, the intersectioncohomology, or the topologist’s intersection cohomology groups. We have the followingcommutative diagram of filtered finite dimensional rational vector spaces, where the hori-zontal arrows are the natural restriction morphisms, the vertical arrows are the well-definedperverse-filtered specialization morphisms and they are both isomorphisms: (cid:0) H • ( X t ) , P f t (cid:1) / / (cid:0) H • ( X ot ) P f t (cid:1)(cid:0) H • ( X s ) , P f s (cid:1) / / sp ∗ Pf ≃ O O (cid:0) H • ( X ot ) , P f s (cid:1) . sp ∗ Pf ≃ O O (83) In the case of (the topoligist’s) intersection cohomology, the arrows are direct sum arrowsfor the canonical decompositions into irreducible components (77).Proof.
By Lemma 4.3.3, coupled with Theorem 3.3.7.(1), we have that the perverse filteredspecialization morphism on the l.h.s. of (83) is defined and it is an isomorphism. We wishto apply Proposition 3.4.2.(A) and its Corollary 3.4.6. We need to verify that φb ∗ b ∗ F = 0and that φf ∗ b ∗ F = 0. Both follow, again, from Lemma 4.3.3. This proves diagram (83).That the arrows are direct sum arrows follows from Lemma 4.3.2. (cid:3) The following result identifies the filtered cone of the restriction morphism in (83). Westate the following proposition using morphisms of long exact sequences in cohomology.The reader should have no difficulty re-writing the stronger version of (84) that uses( R Γ( − , − ) , P f ) and distinguished triangles in DF ( pt ). Theorem 4.4.2.
Let H ( − ) denote either the singular cohomology, the intersection coho-mology, or the topologist’s intersection cohomology groups. In what follows, for intersectioncohomology, when dealing with Z s , Z t , replace P f (1) with P f and • − with • − . Wehave as isomorphism of long exact sequences of filtered morphims: . . . / / (cid:0) H •− ( Z t ) , P f t (1) (cid:1) / / (cid:0) H • ( X t ) , P f t (cid:1) / / (cid:0) H • ( X ot ) P f t (cid:1) / / . . .. . . / / (cid:0) H •− ( Z s ) , P f s (1) (cid:1) / / sp ∗ Pf ≃ O O (cid:0) H • ( X s ) , P f s (cid:1) / / sp ∗ Pf ≃ O O (cid:0) H • ( X ot ) , P f s (cid:1) sp ∗ Pf ≃ O O / / . . . . (84) For (the topologist’s) intersection cohomology, the morphisms are direct sum morphismsfor the decomposition according to irreducible components (77).
Proof.
The plan is to use (66) in Theorem 3.5.4, which stems from (67), and plug (76)into the result.In order to use (66), we need to verify that the conditions (I) in Theorem 3.5.4 are met.Even if it is not necessary, let us note that conditions (II) and (III) are also met in thecase where we deal with intersection cohomology via the use of IC X ; however, they arenot necessarily met if we deal with singular cohomology via the use of the non semisimple Q X .Conditions (Ia,Ib) are met by virtue of Lemma 4.3.4. Conditions (Ic,Id) are met byvirtue of Lemma 4.3.3.Plug F = Q X [1] in (67), use i ∗ Q X = Q X s and i ∗ a ! Q X s = Q Z s [ −
2] (cf. 76), and similarlyfor t ∗ , and get the system of isomorphisms of distinguished triangles: p τ ≤• +1 f ∗ Q Z t [ − / / p τ ≤• f ∗ Q X t / / b ∗ p τ ≤• f ∗ Q X ot / / /o/o/o p τ ≤• +1 f ∗ Q Z s [ − / / ≃ O O p τ ≤• f ∗ Q X s / / ≃ O O b ∗ p τ ≤• f ∗ Q X os / / /o/o/o ≃ O O . (85)Then the reader can use (4) and (3), to observe that P f ( −
1) in (66) then becomes P f (1)as in (84), which is thus proved when H is singular cohomology.Plug F = IC X in (67), use i ∗ [ − IC X = IC X s and i ∗ a ! IC X = IC Z s [ −
1] (cf. 76), andsimilarly for t ∗ , and get the system of isomorphisms of distinguished triangles: p τ ≤• +1 f ∗ IC Z t [ − / / p τ ≤• f ∗ IC X t / / b ∗ p τ ≤• f ∗ IC X ot / / /o/o/o p τ ≤• +1 f ∗ IC Z s [ − / / ≃ O O p τ ≤• f ∗ IC X s / / ≃ O O b ∗ p τ ≤• f ∗ IC X os / / /o/o/o ≃ O O . (86)Then the reader can use (4) and (3) to observe that P f ( −
1) in (66) becomes P f as in (84)-and that • − • − H is intersectioncohomology. That the morphisms are direct sum morphisms for the decompositions intoirreducible components follows from (77).Similarely, if we plug IC X = ⊕ j IC X j [ − dim X j ], we end up with the desired conclusion. (cid:3) Remark 4.4.3. (1)
The results of this section hold if we replace the Dolbeault moduli space of Higgsbundles for G = GL n , SL n , P GL n for families of curves with their twisted coun-terparts of [de-2018, Remark 2.1.2] , where the degree and rank of the Higgs bundlesare coprime. (2) If G = GL n , SL n , these twisted Dolbeault moduli spaces are nonsingular, their com-pactifications in § S ; see [de-2018, Thm. 3.1.1.(6)] .What follows is certainly redundant in our set-up of the Hitchin morphism, butmay be useful in other set-ups: the proof of Theorem 4.4.2 uses conditions (I)in Theorem 3.5.4; since we are in a simple normal crossing divisor situation, inview of Corollary 2.4.2, we could use the variant of conditions (II,III) in Theorem ERVERSE LERAY FILTRATION AND SPECIALIZATION 47 f ∗ i ∗ F [ − satisfies the conclusion of the HardLefschetz Theorem. We leave the details to the reader. References [Be-Be-De-1982]
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Mark Andrea A. de Cataldo, Stony Brook University
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