Introduction↩︎

The Zucker conjecture states that for a Shimura variety \(X\) its square integrable (=\(L_2\)) cohomology is naturally isomorphic with the intersection cohomology of its Baily-Borel compactification \(X^*\) and that this is even true if instead of constant coefficients we let these cohomology groups take their values in a locally homogeneous polarized variation of Hodge structure: \(\operatorname{H}^{\scriptscriptstyle\bullet}_{(2)}(X,{\mathbb{E}})\cong \operatorname{IH}^{\scriptscriptstyle\bullet}(X,{\mathbb{E}})\). This conjecture was settled a long time ago ([1], [3], [2]), yet one aspect remained open. Each side of the isomorphism comes with its Hodge decomposition: the representation of \(L_2\)-cohomology by harmonic forms define a Hodge decomposition of \(\operatorname{H}^{\scriptscriptstyle\bullet}_{(2)}(X,{\mathbb{E}})\) and the theory of Morihiko Saito [4] puts one on \(\operatorname{IH}^{\scriptscriptstyle\bullet}(X,{\mathbb{E}})\), but it was not established that under this isomorphism the two coincide. Harris and Zucker, who stated this as a conjecture in [5], 5.3, proved that the natural map \(\operatorname{H}^{\scriptscriptstyle\bullet}_{(2)}(X,{\mathbb{E}})\to \operatorname{H}^{\scriptscriptstyle\bullet}(X, {\mathbb{E}})\) is in fact a morphism of mixed Hodge structures ([5], Thm.). The purpose of this note is to show that \(\operatorname{H}^{\scriptscriptstyle\bullet}_{(2)}(X,{\mathbb{E}})\cong \operatorname{IH}^{\scriptscriptstyle\bullet}(X,{\mathbb{E}})\) is indeed an isomorphism of Hodge structures.

The proofs of the Zucker conjecture mentioned above took a local form, namely as an equality in the derived category of bounded below complexes of \({\mathbb{C}}\)-vector spaces on \(X^*\). The refinement which we prove here is of a similar nature, the derived category in question now being one of filtered complexes that encode in a local manner the Hodge filtration. This implies that the cohomology sheaves of successive quotients will be coherent \({\mathcal{O}}_{X^*}\)-modules, rather than being locally constant. Our main tool are the local Hecke operators that were introduced in [1] and further developed in [2] with the express goal to prove the Zucker conjecture as stated above. Indeed, we will show that a minor modification can do the same job in a filtered setting. This could well have been done at the time, because the underlying technique was then already available. This also explains why much of this paper consists of revisiting and recalling work from that era.

Since we posted the first version of this paper, Mingyu Ni posted another proof of our main result [6]. This preprint also contains several other related results of interest.

1 Hodge modules↩︎

1.1 Polarizable Hodge modules↩︎

Let \(X\) be a complex manifold and \({\mathbb{E}}\) a polarizable variation of Hodge structures on \(X\) of weight \(w\). We here think of \({\mathbb{E}}\) as local system of finite dimensional \({\mathbb{Q}}\)-vector spaces which contains a lattice (this guarantees that the monodromy can be given by integral matrices and is quasi-unipotent when restricted to a punctured disk), but we do not consider that lattice as part of the data. Recall that the underlying holomorphic vector bundle \({\mathcal{E}}:={\mathcal{O}}_X\otimes_{\mathbb{Q}}{\mathbb{E}}\) then comes with a flag of holomorphic subbundles \({\mathcal{F}}^{\scriptscriptstyle\bullet}{\mathcal{E}}\) (the Hodge filtration) satisfying Griffiths transversality, i.e., the property that the flat connection \(\nabla=d\otimes_{\mathbb{Q}}1_{\mathbb{E}}\) on \({\mathcal{E}}\) takes \({\mathcal{F}}^p{\mathcal{E}}\) to \(\Omega^1_X\otimes_{{\mathcal{O}}_X}{\mathcal{F}}^{p-1}{\mathcal{E}}\). This connection extends to a derivation of degree 1 of square zero in \(\Omega^{\scriptscriptstyle\bullet}({\mathcal{E}})=\Omega^{\scriptscriptstyle\bullet}_X\otimes_{{\mathcal{O}}_M}{\mathcal{E}}=\Omega^{\scriptscriptstyle\bullet}_M\otimes_{\mathbb{Q}}{\mathbb{E}}\) and furnishes the holomorphic De Rham resolution of \({\mathbb{E}}_{\mathbb{C}}:={\mathbb{C}}\otimes_{\mathbb{Q}}{\mathbb{E}}\), \[{\mathbb{E}}_{\mathbb{C}}\to (\Omega^{\scriptscriptstyle\bullet}_X({\mathcal{E}}), \nabla).\] Note that this resolution comes with a (Hodge) filtration \({\mathcal{F}}^{\scriptscriptstyle\bullet}\Omega^{\scriptscriptstyle\bullet}_X({\mathcal{E}})\) by subcomplexes whose \(p\)th term is \(\sum_{r+s\ge p}\Omega^r_X({\mathcal{F}}^s{\mathcal{E}})\).

We can instead consider \(({\mathcal{E}}, {\mathcal{F}}^{\scriptscriptstyle\bullet}{\mathcal{E}})\) as a filtered \({\mathscr D}_X\)-module, where \({\mathscr D}_X\) stands for the sheaf of holomorphic differential operators \({\mathcal{O}}_X\to {\mathcal{O}}_X\) (itself filtered by order). It thus determines an object of a derived category of filtered \({\mathscr D}_X\)-modules: this is the ‘Riemann-Hilbert incarnation’ of the pair \(({\mathcal{F}}^{\scriptscriptstyle\bullet}{\mathcal{E}}, \nabla)\). If we ignore the filtration, then both represent \({\mathbb{E}}_{\mathbb{C}}\) in \(D_c^b(X,{\mathbb{C}})\), the derived category of constructible \({\mathbb{C}}_X\)-modules with bounded cohomology.

The Dolbeault resolution gives a fine resolution (meaning that the sheaves admit partitions of unity relative to any given open cover) of the Hodge filtered complex \((\Omega^{\scriptscriptstyle\bullet}_X({\mathcal{E}}), \nabla)\) by the smooth bigraded de Rham complex \[({\mathscr A}^{{\scriptscriptstyle\bullet},{\scriptscriptstyle\bullet}}({\mathbb{E}}_{\mathbb{C}}), d=\partial+\bar\partial)\] with its Hodge filtration defined in an obvious manner. So if \(U\subset X\) is open, then this puts on \(\operatorname{H}^k(U; {\mathbb{E}})\) a filtration \(F^{\scriptscriptstyle\bullet}\operatorname{H}^k(U; {\mathbb{E}})\), with \(F^p\operatorname{H}^k(U; {\mathbb{E}})\) the part representable by a \(d\)-closed section of \({\mathcal{F}}^p{\mathscr A}_X^{{\scriptscriptstyle\bullet}, {\scriptscriptstyle\bullet}}({\mathbb{E}})\) over \(U\). The meaning of this filtration is however usually obscure unless \(U\) is quite special.

This illustrates the notion of a (polarized) Hodge module in the analytic setting in its most basic form (see [4], [7]), provided we make a degree shift by putting \({\mathbb{E}}\) in degree \(-\dim_{\mathbb{C}}X\) (this is denoted \({\mathbb{E}}[\dim_{\mathbb{C}}X]\)).
Given a complex-analytic variety \(Z\), then a Hodge module \(M\) on \(Z\) has two basic ingredients, the first being a particular type of element \({\mathbb{M}}\) of the derived category \(D_c^b(Z,{\mathbb{Q}})\) of constructible \({\mathbb{Q}}_Z\)-modules with bounded constructible cohomology, called a perverse sheaf. A form of the Riemann-Hilbert correspondence asserts that its complexification \({\mathbb{M}}_{\mathbb{C}}\in D_c^b(Z,{\mathbb{C}})\) is representable by a regular holonomic \({\mathscr D}_Z\)-module \({\mathcal{M}}\). The second consists of a filtration on (the de Rham resolution of) \({\mathcal{M}}\) and should be regarded as a way of representing the Hodge filtration. This puts for any open subset \(U\subset Z\), a Hodge filtration on \(\operatorname{H}^k(U;{\mathbb{M}}_{\mathbb{C}})\), but here again, its meaning is unclear in general. Yet, as we will see below, there is something of interest to say if we let \(U\) run over a neighbourhood basis of a fixed point in \(Z\). It should be clear from this sketchy description that the complexification \(M_{\mathbb{C}}\) of a Hodge module is entirely given by the filtered object \({\mathcal{M}}\). We denote the category of Hodge modules on \(Z\) by \(\operatorname{MF}(Z,{\mathbb{Q}})\) and its complexification (so as a quotient of \(\operatorname{MF}(Z,{\mathbb{Q}})\)) by \(\operatorname{MF}(Z,{\mathbb{C}})\).

The notion of a polarization of a Hodge module is couched in similar terms (see [4]) and a Hodge module is called polarizable if one exists. The polarizable Hodge modules are the objects of an abelian category \(\operatorname{MF}(Z,{\mathbb{Q}})^p\) and its complexification by \(\operatorname{MF}(Z,{\mathbb{C}})^p\).

The polarizable Hodge modules have as their building blocks the intermediate extensions of (degree shifted) polarizable variations of Hodge structures. To explain, assume that we are given an irreducible subvariety \(X^*\subset Z\) of complex dimension \(m\) and a smooth subvariety \(X\subset X^*\) whose complement in \(X^*\) is a proper (closed) subvariety (which makes \(X\) open-dense in \(X^*\)) and a polarized variation of Hodge structure \({\mathbb{E}}\) on \(X\) as before. Writing \(j: X\subset Z\) for the inclusion, then the intermediate extension \(j_{!*}{\mathbb{E}}[m]\) of \({\mathbb{E}}[m]\) is defined as an object of \(\operatorname{MF}(Z, {\mathbb{Q}})^p\). It gets its name from the fact that we have a natural factorization \[j_!{\mathbb{E}}[m]\to j_{!*}{\mathbb{E}}[m]\to j_{*}{\mathbb{E}}[m]\] in the category \(\operatorname{MF}(Z,{\mathbb{Q}})^p\) and that it can be considered in the abelian category \(\operatorname{MF}(Z,{\mathbb{Q}})^p\) as an image: it has the property that it has no \(\operatorname{MF}(Z,{\mathbb{Q}})\)-subquotients supported by a proper subvariety of \(X^*\).

Every polarizable Hodge modules on \(Z\) is isomorphic with a direct sum of polarizable Hodge modules of this type. We therefore assume in what follows that \(Z=X^*\).

The corresponding extension of \({\mathbb{E}}\) over \(X^*\) without the shift is the intersection complex \({\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}})\). One of Saito’s theorems implies that if \(X^*\) is projective, then the intersection cohomology \[\operatorname{IH}^k(X^*; {\mathbb{E}}):=\operatorname{H}^k(X^*, {\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}))\] has a natural polarizable Hodge structure of weight \(w+k\). it also tells us that for any locally closed subvariety \(i_S: S\subset X^*\), the cohomology sheaves \(R^k i^*_S{\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}))\) resp. \(R^k i^!_S{\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}))\) are variations of mixed Hodge structure over an smooth open-dense subset of \(S\)

Representation by square integrable forms↩︎

Let \({\mathbb{E}}\) and \(j:X\subset X^*\) be as above and suppose that \(X\) is endowed with a Kähler metric. The polarization and the Hodge star operator define an anti-linear map \[\star: {\mathscr A}^{p,q}_X({\mathbb{E}}_{\mathbb{C}})\to {\mathscr A}^{m-p,m-q}_X({\mathbb{E}}_{\mathbb{C}}^\vee)\] such that for every open \(U\subset X\), the hermitian pairing \[\langle\;,\; \rangle: (\alpha, \beta)\in {\mathscr A}^{p,q}_X(U,{\mathbb{E}}_{\mathbb{C}})\times {\mathscr A}^{p,q}_X(U,{\mathbb{E}}_{\mathbb{C}})\mapsto (\sqrt{-1})^{w+m} \alpha\cup\star\beta\in {\mathscr A}^{m,m}_X(U),\] is positive with respect to the orientation (here the expression \(\alpha\cup\star\beta\) combines the cup product and the natural pairing of \({\mathbb{E}}_{\mathbb{C}}\) with its dual). This defines a notion of square integrability: for \(\alpha\in {\mathscr A}^k_X(U,{\mathbb{E}}_{\mathbb{C}})\) is square integrable if \(\langle\alpha, \alpha\rangle\) is integrable over \(U\). The cohomology of the subcomplex of \({\mathscr A}^{\scriptscriptstyle\bullet}(X,{\mathbb{E}}_{\mathbb{C}})\) of forms that are together with their image under \(\nabla\) square integrable, denoted \(\operatorname{H}^{\scriptscriptstyle\bullet}_{(2)}(X, {\mathbb{E}}_{\mathbb{C}})\), is what is called the square integrable cohomology of \({\mathbb{E}}_{\mathbb{C}}\). Zucker observed in [8] that if the Kähler metric on \(X\) is complete and the square integrable cohomology of \({\mathbb{E}}_{\mathbb{C}}\) is finite dimensional, then the classical Hodge theory remains valid in this (possibly) noncompact setting. In particular, \(\operatorname{H}^{\scriptscriptstyle\bullet}_{(2)}(X, {\mathbb{E}}_{\mathbb{C}})\) is harmonically represented and the bigrading subsists and defines a Hodge structure on \(\operatorname{H}^k_{(2)}(X, {\mathbb{E}}_{\mathbb{C}})\) of weight \(k+w\). So the spectral sequence for the Hodge filtration \[\operatorname{H}^q_{(2)}(X,\operatorname{Gr}_{\mathcal{F}}^p \Omega_X^{\scriptscriptstyle\bullet}({\mathcal{E}}))\Rightarrow \operatorname{H}_{(2)}^{\scriptscriptstyle\bullet}(X, {\mathbb{E}}_{\mathbb{C}}),\] where \(\operatorname{Gr}_{\mathcal{F}}^p \Omega_X^{\scriptscriptstyle\bullet}({\mathcal{E}})=\oplus_{p'+p''=p} \Omega_X^{p'}(\operatorname{Gr}_{\mathcal{F}}^{p''}{\mathcal{E}})\), degenerates and endows \(\operatorname{H}_{(2)}^{\scriptscriptstyle\bullet}(X, {\mathbb{E}}_{\mathbb{C}})\) with its Hodge filtration.

A presheaf complex on \(X^*\) is defined by assigning to an open subset \(U\) of \({X^*}\) the subcomplex of \({\mathscr A}^{\scriptscriptstyle\bullet}(U\cap X,{\mathbb{E}}_{\mathbb{C}})\) consisting of forms that together with their image under \(\nabla\) are square integrable. Its sheafication gives a subcomplex of \(j_* {\mathscr A}_X^{\scriptscriptstyle\bullet}({\mathbb{E}}_{\mathbb{C}})\) which we shall denote by \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}}_{\mathbb{C}})\). This complex comes filtered by the Hodge filtration: \[\textstyle {\mathcal{F}}^p{\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}}_{\mathbb{C}})={\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}}_{\mathbb{C}})\cap j_* {\mathcal{F}}^p{\mathscr A}^{{\scriptscriptstyle\bullet}}_X({\mathbb{E}}_{\mathbb{C}}).\] Our goal is to give conditions which imply that this Hodge filtered complex is an incarnation of \({\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}_{\mathbb{C}})\in \operatorname{MF}(X, {\mathbb{C}})^p[-m]\). The following is, in view of Zucker’s harmonic representation, merely a tautology.

In the situations we will be dealing with, \(X^*\) has the following property.

This property ensures that the cohomology sheaves of both the derived object \({\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}_{\mathbb{C}})\) and the complex \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}}_{\mathbb{C}})\) are locally trivial on each stratum. In that case the assumption in 1 regarding the Hodge filtration is equivalent for these two properties for to hold for every \(s\in X^*\) and this is in fact the definition which we will use in practice. We will need a similar interpretation in the situation where we only know that these hypotheses are satisfied on the complement of a point.

2 The case of a Shimura variety↩︎

We want to apply this to the case where \(X\) is locally symmetric variety (and hence quasiprojective) and \(X^*\) the Baily-Borel compactification of \(X\). These data involve a reductive linear \({\mathbb{Q}}\)-group \({\mathcal{G}}\). We write \(G\) for the Lie group \({\mathcal{G}}({\mathbb{R}})\) (¹), and assume that its center \(Z(G)\) is compact. So \(G/Z(G)\) is an adjoint group whose symmetric space \(D\) is that of of \(G\). We assume that \(D\) comes with a \(G\)-invariant complex structure, which makes it a hermitian domain. We further assume given a neat arithmetic subgroup \(\Gamma\subset {\mathcal{G}}({\mathbb{Q}})\). Then \(X_\Gamma:=\Gamma\backslash D\) is a quasi-projective complex manifold. We call such an \(X_\Gamma\) a locally symmetric variety. We often write \(X\) for \(X_\Gamma\), when \(\Gamma\) is understood.

We also assume that we are given a finite dimensional \({\mathbb{Q}}\)-representation \(E\) of \({\mathcal{G}}\). This gives rise to a local system \({\mathbb{E}}={\mathbb{E}}_\Gamma\) on \(X\) (given as the quotient of the trivial local system \(E\times D\to D\) by the diagonal action of \(\Gamma\)). It is convenient (and for what follows no loss of generality) to assume that \({\mathbb{E}}\) is \({\mathbb{Q}}\)-irreducible, for this implies that \({\mathbb{E}}\) admits the structure of a polarizable variation of Hodge structure \({\mathbb{E}}\) over \(X\) of some weight \(w\) with fiber \(E\) (in our set-up \(E\) only determines the parity of \(w\); in the Shimura setting this weight is specified by an action on \(E\) of a central extension of \({\mathcal{G}}\) by a \({\mathbb{Q}}\)-split copy of \({\mathbb{G}}_m\)). We shall refer to such an \({\mathbb{E}}\) as a locally homogeneous polarizable variation of Hodge structure on \(X\). The Baily-Borel compactification \(j:X\hookrightarrow X^*\) completes \(X\) to a normal projective variety. Its boundary \(X^*\smallsetminus X\) comes with a stratification, all of whose members are themselves have the structure of a locally symmetric variety. More details of this construction will be recalled below.

The domain \(D\) comes with a complete \(G\)-invariant Kähler metric and this makes \(X\) a locally homogeneous Kähler manifold. Zucker showed that the natural stratification of \(X^*\) is metrically topologically trivial in the sense of Definition 1 and that each of the sheaves \({\mathscr L}^k_{{X^*},(2)}({\mathbb{E}})\) is fine ([8], Prop.(4.4)). A local form of his conjecture (proved in different ways in [1], [3], [2]) asserts that \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}})\) is an incarnation of \({\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}_{\mathbb{C}})\) in \(D^b_c({X^*}, {\mathbb{C}})\). This implies that we have natural isomorphism \(\operatorname{H}^k_{(2)}(X, {\mathbb{E}}_{\mathbb{C}})\cong \operatorname{IH}^k({X^*}, {\mathbb{E}}_{\mathbb{C}})\), but as mentioned in the introduction, does not imply that the harmonic Hodge decomposition equals Saito’s one. By observation 1 this requires that we establish a filtered local version of the Zucker conjecture in the sense that \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}})\) with its Hodge filtration is a cohomological Hodge complex and represents \({\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X^*}({\mathbb{E}})\) in \(\operatorname{MF}(X^*,{\mathbb{C}})[-\dim_{\mathbb{C}}X]\). The main result of this paper says that this is the case.

2.1 Hecke equivariance↩︎

Before we proceed to give the proof, we observe the functorial behaviour in \(\Gamma\). A subgroup \(\Gamma'\subset \Gamma\) of finite index determines a natural map \(F: X_{\Gamma'} \to X_\Gamma\). Since \(\Gamma\) is neat, this map is a finite unramified covering and \({\mathbb{E}}_{\Gamma'}\) can be identified with \(F^*{\mathbb{E}}_\Gamma\). In particular there is a natural map \(F_*{\mathbb{E}}_{\Gamma'}=F_! F^*{\mathbb{E}}_{\Gamma}\to {\mathbb{E}}_{\Gamma}\). The covering \(F\) extends uniquely to a finite projective morphism \(F: X^*_{\Gamma'} \to X^*_\Gamma\) and so we get natural morphisms \[F^*{\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_\Gamma^*}({\mathbb{E}}_\Gamma)\to {\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_{\Gamma'}^*}({\mathbb{E}}_{\Gamma'}) \quad \text{and}\quad F_!{\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_{\Gamma'}^*}({\mathbb{E}}_{\Gamma'})\to {\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_\Gamma^*}({\mathbb{E}}_\Gamma).\] The first arrow is an isomorphism and the composite map \[{\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_\Gamma^*}({\mathbb{E}}_\Gamma)\to F_*F^*{\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_\Gamma^*}({\mathbb{E}}_\Gamma)=F_!F^*{\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_\Gamma^*}({\mathbb{E}}_\Gamma)\to {\mathscr I}{\mathscr C}^{\scriptscriptstyle\bullet}_{X_\Gamma^*}({\mathbb{E}}_\Gamma)\] is simply multiplication by the degree of the covering. After a degree shift, this little diagram is one of polarized Hodge modules. The covering \(F\) is also a local isometry and hence \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X_\Gamma^*},(2)}({\mathbb{E}}_\Gamma)\) and \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X_{\Gamma'}^*},(2)}({\mathbb{E}}_{\Gamma'})\) possess the same properties and induce maps between the associated spaces of harmonic forms. In particular, the validity of Theorem 3 for \(\Gamma'\) implies its validity for \(\Gamma\). Indeed, that theorem can be phrased (and is probably best understood) in adelic language.

This also makes it clear that we have at our disposal an action of the Hecke algebra on both \({\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}_{\mathbb{C}})\) and \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}})\). Concretely, given \(g\in {\mathcal{G}}({\mathbb{Q}})\), then \(g\Gamma g^{-1}\cap \Gamma\) is of finite index in \(\Gamma\) and \(g^{-1}\Gamma g\), and hence gives rise to a finite (Hecke) correspondence \[T_g: X_\Gamma\xleftarrow{F} X_{\Gamma\cap (g^{-1}\Gamma g)} \stackrel{g.}{\cong}X_{(g\Gamma g^{-1})\cap \Gamma}\xrightarrow{F'} X_\Gamma\] where \(F\) and \(F'\) are the natural maps and the middle isomorphism is induced by the action of \(g\) on \(D\). This gives us endomorphisms \(T_g^*\) of the polarized Hodge module \({\mathscr I}{\mathscr C}_{X^*}^{\scriptscriptstyle\bullet}({\mathbb{E}}_{\mathbb{C}})\) (where the polarization gets multiplied by the degree of this correspondence) and the complex \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}})\) with the latter inducing an endomorphism of the space harmonic forms. This is relevant here, because this leads us to impose natural boundary conditions (namely equivariance with respect to local Hecke operators) on the complex \({\mathscr L}^{\scriptscriptstyle\bullet}_{{X^*},(2)}({\mathbb{E}})\) which respect the Hodge filtration and which are inherited by the harmonic forms. These boundary conditions were key in our original proof of the Zucker conjecture [1].

2.2 A brief review of the Baily-Borel compactification↩︎

What follows can be found in [9], but is presented here in a somewhat more geometric spirit, as for example in [10].

We recall that \(X^*\) appears as an orbit space of the Baily-Borel-Satake extension \(D^*\) of the symmetric domain \(D\). That extension is a disjoint union of hermitian domains endowed with the so-called horocyclic topology. It contains \(D\) as an open subset and \(D^*\smallsetminus D\) consists of the rational boundary components of \(D\). The closure of each rational boundary component in \(D^*\) yields its own Baily-Borel-Satake extension. The action of \(G\) on \(D\) does not extend to \(D^*\), but the action of \({\mathcal{G}}({\mathbb{Q}})\) does.

The rational boundary components are in bijective correspondence with the maximal proper \({\mathbb{Q}}\)-parabolic subgroups of \({\mathcal{G}}\). We first recall how this comes about.

Let \({\mathcal{P}}\subsetneq {\mathcal{G}}\) be a maximal proper \({\mathbb{Q}}\)-parabolic subgroup. The Lie group \(P\) underlying \({\mathcal{P}}({\mathbb{R}})\) acts transitively on \(D\). The unipotent radical \({\mathcal{R}}_u({\mathcal{P}})\) is nontrivial, but at most 2-step nilpotent: the center \({\mathcal{U}}_{\mathcal{P}}\) of \({\mathcal{R}}_u({\mathcal{P}})\) is nontrivial and \({\mathcal{V}}_{\mathcal{P}}:={\mathcal{R}}_u({\mathcal{P}})/{\mathcal{U}}_{\mathcal{P}}\) is abelian. So these are vector groups and the commutator map produces a surjective map \(\wedge_{\mathbb{R}}^2\operatorname{Lie}(V_{\mathcal{P}})\to \operatorname{Lie}(U_{\mathcal{P}})\). The reductive (Levi) quotient \({\mathcal{L}}_{\mathcal{P}}:={\mathcal{P}}/{\mathcal{R}}_u({\mathcal{P}})\) acts on \(\operatorname{Lie}(V_{\mathcal{P}})\) and \(\operatorname{Lie}(U_{\mathcal{P}})\) and the commutator map is equivariant for these actions.

The center of \({\mathcal{L}}_{\mathcal{P}}\) contains a distinguished \({\mathbb{Q}}\)-split copy \({\mathcal{A}}_{\mathcal{P}}\) of \({\mathbb{G}}_m\). This copy comes with a (character) isomorphism \(\chi_{\mathcal{P}}: {\mathcal{A}}_{\mathcal{P}}\cong {\mathbb{G}}_m\) such that its action on \(\operatorname{Lie}({\mathcal{U}}_{\mathcal{P}})\) resp. \(\operatorname{Lie}({\mathcal{V}}_{\mathcal{P}})\) is with the character \(\chi_{\mathcal{P}}^2\) resp.\(\chi_{\mathcal{P}}\) We denote the inverse of \(\chi_{\mathcal{P}}\) (when regarded as a morphism) by \(\alpha_{\mathcal{P}}:{\mathbb{G}}_m\cong {\mathcal{A}}_{\mathcal{P}}\subset {\mathcal{L}}_{\mathcal{P}}\).

The adjoint action \({\mathcal{L}}_{\mathcal{P}}\) on \({\mathcal{U}}_{\mathcal{P}}\) leads to an almost decomposition (in the sense that intersections of factors are finite and central) \[{\mathcal{L}}_{\mathcal{P}}={\mathcal{M}}^\ell_{\mathcal{P}}{\mathcal{A}}_{\mathcal{P}}{\mathcal{M}}^h_{\mathcal{P}},\] where \({\mathcal{M}}^h_{\mathcal{P}}\) acts trivially on \({\mathcal{U}}_{\mathcal{P}}\), whereas \({\mathcal{M}}^\ell_{\mathcal{P}}\) acts on \({\mathcal{U}}_{\mathcal{P}}\) with finite kernel. The Lie group \(M^h_{\mathcal{P}}={\mathcal{M}}^h_{\mathcal{P}}({\mathbb{R}})\) has compact center and its symmetric space is in a natural manner a hermitian domain \(D_{\mathcal{P}}\) (the complex structure is preserved by \(M^h_{\mathcal{P}}\)): this is the boundary component associated with \({\mathcal{P}}\) as mentioned above. The complex manifold \(D_{\mathcal{P}}\) is also charactrized by the property that there is a natural \(P\)-equivariant holomorphic surjection \[\pi_{\mathcal{P}}: D\to D_{\mathcal{P}}\] whose fibers are the \(M^\ell_{\mathcal{P}}A_{\mathcal{P}}^\circ\)-orbits, where \(A_{\mathcal{P}}^\circ\) is the identity component of \(A_{\mathcal{P}}={\mathcal{A}}_{\mathcal{P}}({\mathbb{R}})\) (so \(A_{\mathcal{P}}^\circ\) is isomorphic with the Lie group \({\mathbb{R}}_{>0}\), the positive reals regarded as a multiplicative group). The group \(A_{\mathcal{P}}^\circ\) is also responsible for the geodesic flow in the fibers of \(\pi_{\mathcal{P}}\) towards \(D_{\mathcal{P}}\): for every \(z\in D\), there is a unique Lie lift \(\alpha_{{\mathcal{P}},z}: {\mathbb{R}}_{>0}\to P\) of \(\alpha_{\mathcal{P}}:{\mathbb{R}}_{>0}\cong A_{\mathcal{P}}^\circ\) whose orbit through \(z\) is a geodesic which has \(\pi_{\mathcal{P}}(z)\) as its limit at \(+\infty\).

A central role in the definition of the horocyclic topology is played by a nonempty strictly convex self-dual cone \(C_{\mathcal{P}}\) in the Lie algebra \(\operatorname{Lie}(U_{\mathcal{P}})\). This cone is homogeneous: it is an \(M^\ell_{\mathcal{P}}\)-orbit. There is a natural \(P\)-equivariant projection \(\rho_{\mathcal{P}}: D\to C_{\mathcal{P}}\). We combined this with \(\pi_{\mathcal{P}}\) to get a \(P\)-equivariant map \[\label{eqn:nilfactor0} (\rho_{\mathcal{P}},\pi_{\mathcal{P}}): D\to C_{\mathcal{P}}\times D_{\mathcal{P}}.\tag{2}\] As mentioned, \(P\) acts transitively on \(D\) and it is clear that it acts on \(C_{\mathcal{P}}\times D_{\mathcal{P}}\) via \(L_{\mathcal{P}}\). In fact, the map 2 is a principal \(R_u({\mathcal{U}}_{\mathcal{P}})\) bundle. This also shows that the complex fiber dimension of \(\pi_{\mathcal{P}}: D\to D_{\mathcal{P}}\) is \(\dim\operatorname{Lie}(U_{\mathcal{P}})+{\tfrac{1}{2}}\dim\operatorname{Lie}(V_{\mathcal{P}})\) (and that \(\dim\operatorname{Lie}(V_{\mathcal{P}})\) must be even).

Let \(C^+_{\mathcal{P}}\) stand for the convex hull of the rational points of the closure of \(C_{\mathcal{P}}\) in \(\operatorname{Lie}(U_{\mathcal{P}})\) (or equivalently, in \(\operatorname{Lie}(G)\), because \(\operatorname{Lie}(U_{\mathcal{P}})\subset \operatorname{Lie}(G)\) is subspace defined over \({\mathbb{Q}}\)). This is a convex cone whose faces are also convex cones (and open in their linear span). Any such face not equal to the vertex \(\{0\}\) is of the form \(C_{{\mathcal{P}}'}\) for some maximal proper rational parabolic group \({\mathcal{P}}'\). We shall write \(D_{\mathcal{P}}\le D_{{\mathcal{P}}'}\) if \(C_{{\mathcal{P}}'}\) is a face of \(C^+_{\mathcal{P}}\) (written as \(C_{{\mathcal{P}}'}\le C_{{\mathcal{P}}}\); it is therefore reasonable to think of the vertex \(\{0\}\) as the face associated to \({\mathcal{G}}\) or to \(D\)). Let us note here that since the groups \(A^\circ_{\mathcal{P}}\) and \(A^\circ_{{\mathcal{P}}'}\) both act as scalar multiplication in \(U_{{\mathcal{P}}'}\), they can be naturally identified.

So the union of \(D\) and the rational boundary components \(\ge D_{\mathcal{P}}\) is a subset \(\operatorname{Star}(D_{\mathcal{P}})\) of \(D^*\) whose constituents are indexed by the faces of \(C^+_{\mathcal{P}}\). The projection \((\rho_{\mathcal{P}},\pi_{\mathcal{P}}): D\to D_{\mathcal{P}}\) extends to \((\rho_{\mathcal{P}},\pi_{\mathcal{P}}): \operatorname{Star}(D_{\mathcal{P}})\to C^+_{\mathcal{P}}\times D_{\mathcal{P}}\).
The subgroup \(\Gamma_{\mathcal{P}}\) of \({\mathcal{P}}\) is arithmetic (and hence a lattice in \(P\)) and acts on \(D_{\mathcal{P}}\) via an arithmetic quotient group whose orbit space is a stratum \(S\) of \(X^*\). We denote the kernel of this \(\Gamma_{\mathcal{P}}\)-action by \(\Gamma_{\mathcal{P}}^\ell\subset \Gamma_{\mathcal{P}}\).

We recall from [9] that a core in \(C_{\mathcal{P}}\) is a subset \(K\subset C_{\mathcal{P}}\) which is invariant under the translations in \(C_{{\mathcal{P}}^\ell}\) and for which there exist \(p_0, p_1\in C_{\mathcal{P}}\) such that \(\Gamma_{{\mathcal{P}}}^\ell (p_0+C_{\mathcal{P}})\subset K\subset \Gamma_{{\mathcal{P}}}^\ell (p_0+C_{\mathcal{P}})\). This notion does not depend on \(\Gamma\) and the convex hull of \(K\) is then also a core (see also [11]). In this paper, we shall use this notion in a more restricted sense: we call a core \(K\subset C_{\mathcal{P}}\) a \(\Gamma_{\mathcal{P}}^\ell\)-core if in addition \(K\) is a \(\Gamma_{\mathcal{P}}^\ell\)-invariant, open, convex and has smooth boundary. It is clear that \(\lambda K=\alpha_{\mathcal{P}}(\sqrt{\lambda})K\) is then also a core for every \(\lambda >0\).

A relative version is defined in an obvious manner: a \(\Gamma_{\mathcal{P}}\)-core is an open \(\Gamma_{\mathcal{P}}\)-invariant subset \({\mathcal{K}}\) of \(C_{\mathcal{P}}\times D_{\mathcal{P}}\) with smooth boundary which meets each fiber over \(D_{\mathcal{P}}\) in a \(\Gamma_{\mathcal{P}}^\ell\)-core. So \(\Omega_{\mathcal{K}}:=(\rho_{\mathcal{P}},\pi_{\mathcal{P}})^{-1}{\mathcal{K}}\) is then an open subset of \(D\) which is invariant under both the action of the nilpotent Lie group \(R_u({\mathcal{P}})\) and the geodesic flow towards \(D_{\mathcal{P}}\). We extend it to a subset \(\hat{\Omega}_{\mathcal{K}}\) of \(\operatorname{Star}(D_{\mathcal{P}})\) by stipulating that \(\hat{\Omega}_{\mathcal{K}}\cap D_{{\mathcal{P}}'}=\pi_{{\mathcal{P}}'}(\Omega_{\mathcal{K}})\) when \(D_{\mathcal{P}}'\ge D_{\mathcal{P}}\). Then \(\hat{\Omega}_{\mathcal{K}}\) is also invariant under \(R_u({\mathcal{P}})\) and the geodesic flow towards \(D_{\mathcal{P}}\).

It is a fundamental fact (and in a sense the reason for introducing cores in the first place) that there exists a \(\Gamma_{\mathcal{P}}\)-core \({\mathcal{K}}\) such that every \(\Gamma_{\mathcal{P}}\)-orbit in \(\Omega_{\mathcal{P}}\) is the intersection of \(\hat{\Omega}_{\mathcal{P}}\) with a \(\Gamma\)-orbit. In other words, the orbit set \(\hat{U}_S=\Gamma_{\mathcal{P}}\backslash\Omega\) can then be regarded as a subset of \(X^*\). This leads us to the horocyclic topology on \(D^*\): it is the coarsest topology which for every \({\mathcal{P}}\) makes the subsets \(\{\hat{\Omega}_{\mathcal{K}}\}_{\mathcal{K}}\) a basis of neighborhoods of \(D_{\mathcal{P}}\) in \(D^*\) and the projection \(\pi_{\mathcal{P}}: \operatorname{Star}(D_{\mathcal{P}})\to D_{\mathcal{P}}\) continuous. The justification is that if we endow \(X^*\) with the quotient topology, it becomes a locally compact Hausdorff space and that the sheaf on \(X^*\) of continuous \({\mathbb{C}}\)-valued functions which are analytic on each stratum endows \(X^*\) with the structure of a normal analytic variety

A fundamental regular neighbourhood \(\hat{U}_S=\Gamma_{\mathcal{P}}^\ell\backslash\hat{\Omega}_{\mathcal{K}}\) of \(S\) in \(X^*\) has several nice properties. First of all, it is invariant under a geodesic flow towards \(S\). To be precise, it takes \(\hat{U}_S\) to \(\mu\hat{U}_S:=\Gamma_{\mathcal{P}}\backslash\Omega_{\mu{\mathcal{K}}}\) for every \(\mu\ge 1\). This defines a homeomorphism \((0,\infty)\times \partial\hat{U}_S\cong \hat{U}_S\smallsetminus S\) which extends to a stratified homeomorphism of \(\hat{U}_S\) onto the open mapping cone of \(\pi_S|\partial\hat{U}_S\) (which is here defined as the quotient of \((0, \infty]\times \partial\hat{U}_S\) obtained by collapsing \(\{\infty\}\times \partial\hat{U}_S\cong \partial\hat{U}_S\) along \(\pi_S\)). Secondly, the projection \(\pi_S:\partial\hat{U}_S\to S\) is topologically locally trivial in a strata preserving manner. And thirdly, \(\pi_S: U_S:=\Gamma_{\mathcal{P}}\backslash\Omega_{\mathcal{K}}\to S\) factors over a fibration into compact nilmanifolds \[\label{eqn:nilfactor} U_S\xrightarrow{\rho_S} \Gamma_{{\mathcal{L}}_{\mathcal{P}}}\backslash{\mathcal{K}}\xrightarrow{\pi'_S} S,\tag{3}\] where \(\Gamma_{{\mathcal{L}}_{\mathcal{P}}}\) is the image of \(\Gamma_{\mathcal{P}}\) in \(L_{\mathcal{P}}\). The fiber of \(\rho_S\) through any point of \(U_S\) is naturally identified with the nilmanifold \(R_u({\mathcal{P}})/\Gamma_{R_u({\mathcal{P}})}\) (this is a torus bundle over a torus; in particular, \(\rho_S\) is proper). The fiber of \(\pi'_S\) over \(s\in S\) is the \(\Gamma_{\mathcal{P}}^\ell\)-orbit space of the core \({\mathcal{K}}_s\). There exist metrically locally trivial trivializations of \(\hat{U}_S\to S\) in the sense of Definition 1 compatible with the geodesic flow towards \(S\).

2.3 Proof of the Theorem 3↩︎

We begin with reviewing the proof of the Zucker conjecture as given in [1] and [2]. Every \(g\in {\mathcal{P}}^\ell({\mathbb{Q}})\) induces Hecke correspondence \(T^{\mathcal{P}}_g\) of the germ of \(X^*\) at \(S\), which is proportional to the identity over \(S\). It is a local isometry which lifts to \({\mathbb{E}}\). So if \(i_S: S\subset X^*\) is the inclusion, then \(T^{\mathcal{P}}_g\) induces an endomorphism \(T^{\mathcal{P}}_g{}^*\) of the mixed Hodge module \(i_S^*j_{!*}{\mathbb{E}}\), the intersection complex \(i_S^*{\mathscr I}{\mathscr C}_{\Omega}^{\scriptscriptstyle\bullet}({\mathbb{E}})\) and \(i_S^*{\mathscr L}^{\scriptscriptstyle\bullet}_{\Omega \smallsetminus S,(2)}({\mathbb{E}})\).

In case \(g\) is such that \(g\Gamma_{\mathcal{P}}g^{-1}\subset\Gamma_{\mathcal{P}}\), then it is the identity over \(S\) and we can choose the \(\Gamma\)-core \({\mathcal{K}}\) such that \(g\) maps a \({\mathcal{K}}\) to itself and hence defines in fact a finite map from a fundamental regular neighborhood \(\hat{U}_S\) as above onto neighborhood of \(S\) contained in \(\hat{U}_S\).

For example, if we take \(g\in {\mathcal{R}}_u({\mathcal{P}})({\mathbb{Q}})\), then \(T^{\mathcal{P}}_g\) preserves each fiber of \(\rho_S\). Insofar the action on differential forms is concerned, we may take these forms in such a manner that their pull-back to \(D\) is invariant under the nilpotent Lie group \(R_u({\mathcal{P}})\) (the harmonic forms will automatically have this property). The Hecke action of \(T^{\mathcal{P}}_g{}^*\) on such forms is trivial. This explains why for most of its uses, \(T^{\mathcal{P}}_g{}^*\) only depends on the image \(\bar g\) of \(g\) in the Levi quotient.

Of special interest here is the case when \(\bar g\in {\mathcal{A}}_{\mathcal{P}}({\mathbb{Q}})\). If \(\chi_{\mathcal{P}}(\bar g)=\sqrt{q}\) (or equivalently, \(\alpha_{\mathcal{P}}(\sqrt{q})=\bar g\)), then \(g\Gamma_{\mathcal{P}}g^{-1}\subset\Gamma_{\mathcal{P}}\), provided \(q\) is a positive square such that \(q-1\) is sufficiently divisible. We assume this is the case and then write \(T^S(q)\) for \(T^{\mathcal{P}}_g\). The action of \(T^S(q)\) on \(U_S\) is geometrically easy to understand in terms of the factorization 3 : since \(g\) acts as multiplication by \(q\) resp. \(\sqrt{q}\) on \(\operatorname{Lie}(U_{\mathcal{P}})\) resp. \(\operatorname{Lie}(V_{\mathcal{P}})\), the map \(T^S(q)\) takes \([\Gamma_{\mathcal{P}}^\ell y]\in \Gamma_{\mathcal{P}}^\ell\backslash{\mathcal{K}}\) to \([\Gamma_{\mathcal{P}}^\ell qy]\in \Gamma_{\mathcal{P}}^\ell\backslash{\mathcal{K}}\) with the map on their nilmanifold fibers being an isogeny: recall that the nilmanifold is a torus bundle and that over a torus and \(T^S(q)\) acts in the base torus as multiplication by \(\sqrt{q}\) and in the fiber torus as multiplication by \(q\). So \(T^S(q)\) induces on the germ of \(X^*\) at \(S\) a morphism of degree \[q^{\dim\operatorname{Lie}(U_{\mathcal{P}})} (\!\sqrt{q})^{\dim\operatorname{Lie}(V_{\mathcal{P}})}=q^{\dim\operatorname{Lie}(U_{\mathcal{P}})+{\tfrac{1}{2}}\dim\operatorname{Lie}(V_{\mathcal{P}})}=q^d,\] where \(d\) is the complex codimension of \(S\) in \(X^*\). The geodesic flow away from \(S\) produces an isotopy of \(T^S(q)\) with a self map of \(\hat{U}_S\) which is the identity over \(\Gamma_{{\mathcal{L}}_{\mathcal{P}}}\backslash{\mathcal{K}}\). This self map then induces in each in each fiber over \(\Gamma_{{\mathcal{L}}_{\mathcal{P}}}\backslash{\mathcal{K}}\) an isogeny as above. Both the singular cochain complex and the \({\mathbb{E}}\)-valued de Rham complex of the fiber through a given point are naturally quasi-isomorphic with the Lie algebra subcomplex \[\label{eqn:} \operatorname{Hom}_{\mathbb{R}}( \wedge^{\scriptscriptstyle\bullet}\operatorname{Lie}(R_u({\mathcal{P}}), E).\tag{4}\] The action of \(T^S(q)\) is in this subcomplex given by the lift \(g\in {\mathcal{P}}({\mathbb{Q}})\) of \(\bar g=\alpha_{\mathcal{P}}(\sqrt{q})\) and hence is semisimple with eigenvalues powers of \(\sqrt{q}\). This is of course merely the evaluation of the action of a one-parameter subgroup on this Lie algebra complex. In this form it already appeared earlier in the work of Borel [12] and Zucker [13]. It shows that the total direct image of \({\mathbb{E}}|U_S\) on \(\Gamma_{{\mathcal{L}}_{\mathcal{P}}}\backslash{\mathcal{K}}\) decomposes in the derived category according to the integral powers of \(\sqrt{q}\). This implies the semisimplicity of \(T^S(q)\)-actions on various cohomology groups.

Let \(s\in S\), denote by \(U_s\subset \hat{U}_s\) the fiber of \(U_S\subset \hat{U}_S\) over \(s\) and put \({\mathbb{E}}_s:={\mathbb{E}}|U_s\). By the defining properties of the intersection complex, \[\operatorname{IH}^k(\hat{U}_s, {\mathbb{E}}_s )= \begin{cases} \operatorname{IH}^k(\hat{U}_s\smallsetminus\{s\}, {\mathbb{E}}_s)&\text{if k<d},\\ 0 & \text{if k\ge d.} \end{cases}\] \[\operatorname{IH}_c^k(\hat{U}_s, {\mathbb{E}}_s)= \begin{cases} \operatorname{IH}_c^{k}(\hat{U}_s\smallsetminus\{s\}, {\mathbb{E}}_s)&\text{if k>d},\\ 0 & \text{if k\le d} \end{cases}\] and the polarization pairing on \({\mathbb{E}}_s\) induces the Poincaré duality pairing \[\operatorname{IH}^k(\hat{U}_s\smallsetminus\{s\}, {\mathbb{E}}_s)\times \operatorname{IH}_c^{2d-k}(\hat{U}_s\smallsetminus\{s\}, {\mathbb{E}}_s)\to \operatorname{IH}_c^{2d}(\hat{U}_s\smallsetminus\{s\}; {\mathbb{Q}}) (\cong {\mathbb{Q}}).\] The pair \((\hat{U}_s, {\mathbb{E}}_s)\) comes with an endomorphism \(T^s(q)\) of degree \(q^d\) which acts semisimply on \(\operatorname{IH}^{\scriptscriptstyle\bullet}(\hat{U}_s\smallsetminus\{s\},{\mathbb{E}}_s)\) with eigenvalues powers of \(\sqrt{q}\). The pairing is \(T^s(q)\)-equivariant, where \(T^s(q)\) acts on \(\operatorname{IH}^{2d}(\hat{U}_s\smallsetminus\{s\}; {\mathbb{Q}})\) as multiplication by \(q^d\). We restate this in terms of (a derived category of) sheaf complexes: letting \(j_S\) stand for the inclusion \(\hat{U}_S\smallsetminus S\subset \hat{U}_S\), then \[\label{eqn:dec} i_S^*j_{S*}{\mathscr I}{\mathscr C}_{\hat{U}_S\smallsetminus S}^{\scriptscriptstyle\bullet}({\mathbb{E}})= i_S^*{\mathscr I}{\mathscr C}_{\hat{U}_S}^{\scriptscriptstyle\bullet}({\mathbb{E}})\oplus i_S^!{\mathscr I}{\mathscr C}_{\hat{U}_S}^{\scriptscriptstyle\bullet}({\mathbb{E}})[-1],\tag{5}\] with the first summand living in degrees \(<d\) and the second in degrees \(>d\). This comes a perfect duality pairing (in a derived category) \[i_S^*j_{S*}{\mathscr I}{\mathscr C}_{\hat{U}_S\smallsetminus S}^{\scriptscriptstyle\bullet}({\mathbb{E}})\otimes_{{\mathbb{Q}}_S} i_S^!j_{S*}{\mathscr I}{\mathscr C}_{\hat{U}_S\smallsetminus S}^{\scriptscriptstyle\bullet}({\mathbb{E}})\to {\mathbb{Q}}_S(-d)[-2d].\]

The following is proved in [1] (Prop.) (see also [2], Thm.). The semisimplicity of the action of \(T^S(q)\) on \(R^{\scriptscriptstyle\bullet}i_S^*j_{S*}{\mathscr I}{\mathscr C}_{\hat{U}_S\smallsetminus S}^{\scriptscriptstyle\bullet}({\mathbb{E}})\) is not stated there explicitly, but follows from the preceding discussion.

The proof of our main theorem will also use the following simple linear algebra observation, stated here as a lemma.

References↩︎