2.1.1. Let k be a field of characteristic 0, fixed for the whole article. Let $ {\mathbb {M}_m} $ be $ \mathbb {A} ^1$ with its unital multiplicative monoid structure over k and, as usual, let $ {\mathbb {G}_m} \hookrightarrow {\mathbb {M}_m} $ be the open subscheme of the multiplicative group. Denote by $\varepsilon $ the unit of $ {\mathbb {M}_m} $ or $ {\mathbb {G}_m} $ and by $\mu $ the multiplication.

Let n be a positive integer, and let X be a formal scheme over k with an action of $ {\mathbb {M}} _m^n$ , that is with a given morphism

such that the diagram

is commutative and such that the composition

is the identity. By restriction along $ \mathbb {G} _m^n \hookrightarrow {\mathbb {M}} _m^n$ , there is, in particular, also a $ \mathbb {G} _m^n$ -action on X.

2.1.4. On a toroidal formal scheme X, we also have a ring-sheaf $ {\cal O} _{X_0}$ which locally gives the $R_0$ ’s and the $ {\cal O} _{X,v}$ which are coherent $ {\cal O} _{X_0}$ -submodules of $ {\cal O} _X$ . The topological space X together with $ {\cal O} _{X,0}$ is a scheme, and it is isomorphic to the categorical quotient (in the category of formal schemes) of X with respect to the action of $ {\mathbb {M}} _m^n$ . It is denoted by $X_0$ . Furthermore, there is an obvious section (a closed embedding) $X_0 \hookrightarrow X$ .

Proof (sketch).

$1 \leftrightarrow 2$ : Given a module M, the associated $M_v$ is just the $ {\cal O} _{X_0}$ -submodule of elements transforming with weight v under $ \mathbb {G} _m^n$ . Conversely, the module M is given as the product of the modules $M_v$ .

$2 \leftrightarrow 3$ : A collection $M_v$ is associated with the functor $v \mapsto M(v) := M_v \otimes {\cal O} _{X,-v}$ . Here, for arbitrary $v \in \mathbb {Z} ^n$ , we set

$$\begin{align*}{\cal O} _{X,v} := \bigotimes_i {\cal O} _{X,e_i}^{\otimes v_i}. \end{align*}$$

A morphism $v \rightarrow w$ in $ \mathbb {Z} ^n$ is mapped to the morphism

$$\begin{align*}M_v \otimes_{ {\cal O} _{X,0}} {\cal O} _{X,-v} \rightarrow M_w \otimes_{ {\cal O} _{X,0}} {\cal O} _{X,-w} \end{align*}$$

induced by

$$\begin{align*}{\cal O} _{X,w-v} \otimes_{ {\cal O} _{X,0}} M_v \rightarrow M_{w}. \end{align*}$$

The functoriality of the functor M is equivalent to the associativity of the multiplication on the module M.

Proof. Left to the reader.

2.2.5. Let M be a coherent sheaf on X with a compatible action of $ \mathbb {G} _m^{n}$ . We have its associated functor $M: \mathbb {Z} ^{n} \rightarrow {[{ {\mathcal {O}_{X_0}}\textbf{-}\mathbf {coh} }]} $ . As said, there is an N, such that $M(\sum \alpha _i e_i)$ is (essentially) constant in $\alpha _i$ if $\alpha _i> N$ . We denote this sheaf by $\lim _{\alpha \rightarrow \infty } M(v + \alpha e_i)$ . Note that also expressions like $\lim _{\alpha _{1} \rightarrow \infty , \dots , \alpha _{j} \rightarrow \infty } M(v + \alpha _1 e_{i_1} + \cdots + \alpha _j e_{i_j})$ do make sense (up to isomorphism). Given an injection $\beta : [j] \hookrightarrow [n]$ , we will regard this construction with respect to the missing indices in the image of $\beta $ as a functor

$$\begin{align*}\lim_{\beta}: \operatorname{\mathrm{Fun}}( \mathbb{Z} ^{n}, {[{ {\mathcal{O}_{X_0}}\textbf{-}\mathbf{coh} }]} )^{f.g.} \rightarrow \operatorname{\mathrm{Fun}}( \mathbb{Z} ^{j}, {[{ {\mathcal{O}_{X_0}}\textbf{-}\mathbf{coh} }]} )^{f.g.}. \end{align*}$$

We just write ‘ $\lim $ ’ for this construction with respect to all indices.

2.2.6. For coherent, torsion-free sheaves M and N, we can describe the tensor product $M \otimes N$ with its natural $ {\mathbb {M}} _m^n$ action by the functor

$$\begin{align*}(M \otimes N)(v) = \sum_{v_1+v_2=v} M(v_1)\otimes N(v_2) ,\end{align*}$$

where the sum is formed in $(\lim M) \otimes (\lim N)$ .

2.2.7. For any injection $\beta : [j] \hookrightarrow [n]$ , define a sheaf $ {\cal O} _X[\beta ^{-1}]$ as the sheafification of the presheaf defined (for small enough U) by

$$\begin{align*}U \mapsto {\cal O} _X(U)[x_{k_1}^{-1}, \dots, x_{k_{n-j}}^{-1}], \end{align*}$$

where $\{k_1, \dots , k_{n-j}\}$ is the complement of $\mathrm {im}(\beta )$ and the $x_i$ are generators of $ {\cal O} _{X,e_i}$ . To a coherent (in the sense of modules on ringed spaces) $ {\cal O} _X[\beta ^{-1}]$ -module with $ \mathbb {G} _m^n$ -action, we may still associate (in the same way as in Proposition 2.2.1) a functor in $\operatorname {\mathrm {Fun}}( \mathbb {Z} ^{n}, {[{ {\mathcal {O}_{X_0}}\textbf{-}\mathbf {coh} }]} )$ . This yields a fully-faithful functor

$$\begin{align*}{[{ {\mathcal{O}_X[\beta^{-1}]}\textbf{-}\mathbf{tcoh} }]} \rightarrow \operatorname{\mathrm{Fun}}( \mathbb{Z} ^{n}, {[{ {\mathcal{O}_{X_0}}\textbf{-}\mathbf{coh} }]} ) \end{align*}$$

which has the property that the functors in the image are essentially constant in the direction of the $e_{k_i}$ .

The corresponding localisation for modules is given by the $\lim $ -construction of Section 2.2.5. More precisely, the diagram

is 2-commutative. Here, $p_{\beta }^*$ is the pullback induced by the projection $p_{\beta }: \mathbb {Z} ^n \rightarrow \mathbb {Z} ^{j}$ induced by $\beta $ . The sheaf $ {\cal O} _X[\beta ^{-1}]$ can be completed afterwards with respect to any of the ideals generated by $ {\cal O} _{X,e_i}$ for $i \in \mathrm {im}(\beta )$ (for $i \not \in \mathrm {im}(\beta )$ the completion would be zero). This process of inverting elements and completion might be repeated. Any sheaf R of $ {\cal O} _X$ -algebras so obtained (which still carries an action of $ \mathbb {G} _m^n$ ) still yields a fully-faithful functor

$$\begin{align*}{[{ {R}\textbf{-}\mathbf{tcoh} }]} \rightarrow \operatorname{\mathrm{Fun}}( \mathbb{Z} ^{n}, {[{ {\mathcal{O}_{X_0}}\textbf{-}\mathbf{coh} }]} ) \end{align*}$$

whose image consists of functors that are constant in the direction of the $e_{\beta (i)}$ for those i, such that (locally) a generator $x_i$ has been inverted. An inverse functor on the essential image might be quite complicated to describe. Its values are given as a subset of the infinite product that was considered in Proposition 2.2.1, but the sequences might be for instance bounded below in some direction, point-wise with respect to another direction. Since we will not need it, we will not elaborate on this.

A $ \mathbb {G} _m^n$ -equivariant coherent $ {\cal O} _X[\emptyset ^{-1}]$ -module $\widetilde {M}$ (where $\emptyset : [0] \rightarrow [n]$ is the inclusion of the empty set) is equivalent to just an $ {\cal O} _{X_0}$ -module via $\widetilde {M} \mapsto \widetilde {M}(0)$ . Each $ {\cal O} _{X_0}$ -module $M_0$ in turn has a canonical extension to an $ {\cal O} _X$ -Module with $ {\mathbb {M}} _m^n$ -action, given by means of the functor

$$\begin{align*}M_0(v) = \begin{cases} M_0 & \text{if }v\in \mathbb{Z} ^n_{\ge 0}, \\ 0 & \text{otherwise,}\end{cases} \end{align*}$$

or equivalently by $M := M_0 \otimes _{ {\cal O} _{X,0}} {\cal O} _X$ with its natural $ {\mathbb {M}} _m^n$ -action. We denote the full subcategory of $ {[{ {\mathcal {O}_X}\textbf{-}\mathbf {tcoh} }]} $ consisting of canonical extensions by $ {[{ {\mathcal {O}_X}\textbf{-}\mathbf {tcoh}\textbf{-}\mathbf{can} }]} $ .

2.2.8. There is the following exact sequence (equivariant with respect to the action of $ {\mathbb {M}} _m^{n}$ ) of coherent sheaves on X:

where $\sum _{i} {\cal O} _{X,e_i} \otimes _{ {\cal O} _{X_0}} {\cal O} _X $ is isomorphic to the bundle $\widehat {\Omega }_{X/X_0}$ . The bundle $\widehat {\Omega }_X$ is not a canonical extension. There is the larger bundle $\widehat {\Omega }_{X}(\log )$ which is locally generated by $\widehat {\Omega }_{X}$ and by the rational differentials $\frac {\operatorname {\mathrm {d}} x_i}{x_i}$ . The latter are invariant under the action of $ {\mathbb {M}} _m^{n}$ . We proceed to describe the associated functors of the $ {\mathbb {M}} _m^{n}$ -equivariant vector bundles $\widehat {\Omega }_{X}$ and $\widehat {\Omega }_{X}(\log )$ .

Consider the Atiyah extensions on $X_0$ associated with the line bundles $ {\cal O} _{X,e_i}$

and their amalgamated sum

(3)

Then $\widehat {\Omega }_X(\log )$ is just the canonical extension of E, that is, it is given by the functor

$$\begin{align*}\widehat{\Omega}_X(\log)(v) = \begin{cases} E & \text{if }v \ge 0, \\ 0 & \text{otherwise}. \end{cases} \end{align*}$$

2.3.1. Let M be a smooth k-variety. Consider an open embedding $M \hookrightarrow \overline {M}$ into a smooth k-variety (mostly assumed to be proper), such that $D:=\overline {M} \setminus M$ is a divisor with strict normal crossings. Consider the coarsest statification $\overline {M} = \bigcup _{Y \in \mathcal {S}} Y$ into locally closed subsets, such that all components of D are closures of a stratum in the finite set $\mathcal {S}$ . The variety M itself will be the unique open stratum. Denote by $n_Y$ the codimension of $\overline {Y}$ . Consider, furthermore, a toroidal action $\rho _Y$ of $ {\mathbb {M}} _m^{n_Y}$ on the formal completion $M_Y:=C_{\overline {Y}}(\overline {M})$ of $\overline {M}$ along $\overline {Y}$ which hence establishes $\overline {Y}$ as the invariant subscheme $M_{Y,0}$ . For a pair of strata $Y, Z$ , we write $Z \le Y$ if $Z \subset \overline {Y}$ .

2.3.5. Each closed stratum $\overline {Y}$ is itself a (partial) toroidal compactification. The completion $C_{\overline {Z}}(\overline {Y})$ is the following formal subscheme of $C_{\overline {Z}}(\overline {M})$ . Its affine pieces are given (with the notation from Definition 2.1.3) by

modulo the ideal generated by $R_{e_{\beta (1)}}, \dots , R_{e_{\beta (n_Y)}}$ (where $\beta =\beta _{ZY}$ ). The formal scheme $C_{\overline {Z}}(\overline {Y})$ carries an action of $ \mathbb {G} _m^{n_Z - n_Y}$ . Here, the missing indices not in the image of $\beta $ can be numbered in any way. We denote the corresponding injective map by $\beta _{ZY}^\perp : [n_Z-n_Y] \hookrightarrow [n_Z]$ . With the restriction $\beta _{WZ}': [n_Z-n_Y] \hookrightarrow [n_W-n_Y]$ of the transition maps $\beta _{WZ}$ for $W \le Z \le Y$ , the scheme $\overline {Y}$ becomes a toroidal compactification. The following commutative diagram shows the compatibility of the chosen numberings:

Proof. Left to the reader.

2.3.7. For the following, we will work on the topological space underlying $\overline {M}$ itself and consider coherent sheaves $\mathcal {F}$ on $M_Y$ as coherent $C_{\overline {Y}}( {\cal O} _{\overline {M}})$ -modules (in the sense of ringed spaces) on $\overline {M}$ . Note that we have

$$\begin{align*}(C_{\overline{Y}}( {\cal O} _{\overline{M}}))(U) = {\cal O} _{M_Y}(U \cap \overline{Y}). \end{align*}$$

Note that this is not quasi-coherent as $ {\cal O} _{\overline {M}}$ -module, except for the open stratum M itself. We write $C_{\overline {Y}}( {\cal O} _{\overline {M}})|_Y$ for the sheaf

$$\begin{align*}U \mapsto {\cal O} _{M_Y}(Y \cap U) \end{align*}$$

and similarly for a sheaf of $ {\cal O} _{M_Y}$ -modules $\mathcal {F}$ on $M_Y$ , we will write $\mathcal {F}|_Y$ for the so-defined restriction considered as a sheaf on $\overline {M}$ .

Proof. We apply (Reference HörmannHör20, Main Theorem 7.6). The sheaves of $ {\cal O} _{\overline {M}}$ -algebras $R_Y$ of (Reference HörmannHör20, 7.2) are isomorphic to the restriction of the sheaf $C_{\overline {Y}}( {\cal O} _{\overline {M}})$ to any open subset $U \subset \overline {M}$ , such that $U \cap \overline {Y} = Y$ , the sheaf that we denote by $C_{\overline {Y}}( {\cal O} _{\overline {M}})|_Y$ .

For any pair of strata $Z \le Y$ , the sheaf of $ {\cal O} _{\overline {M}}$ -algebras $R_{Y,Z}$ of (Reference HörmannHör20, 7.2) is, by definition, equal to $C_{\overline {Y}}(R_Z \otimes _{ {\cal O} _{\overline {M}}} {\cal O} _U)$ , where U is any open subset, such that $U \cap \overline {Y} = Y$ and where the tensor product is formed in the category of ring sheaves. The sheaf of $ {\cal O} _{\overline {M}}$ -algebras $C_{\overline {Y}}(R_Z \otimes _{ {\cal O} _{\overline {M}}} {\cal O} _U)$ is also isomorphic to a completion of the localisation $C_{\overline {Z}}( {\cal O} _X)[\beta _{YZ}^{-1}]$ since $\overline {Y} \setminus Y$ is given in formal local coordinates in $C_{\overline {Z}} (\overline {Y})$ by the zero locus of $x_{k_1}, \dots , x_{k_{j}}$ , where $\{k_1, \dots , k_{j}\}$ is the complement of $\mathrm {im}(\beta )$ .

By the nature of toroidal compactification of $\overline {M}$ , we have an action of $ \mathbb {G} _m^{n_Y}$ on $R_Y$ and an action of $ \mathbb {G} _m^{n_Z}$ on $R_{Y,Z}$ which are compatible (via $\beta _{ZY}$ ) with the inclusion

$$\begin{align*}R_Y \hookrightarrow R_{Y,Z}. \end{align*}$$

The category of $R_Y$ -coherent sheaves with $ \mathbb {G} _m^{n_Y}$ -action is equivalent to the category

$$\begin{align*}\operatorname{\mathrm{Fun}}( \mathbb{Z} ^{n_Y}, {[{ {\mathcal{O}_{Y}}\textbf{-}\mathbf{tcoh} }]} )^{f.g.}. \end{align*}$$

Hence, the given collection of functors $\{F_Y\}_Y$ gives such objects by restricting $F_Y$ to Y.

From the category of $R_{Y, Z}$ -coherent sheaves with $ \mathbb {G} _m^{n_Z}$ -action, we have still a fully faithful embedding into the subcategory of

$$\begin{align*}\operatorname{\mathrm{Fun}}( \mathbb{Z} ^{n_Z}, {[{ {\mathcal{O}_{Z}}\textbf{-}\mathbf{tcoh} }]} ) \end{align*}$$

consisting of the functors which are constant in the directions $e_i$ for $i \not \in \mathrm {im}(\beta _{ZY})$ . The glueing datum required by (Reference HörmannHör20, Lemma 7.5) can therefore be given by diagram (5). Hence, (Reference HörmannHör20, Main Theorem 7.6) provides the requested sheaf of $ {\cal O} _{\overline {M}}$ -modules, which is by construction an object in $ {[{ {\mathcal {O}_{\overline {M}}}\textbf{-}\mathbf {tcoh} }]} $ .

2.4.1. The standard examples of abstract toroidal compactifications in the sense of Definition 2.3.2 are toroidal compactifications of Shimura varieties (Reference Ash, Mumford, Rapoport and TaiAMRT10). Since we are interested only in the situation over a field, we can use the theory of canonical models of toroidal compactifications of mixed Shimura varieties due to Pink (Reference PinkPin90, 2.1). We will use the language of (Reference HörmannHör10) (cf. also (Reference HörmannHör14, 2.5)), which is concerned with extensions of the theory over the integers (in the case of good reduction of Hodge type mixed Shimura varieties). For the automorphic data referred to in the next section, we rely on (Reference HörmannHör14, 2.5) also for the rational case. In that case, the ideas for the proofs of the theorems in (Reference HörmannHör14, 2.5.) (which are given in (Reference HörmannHör10)) are essentially due to Harris (Reference HarrisHar85; Reference HarrisHar86; Reference Harris and ZuckerHZ94).

2.4.2. For each pure (or mixed) rational Shimura datum $\mathbf {X}=(G_{\mathbf {X}}, \mathbb {D}_{\mathbf {X}}, h_{\mathbf {X}})$ in the sense of (Reference HörmannHör14, 2.2.3)Footnote ⁴ or (Reference PinkPin90, 2.1), and for each sufficiently small compact open subgroup $K \subset G_{\mathbf {X}}( {\mathbb {A}^{(\infty )}} )$ , there is an associated Shimura variety $\operatorname {\mathrm {M}}({}^K \mathbf {X})$ which is a smooth quasi-projective variety defined over the reflex field $E(\mathbf {X})$ .

Furthermore, for each smooth K-admissible rational polyhedral cone decomposition $\Delta $ for $\mathbf {X}$ (cf. (Reference HörmannHör14, 2.2.23)), there is a (partial) toroidal compactification $\operatorname {\mathrm {M}}({}^K_\Delta \mathbf {X})$ which contains $\operatorname {\mathrm {M}}({}^K \mathbf {X})$ as an open subvariety whose complement is a divisor with strict normal crossings, if K is sufficiently small. This and the following is a summary of (Reference HörmannHör14, Main Theorem 2.5.9). If $\Delta $ is chosen (and this is always possible) to be projective and complete, then $\operatorname {\mathrm {M}}({}^K_\Delta \mathbf {X})$ is a smooth projective variety defined over the reflex field $E(\mathbf {X})$ . This situation thus gives rise to a stratification of $\operatorname {\mathrm {M}}({}^K_\Delta \mathbf {X})$ as considered in Section 2.3.1. Each stratum corresponds, furthermore, to an orbit of rational polyhedral cones in $\Delta $ . For each stratum Y in this stratification, there is a mixed Shimura datum $\mathbf {Y}=(G_{\mathbf {Y}}, \mathbb {D}_{\mathbf {Y}}, h_{\mathbf {Y}})$ , such that $G_{\mathbf {Y}}$ is a subgroup of $G_{\mathbf {X}}$ (if $\mathbf {X}$ is pure, this is a certain normal subgroup of the $ \mathbb {Q} $ -parabolic of $G_{\mathbf {X}}$ describing the corresponding boundary component in the Baily-Borel compactification). The boundary component $\mathbf {Y}$ is determined only up to conjugation. Furthermore, $\Delta $ restricts to a rational polyhedral cone decomposition $\Delta _Y$ for $\mathbf {Y}$ . The partial toroidal compactification of the mixed Shimura variety $\operatorname {\mathrm {M}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ has a matching stratum $\widetilde {Y}$ and there is an isomorphism of formal schemes (assuming that K is small enough)

$$\begin{align*}C_{\overline{Y}} \operatorname{\mathrm{M}}({}^{K}_{\Delta} \mathbf{X}) \cong C_{\overline{\widetilde{Y}}} \operatorname{\mathrm{M}}({}^{K_Y}_{\Delta_Y} \mathbf{Y}). \end{align*}$$

Furthermore, the mixed Shimura variety $\operatorname {\mathrm {M}}({}^{K_Y} \mathbf {Y})$ is a torus torsor over another mixed Shimura variety $\operatorname {\mathrm {M}}({}^{K_Y'} \mathbf {Y}/U)$ , where U is a subgroup of $U_{\mathbf {Y}}$ (a subgroup of the centre of the unipotent radical of $G_{\mathbf {Y}}$ determined by the mixed Shimura datum) and the action of the torus extends to $\operatorname {\mathrm {M}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ (cf. (Reference HörmannHör14, Paragraph 2.5.8)). The acting torus gets canonically identified with $ \mathbb {G} _m^{n_Y}$ (up to numbering of the coordinates) by means of the integral basis of the $n_Y$ -dimensional rational polyhedral cone describing Y. By construction of the toroidal compactification, this action extends to $ {\mathbb {M}} _m^{n_Y}$ in such a way that $C_{\overline {\widetilde {Y}}} \operatorname {\mathrm {M}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ becomes a toroidal formal scheme in the sense of Definition 2.1.3. The functoriality of the theory implies that the actions of the tori match for pairs of strata $Z \le Y$ . Thus, $\overline {M}:=\operatorname {\mathrm {M}}({}^K_\Delta \mathbf {X})$ is an abstract toroidal compactification in the sense of Definition 2.3.2.

3.1.2. The diagram in Definition 3.1.1, iii. for $Y=M$ can be equivalently described by a morphism of Artin stacks (omitting the subscripts Y)

$$\begin{align*}\Xi: \overline{M} \rightarrow \left[ G \backslash M^\vee \right]. \end{align*}$$

Let $\mathcal {E}$ be a vector bundle on $\left [ G \backslash M^\vee \right ]$ , that is, a G-equivariant vector bundle on $M^\vee $ . The pullback $\Xi ^*\mathcal {E}$ is called the automorphic vector bundle associated with $\mathcal {E}$ . It can be explicitly described as follows: Note that there is an equivalence of categories between G-equivariant vector bundles on B and vector bundles on $\overline {M}$ . The vector bundle $\Xi ^*\mathcal {E}$ is the vector bundle on $\overline {M}$ corresponding to the G-equivariant vector bundle $p^*\mathcal {E}$ . This construction will be generalised in Section 3.4 (cf. Example 3.4.4 for the special case).

3.1.3. Consider the following sequence of vector bundles on $B_Y$ (which are all $ {\mathbb {M}} _n^{n_Y}$ -equivariant and canonical extensions). We assume given a logarithmic Ehresmann connection on $B_Y$ , that is a section $s_Y$ which is $G_Y$ -equivariant and $ {\mathbb {M}} _n^{n_Y}$ -equivariant:

Note that $G_Y$ acts on $ {\cal O} _{B_Y}$ by translation and on $\operatorname {\mathrm {Lie}}(G_Y)$ via $\operatorname {\mathrm {Ad}}$ . Since everything is $ {\mathbb {M}} _n^{n_Y}$ -equivariant and a canonical extension, this is equivalent to giving a $G_Y$ -equivariant section of the sequence

(6)

Furthermore, these sections are supposed to be compatible with respect to the relation $Z \le Y$ on strata. Such a datum will be called automorphic data with logarithmic connection on the toroidal compactification $\overline {M}$ .

3.1.4. We define the $G_Y$ -subvector bundle $T_{B_Y}^{\mathrm {horz}}$ as the image of $s_Y$ , and get a $G_Y$ -equivariant decomposition:

$$\begin{align*}T_{B_Y}(\log) = T_{B_Y}^{\pi-\mathrm{vert}} \oplus T_{B_Y}^{\mathrm{horz}}. \end{align*}$$

The connection is called flat, if

1. (F) $T_{B_Y}^{\mathrm {horz}}$ is closed under the Lie bracketFootnote ⁵ .

We denote the corresponding projection operators by $P_\pi ^{\mathrm {vert}}$ and $P_\pi ^{\mathrm {horz}}$ . If $s_Y$ is flat, it induces a homomorphism of ring-sheaves

(7) $$ \begin{align} \nu: \pi^{-1} \mathcal{D}_{M_Y}(\log) \rightarrow \mathcal{D}_{B_Y}(\log). \end{align} $$

3.1.6. Note that, by the structure of toroidal compactification, we have a sequence dual to sequence (3)

where $\mathrm {can}_{i,M_Y}$ are the fundamental vector fields for the $ \mathbb {G} _m^{n_Y}$ -action on $M_Y$ , and $\Pi $ is the projection to $\overline {Y}$ . Similarly for $B_Y$ .

Since $\operatorname {\mathrm {can}}_{i,B_Y} |_{\pi ^{-1}\overline {Y}} = \xi _{i,Y}'$ , we have therefore

$$\begin{align*}\operatorname{\mathrm{Res}}_{D_i}(s_Y) = P^{\mathrm{vert}}_\pi(\mathrm{can}_{i,B_Y}) |_{\pi^{-1}\overline{Y}}. \end{align*}$$

The following axiom will be called the unipotent monodromy condition:

1. (M) For any i, we have $P^{\mathrm {vert}}_\pi (\mathrm {can}_{i,B_Y}) \in \operatorname {\mathrm {Lie}}(U^{(i)}) \otimes {\cal O} _{B_Y}$ , where $\operatorname {\mathrm {Lie}}(U^{(i)})$ is a Lie subalgebra of $\operatorname {\mathrm {Lie}}(G_Y)$ given by a one-dimensional normal unipotent subgroup $ \mathbb {G} _a \cong U^{(i)} \subset G_Y$ .

Since everything is $ {\mathbb {M}} _n$ -equivariant, we could state the condition equivalently as $\operatorname {\mathrm {Res}}_{D_i}(s_Y) \in \mathfrak {u}_Y^{(i)} \otimes {\cal O} _{\pi ^{-1}\overline {Y}}$ .

3.1.9. The automorphic data satisfy Torelli Footnote ⁶ , if we have in addition

1. (T) a direct sum decomposition

   $$\begin{align*}T_{B_Y}(\log) = T_{B_Y}^{p-\mathrm{vert}}(\log) \oplus T_{B_Y}^{\mathrm{horz}}, \end{align*}$$
   where $T_{B_Y}^{p-\mathrm {vert}}(\log )$ is the intersection of $T_{B_Y}^{p-\mathrm {vert}}$ with $T_{B_Y}(\log )$ in $T_{B_Y}$ .

Since the morphism $\pi ^{-1}\overline {Y} \rightarrow M_Y^\vee $ is a submersion (because P maps $\pi ^{-1}\overline {Y}$ into itself) $T_{B_Y}(\log ) \to p^* T_{M^\vee _Y}$ is still surjective, and we have again an exact sequence with section

whose image is $T_{B_Y}^{\mathrm {horz}}$ .

Hence, Torelli (T) induces an isomorphism

$$\begin{align*}p^* T_{M^\vee} \cong \pi^* T_M(\log) \end{align*}$$

and in the same way as before, if $s_Y$ is in addition flat, it induces a homomorphism of ring-sheaves

(9) $$ \begin{align} \mu: p^{-1} \mathcal{D}_{M_Y^\vee} \rightarrow \mathcal{D}_{B_Y}(\log). \end{align} $$

3.1.10. We also consider the following axiom (called the boundary vanishing condition):

1. (B) For all strata $Y \not = M$ , we have: $H^i(\left [ M^\vee _Y / G_Y\right ], \omega _{M_Y^\vee }) = 0$ for $i \ge \dim (Y)$

(cf. Section 3.2 for the notation). Here, $\omega _{M_Y^\vee } = \Omega ^{n}_{M^\vee _Y} $ is the highest power of the $G_Y$ -equivariant sheaf of differential forms on $M^\vee _Y$ .

3.2.1. For a linear algebraic group G and a quasi-parabolic subgroup Q, we have several functors between Q-representations, G-representations and (equivariant) coherent sheaves on the quasi-projective variety $M^\vee = Q \backslash G$ (generalized flag variety)Footnote ⁷ . These functors are best understood in the language of Artin stacks. We will not use this theory explicitly but mention it as a guiding principle because it so much clarifies the relations. All representations are, of course, understood to be algebraic. We have the following diagram of morphisms of Artin stacks, where all stacks are quotient stacks (even schemes in the right-most column):

(10)

We denote the categories of (quasi-)coherent sheaves on a stack X by $ {[{ {X}\textbf{-}\mathbf {(q)coh} }]} $ or sometimes by $ {[{ {\mathcal {O}_X}\textbf{-}\mathbf {(q)coh} }]} $ . For the particular stacks above, we get

$$\begin{align*}\begin{array}{rl} {[{ {\left[ \cdot / Q \right]}\textbf{-}\mathbf{coh} }]} & \text{category of finite-dimensional algebraic }Q\text{-representations in }k\text{-vector}\\ &\text{spaces;} \\ {[{ {\left[ \cdot / G \right]}\textbf{-}\mathbf{coh} }]} & \text{category of finite-dimensional algebraic }G\text{-representations in }k\text{-vector}\\ &\text{spaces;} \\ {[{ {\left[ M^\vee / G \right]}\textbf{-}\mathbf{coh} }]} & \text{category of }G\text{-equivariant finite dimensional vector bundles on }M^\vee; \\ {[{ {M^\vee}\textbf{-}\mathbf{coh} }]} & \text{category of coherent sheaves on }M^\vee; \\ {[{ {\operatorname{\mathrm{spec}}(k)}\textbf{-}\mathbf{coh} }]} & \text{category of finite-dimensional }k\text{-vector spaces,} \end{array} \end{align*}$$

and similarly for the categories of quasi-coherent sheaves.

The corresponding pullback and (derived) pushforward functors between the categories of (quasi-)coherent sheaves are given as follows.

• $a_*$ associates with a Q-representation V, a locally free G-equivariant sheaf on $M^\vee $ . The total space can be described as $(V \times G) / Q$ , where Q acts on V and G. It defines an equivalence of the category of finite-dimensional Q-representations and coherent G-equivariant sheaves on $M^\vee $ .

• $a^*$ is the inverse of $a_*$ , evaluation at the chosen base point of $M^\vee $ .

• $b_*$ global sections on $M^\vee $ , remembering the induced G-action. The right derived functors give the cohomology on $M^\vee $ equipped with the induced G-action.

• $b^*$ associates with a G-representation V, the coherent sheaf $V \otimes {\cal O} _{M^\vee }$ with the natural G-action.

• $c^*$ forgets the G-action.

• $d_*$ global sections on $M^\vee $ . The right derived functors are the cohomology on $M^\vee $ .

• $d^*$ associates with a vector space V the coherent sheaf $V \otimes {\cal O} _{M^\vee }$ .

• $e_*$ induction $\operatorname {\mathrm {Ind}}_{\{e\}}^G(-)$ , associates with a vector space V, the G-representation $V \otimes {\cal O} (G)$ .

• $e^*$ forgets the G-action.

• $f_*$ associates with a G-representation, the vector space of G-invariants. This functor is exact if G is reductive. Otherwise, the right derived functors are the (Hochschild) group cohomology of G with values in the respective representation.

• $f^*$ equips a vector space V with the trivial G-representation.

The composed functor $a^* b^*$ is the forgetful functor considering a G-representation as a Q-representation. Its right adjoint, the composed functor $b_* a_*$ , is therefore also called $\operatorname {\mathrm {Ind}}_{Q}^G(-)$ , but it is not exact in general.

Proof. See, for example (The14, Tag 070A). Cf. also (Reference JantzenJan03) for more elementary statements regarding the stacks appearing in this section.

3.3.1. We start with a general discussion of jet bundles and differential operators. Let X be a smooth k-variety and $X^{(n)}$ the n-th diagonal, that is

$$\begin{align*}X^{(n)} \hookrightarrow X \times X \end{align*}$$

is the subscheme defined by $\mathcal {J}^n$ , where $\mathcal {J}$ is the ideal sheaf of the diagonal. Let $\mathcal {E}$ be a vector bundle on X.

We have the two projections:

One defines the ${n}$ -th jet bundle $J^n \mathcal {E}$ by

$$\begin{align*}J^n \mathcal{E} = \operatorname{\mathrm{pr}}_{1,*} \operatorname{\mathrm{pr}}_2^*\mathcal{E} \end{align*}$$

which is always equipped with a surjective map

$$\begin{align*}J^n \mathcal{E} \rightarrow \mathcal{E}, \end{align*}$$

induced by the unit $\mathcal {E} \rightarrow \Delta _* \Delta ^*\mathcal {E}$ , where $\Delta : X \hookrightarrow X^{(n)}$ is the diagonal. Since $ {\cal O} _{X^{(n)}} = \operatorname {\mathrm {pr}}_1^* {\cal O} _X = \operatorname {\mathrm {pr}}_2^* {\cal O} _X$ , there is also a splitting of this map in the case $\mathcal {E} = {\cal O} _X$ :

$$\begin{align*}{\cal O} _X \rightarrow J^n {\cal O} _X. \end{align*}$$

3.3.2. For two vector bundles $\mathcal {E}$ and $\mathcal {F}$ , the sheaf of differential operators (of degree $\le n$ ) is defined as

$$\begin{align*}\mathcal{D}^{\le n}(\mathcal{E}, \mathcal{F}) := \mathcal{HOM}_{ {\cal O} _X}( J^n \mathcal{E}, \mathcal{F}). \end{align*}$$

The bundle $J^n\mathcal {E}$ has a second $ {\cal O} _X$ -module structure coming from $\operatorname {\mathrm {pr}}_2$ , which we denote as an action on the right. We have

$$\begin{align*}J^n {\cal O} _X \otimes \mathcal{E} \cong J^n \mathcal{E} ,\end{align*}$$

where the tensor-product is formed with respect to this second $ {\cal O} _X$ -module structure.

3.3.3. There is an inclusion

$$\begin{align*}\mathcal{D}^{\le n}(\mathcal{E}, \mathcal{F}) \hookrightarrow \mathcal{HOM}_k ( \mathcal{E}, \mathcal{F}) \end{align*}$$

into the sheaf of k-linear (not $ {\cal O} _X$ -linear) morphisms of sheaves. For an open subset $U \subset X$ , a section $s \in H^0(U, \mathcal {E})$ here is considered to be a morphism

$$\begin{align*}{\cal O} _U \rightarrow \mathcal{E}_U \end{align*}$$

and the composition

$$\begin{align*}{\cal O} _U \rightarrow \operatorname{\mathrm{pr}}_{1,*} \operatorname{\mathrm{pr}}_1^* {\cal O} _U = \operatorname{\mathrm{pr}}_{1,*} \operatorname{\mathrm{pr}}_2^* {\cal O} _U \rightarrow \operatorname{\mathrm{pr}}_{1,*} \operatorname{\mathrm{pr}}_2^* \mathcal{E}_U = J^n \mathcal{E}_U \end{align*}$$

yields a section in $H^0(U, J^n \mathcal {E})$ and then, via application of an element of $H^0(U, \mathcal {HOM}( J^n \mathcal {E}, \mathcal {F}))$ , a section in $H^0(U, \mathcal {F})$ . The second $ {\cal O} _X$ -module structure on $J^n\mathcal {E}$ here dualises to precomposition with a section of $ {\cal O} _X$ . We write $\mathcal {D}_X^{\le n} := \mathcal {D}^{\le n}( {\cal O} _X, {\cal O} _X)$ . The ring sheaf $\mathcal {D}_X := \operatorname {\mathrm {colim}}_n \mathcal {D}_X^{\le n}$ is generated by $ {\cal O} _X$ and $\mathcal {T}_X$ with the only relations coming from the Lie bracket of vector fields and differentiation of functions.

Similarly to the case of jet bundles, we have

$$\begin{align*}\mathcal{D}^{\le n}(\mathcal{E}, {\cal O} ) = \mathcal{D}_X^{\le n} \otimes \mathcal{E}^* , \end{align*}$$

where the tensor product is formed with respect to the right- $ {\cal O} _X$ -module structure.

3.3.4. In the special case $X = G$ , where G is an algebraic group, we have a natural isomorphism (compatible with the filtration by degree):

$$\begin{align*}\mathcal{D}_P = \operatorname{\mathrm{colim}}_{ n } \mathcal{D}^{\le n}_G \cong {\cal O} _G \otimes U(\operatorname{\mathrm{Lie}}(G)), \end{align*}$$

where $U(\operatorname {\mathrm {Lie}}(G))$ is the universal enveloping algebra of the Lie algebra $\operatorname {\mathrm {Lie}}(G)$ . Elements of $\operatorname {\mathrm {Lie}}(G)$ are considered to be vector fields using the action by left-translation. They are invariant under the action of G on G by right-translation. The isomorphism is hence G-equivariant under right-translation, where G acts on the right-hand side only on $ {\cal O} _G$ . It is G-equivariant under left-translation if G on the right-hand side acts on $ {\cal O} _G$ by left-translation and via $\operatorname {\mathrm {Ad}}$ on $\operatorname {\mathrm {Lie}}(G)$ .

3.3.5. The construction in Section 3.3.4 is a special case of the following. Let G be an algebraic group and $X= Q \backslash G$ , where Q is a quasi-parabolic subgroup of G. These are the generalised flag varieties, denoted $M^\vee _Y$ in the last section, thus assuming here that they have a k-rational point $[Q]$ in the sequel. Denote by $\pi : G \rightarrow Q \backslash G$ the projection.

Proof. Sections on $U \subset Q \backslash G$ of the bundle $Q \backslash (G \times (U(\operatorname {\mathrm {Lie}}(G)) \otimes _{U(\operatorname {\mathrm {Lie}}(Q))} E))$ can be considered as Q-invariant sections on $\pi ^{-1}U$ of the constant bundle $U(\operatorname {\mathrm {Lie}}(G)) \otimes _{U(\operatorname {\mathrm {Lie}}(Q))} E$ , and similarly, sections on U in $\mathcal {E}^*$ are Q-invariant sections of the constant bundle $E^*$ on $\pi ^{-1}U$ . The action

$$\begin{align*}H^0(\pi^{-1}U, U(\operatorname{\mathrm{Lie}}(G)) \otimes_{U(\operatorname{\mathrm{Lie}}(Q))} E) \times H^0(\pi^{-1}U, E^*) \rightarrow H^0(\pi^{-1}U, E^*) \end{align*}$$

given by

$$\begin{align*}g (X \otimes v) \cdot f \mapsto g (X v(f)), \end{align*}$$

where X acts as differential operator on the function $v(f) \in \mathcal {O}_G({\pi ^{-1}U})$ , is Q-invariant and therefore induces a morphism

$$\begin{align*}H^0(\pi^{-1}U, U(\operatorname{\mathrm{Lie}}(G)) \otimes_{U(\operatorname{\mathrm{Lie}}(Q))} E)^Q \rightarrow \mathcal{D}(\mathcal{E}^*, {\cal O} )(U). \end{align*}$$

Using local coordinates, one checks that it is an isomorphism.

3.3.9. There is a logarithmic version of the sheaves of differential operators defined in the last section. Let $X=\overline {M}$ be a smooth k-variety equipped with a divisor with normal crossings. We define

$$\begin{align*}\mathcal{D}^{\le n}( {\cal O} _X, {\cal O} _X)(\log) \subset \mathcal{D}^{\le n}( {\cal O} _X, {\cal O} _X) \end{align*}$$

as the subsheaf of differential operators generated by $ {\cal O} _X$ and the vector fields in $\mathcal {T}_X(\log )$ , and define $\mathcal {D}^{\le n}(\mathcal {E}, \mathcal {F})(\log )$ similarly. We set

$$\begin{align*}J^n_{\log} \mathcal{E} := \mathcal{D}^{\le n}(\mathcal{E}, \mathcal{ {\cal O} }_X)(\log)^\vee. \end{align*}$$

Proof. Let $\widetilde {V}$ denote the bundle $Q \backslash (G \times V)$ on $Q \backslash G$ . It suffices to show, dually, that the automorphic vector bundle associated with the G-equivariant vector bundle $\mathcal {D}^{\le n}(\widetilde {V}^*, {\cal O} )$ on $Q \backslash G$ is $\mathcal {D}^{\le n}(\log )(\mathcal {V}^*, {\cal O} )$ .

Let Y be a stratum. For the proof, it suffices to take $Y=M$ , however, we will need the more refined discussion later. There are $G_Y$ -equivariant homomorphisms of ring sheaves (which respect the filtrations by degree), cf. Sections 3.1.3–3.1.9:

$$ \begin{align*} \mu: \pi^{-1} \mathcal{D}_{M_Y}(\log)& \rightarrow \mathcal{D}_{B_Y}(\log) \\ \nu: p^{-1} \mathcal{D}_{M^\vee_Y}& \rightarrow \mathcal{D}_{B_Y}(\log) \end{align*} $$

given by the flat connection $s_Y$ (and the Torelli axiom). They are compatible with the left- and right-module structures under $\pi ^{-1} {\cal O} _{M_Y}$ , respectively, $p^{-1} {\cal O} _{M^\vee _Y}$ . Furthermore, we have

$$\begin{align*}{\cal O} _{B_Y} \cdot \nu(p^{-1} \mathcal{D}_{M^\vee_Y}^{\le n}) = \mathcal{D}_{B_Y}^{\mathrm{horz}} = {\cal O} _{B_Y} \cdot \mu(\pi^{-1} \mathcal{D}_{M_Y}^{\le n}(\log)), \end{align*}$$

where $\mathcal {D}_{B_Y}^{\mathrm {horz}}$ is the subring sheaf of $\mathcal {D}_{B_Y}(\log )$ generated by $ {\cal O} _{B_Y}$ and $\mathcal {T}_{B_Y}^{\mathrm {horz}}$ .

The bundle $\mathcal {D}^{\le n}(\widetilde {V}, {\cal O} )$ on $M^\vee _Y$ is isomorphic to

$$\begin{align*}\mathcal{D}_{M^\vee_Y}^{\le n} \otimes_{ {\cal O} _{M^\vee_Y}} \widetilde{V}^*, \end{align*}$$

where the tensor product has been formed with respect to the $ {\cal O} _{M^\vee _Y}$ -right-module structure on $\mathcal {D}_{M^\vee _Y}^{\le n}$ .

Furthermore, we have a $G_Y$ -equivariant isomorphism:

$$\begin{align*}p^*( \mathcal{D}_{M_Y^\vee}^{\le n} \otimes_{ {\cal O} _{M_Y^\vee}} \widetilde{V}) \cong ( {\cal O} _{B_Y} \cdot \mu(\pi^{-1} \mathcal{D}_{M_Y}^{\le n}(\log))) \widehat{\otimes}_{ {\cal O} _{B_Y}} p^* \widetilde{V} \end{align*}$$

(Lemma 3.3.11 below). Now, $G_Y$ acts on $ {\cal O} _{B_Y} \cdot \mu (\pi ^{-1} \mathcal {D}_{M_Y}^{\le n}(\log ))$ exclusively on the first factor, that is

$$\begin{align*}( {\cal O} _{B_Y} \cdot \mu(\pi^{-1}\mathcal{D}_{M_Y}^{\le n}(\log)))^{G_Y} \cong \mathcal{D}_{M_Y}^{\le n}(\log) \end{align*}$$

using the identification of $G_Y$ -invariant sections of a $G_Y$ -bundle on $B_Y$ with the sections of a vector bundle on $M_Y$ . Conclusion:

$$\begin{align*}(p^*( \mathcal{D}_{M_Y^\vee}^{\le n} \otimes_{ {\cal O} _{M^\vee_Y}} \widetilde{V}))^{G_Y} \cong \mathcal{D}_{M_Y}^{\le n}(\log) \widehat{\otimes}_{ {\cal O} _{M_Y}} (p^* \widetilde{V})^{G_Y}.\\[-37pt] \end{align*}$$

Proof. This follows by induction on the degree from the fact that $\nu $ is compatible with the right- $p^{-1} {\cal O} _{M^\vee _Y }$ -module structure.

3.4.3. Obviously the definition of Fourier-Jacobi category mimics the situation for vector bundles on toroidal compactifications, and we now proceed to define an exact functor

$$\begin{align*}\Xi^*: {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} \rightarrow {[{ {\overline{M}}\textbf{-}\mathbf{tcoh} }]} \end{align*}$$

as follows: For each $F_Y(v) \in {[[M_Y^{\vee}/G_Y]\textbf{-}\mathbf{coh}]} $ , we form $p^*(F_Y(v))^{G_Y}|_{\overline {Y}}$ which is a vector bundle on $\overline {Y}$ . It carries an action of $ {\mathbb {M}} _m^{n_Z-n_Y}$ on

$$\begin{align*}C_{\overline{Z}}(p_Y^*(F_Y(v))^{G_Y}|_{\overline{Y}}) \cong (p_Z^*(\alpha_{ZY}^*F_Y(v))^{G_Z})|_{\overline{Y}} \end{align*}$$

which is a canonical extension (cf. Section 2.2.7).

The so-defined functors

$$\begin{align*}F_Y': \mathbb{Z} ^{n_Y} \rightarrow {[{ {\overline{Y}}\textbf{-}\mathbf{tcoh}\textbf{-}\mathbf{can} }]} \end{align*}$$

(where $\overline {Y}$ is equipped with its structure as restricted toroidal compactification) together with the maps induced by the $\mu _{ZY}$ satisfy the requirements of Lemma 2.3.8. Hence, we get a coherent sheaf $\Xi ^*(\{F_Y\})$ on $\overline {M}$ which carries a $ \mathbb {G} _m^{n_Y}$ action on $C_{\overline {Y}}(\Xi ^*(\{F_Y\}))$ .

We call the sheaves in the image of $\Xi ^*$ generalised automorphic sheaves.

Proof. It suffices to see this on the open parts $M_Y|_Y$ of the $M_Y$ . The verification is left to the reader.

3.4.8. For each pair $(Y, v)$ , where Y is a stratum and $v \in \mathbb {Z} ^{n_Y}$ , there exist restriction functors:

$$\begin{align*}\begin{array}{rrcl} (v)_Y^*: & {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}{\ge{}N}\textbf{-}\mathbf{coh} }]} &\rightarrow& {[{ {\left[M^\vee_Y/G_Y\right]}\textbf{-}\mathbf{coh} }]} \\ (v)_Y^*: & {[{ {\overline{M}}\textbf{-}\mathbf{FJ} }]} &\rightarrow& {[{ {\left[M^\vee_Y/G_Y\right]}\textbf{-}\mathbf{qcoh} }]} \\ (v)_Y^*: & {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}{\ge{}N} }]} &\rightarrow& {[{ {\left[M^\vee_Y/G_Y\right]}\textbf{-}\mathbf{qcoh} }]} \end{array} \end{align*}$$

given by $F \mapsto F_Y(v)$ . Those are exact and have each an exact right-adjoint $(v)_{Y,*}$ , which is given as follows. The functor $((v)_{Y,*}V)_Y$ is given by the right Kan-extension $v_*$ , where $v: \{\cdot \} \hookrightarrow \mathbb {Z} ^{n_Y}$ , respectively, $v: \{\cdot \} \hookrightarrow \mathbb {Z} ^{n_Y}_{\ge N}$ also denotes the inclusion of v. In other words, we have

$$\begin{align*}((v)_{Y,*}V)_Y(w) = \begin{cases} V & \text{if }w \le v\text{ (and }w_i \ge N\text{ for all }i\text{ in the }\ge N\text{-cases)} \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

Note that $v \le w$ means that $v_i \le w_i$ for all i. For any stratum $Z \le Y$ , we define

$$\begin{align*}((v)_{Y,*}V)_Z(v) := \alpha^*_{ZY} ((v)_{Y,*}V)_Y(\operatorname{\mathrm{pr}}(v)), \end{align*}$$

where $\operatorname {\mathrm {pr}}: \mathbb {Z} ^{n_Z} \rightarrow \mathbb {Z} ^{n_Y}$ is the projection induced by $\beta _{ZY}$ . In the bounded case, it is set identically zero if $v_i < N$ for some i. For all other strata Z, the functor $((v)_{Y,*}V)_Z$ is set identically zero. The so-defined object $(v)_{Y,*}V$ , together with the obvious isomorphisms, satisfies conditions i and ii of the definition of the Fourier-Jacobi category (Definition 3.4.1).

3.4.9. For each stratum Y and each $N \in \mathbb {Z} $ , there are exact restriction functors

$$\begin{align*}\iota_N^*: {[{ {\overline{Y}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} \rightarrow {[{ {\overline{Y}}\textbf{-}\mathbf{FJ}\textbf{-}{\ge}{}N\textbf{-}\mathbf{coh} }]} \end{align*}$$

which have an exact left-adjoint

$$\begin{align*}\iota_{N,!}: {[{ {\overline{Y}}\textbf{-}\mathbf{FJ}\textbf{-}{\ge}{}N\textbf{-}\mathbf{coh} }]} \hookrightarrow {[{ {\overline{Y}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} \end{align*}$$

which is given by the natural inclusion (or, in other words, by extension by zero or left Kan extension for the individual $F_Z$ ).

Proof. We have in each case a pair of adjoint functors in which the unit, respectively, the counit, is an isomorphism. Since all four functors are exact, they induce functors on the derived categories without modification, and form again pairs of adjoint functors (because the counit/unit-equations still hold). Since also the unit, respectively, the counit, is still an isomorphism, we get the requested fully faithfulness of the left- (respectively, right-) adjoint.

Proof. For any object $F=\{F_Y\}$ , we define an injective resolution by

$$\begin{align*}\prod_{(Y, v),v_i \le N_Y} (v)_{Y,*} I((v)_Y^* F), \end{align*}$$

where $I((v)_Y^* F)$ is an injective resolution of $(v)_Y^* F$ in the category $ {[{ {\left [M^\vee _Y / G_Y\right ]}\textbf{-}\mathbf {qcoh} }]} $ . Note that right-adjoints of exact functors and $\prod $ preserve injective objects. Here, $N_Y$ is some appropriate upper bound for the stratum Y. Note that because of the bound, the product exists (as opposed to general products in $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-} {\ge {}N} }]} $ and $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ} }]} $ ).

Proof. Follows from (the dual of) (Reference Kashiwara and SchapiraKS06, Theorem 13.2.8).

3.5.1. We write as usual $M_Y:= C_{\overline {Y}}(\overline {M})$ and $M_Y|_Y$ for the formal open subscheme on Y. Recall the definition of the vector bundle $\Omega _{\overline {M}}(\log )$ on a variety with a normal crossings divisor. Locally, the bundle $C_{\overline {Y}}( \Omega _{\overline {M}}(\log ))|_Y$ is the bundle $\widehat {\Omega }_{M_Y|_Y}(\log )$ (defined in Section 2.2.8) on the toroidal formal scheme $M_Y|_Y$ , but not on $M_Y$ ! Recall from Section 2.2.8 the description of the associated functor of $\widehat {\Omega }_{M_Y|_Y}(\log )$ on $M_Y|_Y$ .

By Theorem 3.3.10, the vector bundle $\Omega _{\overline {M}}(\log )$ on $\overline {M}$ can therefore be obtained by glueing and is associated with the following element in $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{lf}\textbf{-}\mathbf{coh}} ]} $ :

$$\begin{align*}F_Y: v \mapsto \begin{cases} \Omega_{M_Y^\vee} & \text{if }v \ge 0, \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

Note that for $Z \le Y$ , the restriction $\alpha _{ZY}^* \Omega _{M_Y^\vee }$ is canonically isomorphic to $\Omega _{M_Z^\vee }$ because $\alpha _{ZY}$ is an open embedding by definition.

If the given automorphic data with flat logarithmic connection satisfy the unipotent monodromy condition (M) (cf. Section 3.1.6), then the subbundle $\Omega _{\overline {M}}$ can be described by the following functor

(11) $$ \begin{align} F_Y: v \mapsto \begin{cases} \left\{ \xi \in \Omega_{M_Y^\vee}\ \middle|\ \forall i: v_i = 0 \Rightarrow p_i(\xi) = 0 \right\} & \text{if }v \ge 0, \\ 0 & \text{otherwise.} \end{cases} \end{align} $$

Here, $p_i$ is given as follows: By the unipotent monodromy axiom, there are $G_Y$ -equivariant subbundles $T^{(i)}_{M_Y^\vee } \subset T_{M_Y^\vee }$ given by the Lie algebras $\mathfrak {u}_i$ of one-dimensional normal unipotent subgroups $U_i \subset G_Y$ . The morphism $p_i$ is then defined as the projection dual to this inclusion. By the unipotent monodromy axiom (M), we have $ {\cal O} _{B_Y} \cdot \pi ^{-1}(\operatorname {\mathrm {can}}_{i,M_Y}) \cong p^*(T^{(i)}_{M_Y^\vee })$ under the natural $G_Y$ -equivariant isomorphism

$$\begin{align*}\pi^* \mathcal{T}_{M_Y}(\log) \cong p^* \mathcal{T}_{M_Y^\vee}. \end{align*}$$

It follows therefore from the proof of Theorem 3.3.10 that $\Omega _{\overline {M}}$ is associated with this subfunctor.

3.5.2. Assume for the rest of the section that there exists a k-valued point in $M^\vee $ , and let $Q_M$ be the corresponding quasi-parabolic subgroup of $G_M$ . The discussion in Section 3.5.1 enables us to refine Theorem 3.3.10. Given a $Q_M$ -representation V or equivalently a $G_M$ -equivariant vector bundle $\widetilde {V}$ on $M^\vee $ , we define the object $(J^n\widetilde {V})'$ in $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{lf}\textbf{-}\mathbf{coh}} ]} $ by

$$\begin{align*}(J^n\widetilde{V})_Y': v \mapsto J^n(\widetilde{V})^{v}, \end{align*}$$

where we define a $ \mathbb {Z} ^{n_Y}$ -indexed filtration on $J^n(\widetilde {V})$ induced by the dual of the following $ \mathbb {Z} ^{n_Y}$ -indexed filtration on $( U(\operatorname {\mathrm {Lie}}(G_Y)) \otimes _{U(\operatorname {\mathrm {Lie}}(Q_Y))} V^* )^{\le n}$ : It is given by the tensor product of the trivial filtration on $V^*$ and the filtration on $U(\operatorname {\mathrm {Lie}}(G_Y))$ which is the quotient of the induced filtration on $T(\operatorname {\mathrm {Lie}}(G_Y))$ (tensor algebra) of the following filtration on $\operatorname {\mathrm {Lie}}(G_Y)$ :

$$\begin{align*}\operatorname{\mathrm{Lie}}(G_Y)(v) = \begin{cases} \operatorname{\mathrm{Lie}}(G_Y) & v \ge 0 \\ \mathfrak{u}_i & v_i = -1 \text{ and } v_j \ge 0\ \forall j \not= i \\ 0 & \text{otherwise} \end{cases} \end{align*}$$

(this is essentially the dual of (11)).

3.5.4. Define $\omega _{\overline {M}}(\log ):= \Lambda ^n(\Omega _{\overline {M}}(\log ))$ , where $n= \dim (M)$ . By Proposition 3.3.10, this is an automorphic line bundle associated with $\omega _{M^\vee }$ , and by the above discussion the subbundle $\omega _{\overline {M}} \subset \omega _{\overline {M}}(\log )$ is a generalised automorphic sheaf on $\overline {M}$ given by $\omega = \{\omega _Y\}$ with

$$\begin{align*}\omega_Y: v \mapsto \begin{cases} \omega_{M_Y^\vee} & \text{if }v_i \ge 1\ \forall i, \\ 0 & \text{otherwise}. \end{cases} \end{align*}$$

In other words, it is given by $\iota _{1,!}\ (0)_{M,*}\ \omega _{M^\vee }$ , where $(0)_{M,*}$ is considered as a functor with values in $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-} {\ge 1}\textbf{-}\mathbf {coh} }]} $ . Note that $\omega _{M_Y^\vee }$ is associated with the $Q_Y$ -representation $\Lambda ^n (\operatorname {\mathrm {Lie}}(G_Y) / \operatorname {\mathrm {Lie}}(Q_Y))^*$ . We also define the following generalised automorphic sheaves $\omega _Y$ associated with the functor in $ {[{ {Y}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{coh} }]} $ :

$$\begin{align*}(\omega_Y)_Y: v \mapsto \begin{cases} \omega_{M^\vee_Y} & \text{if } v=0, \\ 0 & \text{otherwise}. \end{cases} \end{align*}$$

It extends (as canonical extension along smaller strata) to an element $\omega _{\overline {Y}}$ in $ {[{ {\overline {Y}}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{coh} }]} $ (cf. Section 3.4.8). In other words, $\omega _{\overline {Y}}$ is given by $\iota _{0,!}\ (0)_{Y,*}\ \omega _{M^\vee _Y}$ , where $(0)_{Y,*}$ is considered as a functor with values in $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-} {\ge {}0}\textbf{-}\mathbf {coh} }]} $ .

Proof. By induction.

3.6.1. The toroidal compactifications of (mixed) Shimura varieties are naturally equipped with automorphic data with logarithmic connection in the sense of Definition 3.1.1. We sketch the relation with the theory of mixed Shimura varieties and their toroidal compactifications in this section, hinting at the reasons for the axioms to be satisfied. The boundary vanishing axiom, will be investigated more in detail.

Firstly, we may fix the particular boundary components $\mathbf {Y}$ (in the sense of mixed Shimura data) in its conjugacy class, such that for $Z \le Y$ , we get a boundary map $\mathbf {Z} \rightarrow \mathbf {Y}$ , that is a closed embedding $G_{\mathbf {Z}} \hookrightarrow G_{\mathbf {Y}}$ together with a compatible open embedding $\mathbb {D}_{\mathbf {Y}} \hookrightarrow \mathbb {D}_{\mathbf {Z}}$ . By (Reference HörmannHör14, Main Theorem 2.5.12), for each of these boundary components $\mathbf {Y}$ , there exists a ‘compact’ dual $\operatorname {\mathrm {M}}^\vee (\mathbf {Y})$ (which is only proper for $\mathbf {Y} = \mathbf {X}$ , i.e. $Y=M$ , if $\mathbf {X}$ is itself pure) defined over the reflex field $E(\mathbf {X})$ . It is of the form $M^\vee _Y$ as in the definition of automorphic data, that is, it is a $G_{\mathbf {Y}}$ -equivariant component in the classifying space of quasi-parabolics for $G_{\mathbf {Y}}$ . For the definition of automorphic data, we will consider all varieties and groups as schemes over the reflex field $E(\mathbf {X})$ .

3.6.2. The following is a summary of (Reference HörmannHör14, Main Theorem 2.5.14). For each stratum Y, there is a $G_{\mathbf {Y},E(\mathbf {X})}$ -principal bundle $\operatorname {\mathrm {B}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ over the mixed Shimura variety $\operatorname {\mathrm {M}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ together with an equivariant map to the ‘compact’ dual:

Because of the functoriality (the torus action comes from a morphism of mixed Shimura data), the morphism p is $ {\mathbb {M}} _m^{n_Y}$ -equivariant and the morphism $\pi $ is $ {\mathbb {M}} _m^{n_Y}$ -invariant. These data are compatible in the sense that if we have strata $Z \le Y$ , then there is a commutative diagram

where the maps are functorial with respect to relations $W \le Z \le Y$ of strata.

The flat logarithmic connection can be defined analytically by means of the flat section $\xi $ on the universal cover given as follows:

It has logarithmic singularities along the extension of $\operatorname {\mathrm {B}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ to $\operatorname {\mathrm {M}}({}^{K}_{\Delta } \mathbf {X})$ and by Géometrie Algébrique et Géométrie Analytique (Reference SerreGAGA) is therefore algebraic. The fact that the corresponding algebraic connection is defined over $E(\mathbf {X})$ can be deduced from (Reference HarrisHar85, Section 3.4). In purely algebraic constructions of Shimura varieties as moduli spaces, it comes from the Gauss-Manin connection on the cohomology bundle and thus can be constructed in a purely algebraic way.

3.6.3. The Torelli axiom (T) follows analytically because the composition

$$\begin{align*}\mathbb{D}_{\mathbf{Y}} \times G_{\mathbf{Y}}( {\mathbb{A}^{(\infty)}} ) / K_Y \rightarrow G_{\mathbf{Y}}( \mathbb{C} ) / Q_{\mathbf{Y}}( \mathbb{C} ) \end{align*}$$

is an open embedding after projection to the first factor (the Borel embedding). In purely algebraic constructions of Shimura varieties, the axiom corresponds to infinitesimal Torelli theorems of the parametrised objects, which can be proven purely algebraically.

3.6.4. The unipotent monodromy axiom (M) is satisfied because the cone $\sigma $ describing a boundary component sits per definition in $U_{\mathbf {Y}, \mathbb {R} }(-1)$ and $U_{\mathbf {Y}} \cong \mathbb {G} _a^u$ is a normal subgroup of $G_{\mathbf {Y}}$ (cf. e.g. (Reference HörmannHör14, Section 2.2) for its definition). By construction, the fundamental vector fields $\operatorname {\mathrm {can}}_i$ of the action of $ \mathbb {G} _m^{n_Y}$ on $\operatorname {\mathrm {M}}({}^{K_Y}_{\Delta _Y} \mathbf {Y})$ lifted to the universal cover correspond to the basis-vectors of $(U_{\mathbf {Y}} \cap K_Y)(-1)$ spanning $\sigma $ . In cases in which the mixed Shimura variety is constructed using a moduli problem of 1-motives, as in (Reference HörmannHör14, Section 2.7), the unipotent monodromy axiom can be read off from the construction.

Proof. Without loss of generality, we may assume that the base field of the category of Q-representations is $ \mathbb {C} $ and that all algebraic groups involved are defined over $ \mathbb {C} $ . We have the following zoo of connected linear algebraic groups (cf. (Reference HörmannHör14, 2.2) or (Reference PinkPin90)):

By definition of a mixed Shimura datum, the Lie algebras of these groups have the following weights under $ \mathbb {S} $ (acting via $\operatorname {\mathrm {Ad}} \circ h$ ):

$$ \begin{align*} \begin{array}{r|l} \operatorname{\mathrm{Lie}}(U) & (-1,-1) \\ \end{array}\qquad \begin{array}{r|l} \operatorname{\mathrm{Lie}}(V^+) & (-1,0) \\ \operatorname{\mathrm{Lie}}(V^-) & (0,-1) \\ \end{array}\qquad \begin{array}{r|l} \operatorname{\mathrm{Lie}}(R^+) & (-1,1) \\ \operatorname{\mathrm{Lie}}(K) & (0,0) \\ \operatorname{\mathrm{Lie}}(R^-) & (1,-1) .\end{array} \end{align*} $$

We have the following sequence of affine morphisms

$$\begin{align*}M^\vee(\mathbf{Y}) = G / (R \cdot V^+) \rightarrow G_0 \cdot V / (R \cdot V^+) \rightarrow G_0/R \end{align*}$$

of relative dimensions $u = \dim (U)$ , and $v = \dim (V^-)$ , respectively. $G_0/R$ is a projective flag variety of dimension $n_0 = \dim (R^-)$ . Note that $\omega $ is isomorphic to the representation (with Q acting via $\operatorname {\mathrm {Ad}}$ on the Lie algebras)

(12) $$ \begin{align} \left( \Lambda^{n_0} \operatorname{\mathrm{Lie}}(R^-) \otimes \Lambda^{v} \operatorname{\mathrm{Lie}}(V^-) \otimes \Lambda^{u} \operatorname{\mathrm{Lie}}(U) \right)^*. \end{align} $$

Step 1: We have

$$\begin{align*}H^i(\left[ \cdot / Q \right], \omega) = H^i(\left[ \cdot / (V^+ \cdot R^+) \right], \omega)^K \end{align*}$$

because K is reductive. Furthermore, since $\omega $ is one-dimensional and hence trivial as a $V^+$ and $R^+$ representation, we have as K-representations

$$\begin{align*}H^i(\left[ \cdot / (V^+ \cdot R^+) \right], \omega) = H^i(\left[ \cdot / (V^+ \cdot R^+) \right], \mathbb{C} ) \otimes \omega. \end{align*}$$

Step 2: The subgroups $V^+$ and $R^+$ commute (because there is no part of the Lie algebra of weight $(-2,1)$ ). Hence, $H^i(\left [ \cdot / (V^+ \cdot R^+) \right ], \mathbb {C} )$ is just the cohomology of $ \mathbb {G} _a^{n_0+v}$ with respect to the trivial representation. Hence, $H^i(\left [ \cdot / V^+ \cdot R^+ \right ], \mathbb {C} ) = \Lambda ^i (\operatorname {\mathrm {Lie}}(V^+)^* \oplus \operatorname {\mathrm {Lie}}(R^+)^*)$ as natural $\mathrm {Aut}(V^+ \cdot R^+)$ -modules (Reference JantzenJan03, p.64, Remark 2). Therefore, $H^i(\left [ \cdot / (V^+ \cdot R^+) \right ], \mathbb {C} ) = 0$ for $i> n_0 + v$ and

$$\begin{align*}H^{n_0+v}(\left[ \cdot / (V^+ \cdot R^+) \right], \mathbb{C} ) = \Lambda^{n_0+v} (\operatorname{\mathrm{Lie}}(V^+)^* \oplus \operatorname{\mathrm{Lie}}(R^+)^*) \cong \mathbb{C}. \end{align*}$$

Step 3: Since the last isomorphism is compatible with respect to the natural $\mathrm {Aut}(V^+ \cdot R^+)$ -actions, we see that $H^{n_0+v}(\left [ \cdot / (V^+ \cdot R^+) \right ], \mathbb {C} )$ is one-dimensional of weight

$$\begin{align*}(v+n_0, -n_0) \end{align*}$$

under $ \mathbb {S} $ . The representation $\omega $ is isomorphic to (12) and hence one-dimensional of weight

$$\begin{align*}( u-n_0,u+v+n_0). \end{align*}$$

Therefore

$$\begin{align*}H^{n_0+v}(\left[ \cdot / (V^+ \cdot R^+) \right], \mathbb{C} ) \otimes \omega \quad \text{ has weight } \quad (u+v,u+v) \end{align*}$$

and thus cannot have any K-invariants as long as $u+v\not =0$ .

4.1.1. Let X be a complex smooth projective variety of dimension n. There are several equivalent ways of constructing Chern classes of vector bundles on X. We follow (Reference AtiyahAti57) and start by discussing the abstract homological algebra behind this construction.

Let $\mathcal {C}$ be a closed monoidal $ \mathbb {Q} $ -linear exact category with exact tensor product $\otimes $ and unit $ {\cal O} $ . Its derived category $D^b(\mathcal {C})$ can be constructed as for Abelian categories and is a tensor triangulated category. An extension of the form

(13)

in which $\mathcal {E}$ is dualisable and where, for the moment, $\Omega ^{1}$ is just any fixed object, induces a morphism

$$\begin{align*}{\cal O} \rightarrow \mathcal{E}^* \otimes \mathcal{E} \otimes \Omega^{1}[1] \end{align*}$$

in $D^b(\mathcal {C})$ and thus an alternatingFootnote ⁹ morphism

$$\begin{align*}{\cal O} \rightarrow (\mathcal{E}^* \otimes \mathcal{E} \otimes \Omega^{1})^{\otimes n}[n]. \end{align*}$$

A homogenous polynomial p of degree n in $ \mathbb {Q} [c_1, c_2, \dots ]$ , in which $c_i$ has degree i, gives rise to a symmetric morphism

$$\begin{align*}\widetilde{p}: (\mathcal{E}^* \otimes \mathcal{E})^{\otimes n} \rightarrow {\cal O}. \end{align*}$$

For example, if $\mathcal {C}$ is a category of vector spaces, we have $\widetilde {c_i}(M^ {\otimes i}) = \sigma _i(\alpha _1, \dots , \alpha _n)$ , where $\sigma _i$ is the i-th elementary symmetric function and the $\alpha _i$ are the eigenvalues of M. Note that $\widetilde {p}$ is a $ \mathbb {Q} $ -linear combination of compositions of various trace maps contracting one of the $\mathcal {E}^*$ -factors against one of the $\mathcal {E}$ -factors and is thus feasible in any closed monoidal category. Together with the alternating morphism (where $\Omega ^n:=\Lambda ^n \Omega ^1$ )

$$\begin{align*}( \Omega^{1})^{\otimes n} \rightarrow \Omega^n, \end{align*}$$

this induces an alternating morphism

$$\begin{align*}(\mathcal{E}^* \otimes \mathcal{E} \otimes \Omega^{1})^{\otimes n} \rightarrow \Omega^{n}. \end{align*}$$

Putting everything together, we obtain a ‘Chern-Weil’ homomorphism of graded commutative algebras

$$\begin{align*}\mathbb{Q} [c_1, c_2, \dots] \to \bigoplus_i \operatorname{\mathrm{Hom}}_{D^b(\mathcal{C})}( {\cal O} , \Omega^i [i]). \end{align*}$$

4.1.2. Let $\mathcal {E}$ be a vector bundle on X. To get the usual Chern classes of $\mathcal {E}$ , we start with the Atiyah extension (where $J^1$ is the first jet bundle (cf. Section 3.3))

and perform the construction in Section 4.1.1 in the category of locally free sheaves on X. This gives a morphism of graded commutative rings

$$\begin{align*}\mathbb{Q} [c_1, \dots, c_n] \to \bigoplus_i \operatorname{\mathrm{Hom}}_{D^b( {[{ {\mathcal{O}_{X}}\textbf{-}\mathbf{coh} }]} )}( {\cal O} _X, \Omega_X^i [i]) \cong \bigoplus_i H^i(X, \Omega^i). \end{align*}$$

By (Reference AtiyahAti57, Theorem 6), this agrees with other constructions of the Chern-Weil homomorphism. To get a numerical statement, we consider a homogenous polynomial p of degree $n = \dim X$ . It yields an element

$$\begin{align*}\widetilde{p}(\mathcal{E}^{\otimes n}) \in H^n(X, \omega) \cong k,\end{align*}$$

where $\omega := \Omega ^n$ (the canonical isomorphism being the trace map of Serre duality).

Proof. Let p be a homogenous polynomial of degree n in $ \mathbb {Q} [c_1,c_2, \dots , c_n]$ , and let $\mathcal {E}$ be a G-equivariant vector bundle in $ {[{ {\left [ M^\vee / G \right ]}\textbf{-}\mathbf {coh} }]} $ . Starting from the sequence

in $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{lf}\textbf{-}\mathbf{coh}} ]} $ (cf. Section 3.5.2), by the procedure described in Section 4.1.1 applied to the exact category $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{lf}\textbf{-}\mathbf{coh}} ]} $ , we can construct an element

$$\begin{align*}\widetilde{p}(\mathcal{E}^{\otimes n}) \in \operatorname{\mathrm{Ext}}^n_{ {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} }( {\cal O} , \omega). \end{align*}$$

Note that in the construction only the tensor product of locally free objects is involved and its exactness on sequences involving those.

Consider the following two (compositions of) functors

They induce linear maps (composing further with $\operatorname {\mathrm {tr}}$ )

$$\begin{align*}\operatorname{\mathrm{Ext}}_{ {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} }^n( {\cal O} , \omega) \rightarrow k \end{align*}$$

which map $\widetilde {p}(\mathcal {E}^{\otimes n})$ to

$$\begin{align*}p(c_1(\Xi^*\mathcal{E}), \dots, c_n(\Xi^*\mathcal{E})), \quad \text{ and } \quad p(c_1(\mathcal{E}), \dots, c_n(\mathcal{E})), \end{align*}$$

respectively. Here, it is used that $\Xi ^*$ is an exact functor which is compatible with the tensor product when restricted to locally free (or even torsion-free) objects — hence, with all of the operations performed in Section 4.1.1 — that by Theorem 3.5.3, the image of $J^1(\mathcal {E})'$ under $\Xi ^*$ is precisely $J^1(\Xi ^*\mathcal {E})$ , and that the image under the second functor is $J^1(\mathcal {E})$ , where the $G_M$ -action on $\mathcal {E}$ is forgotten (by definition of $J^1(\mathcal {E})'$ ).

Since there are nonzero Chern polynomials on $M^\vee $ , it therefore suffices to show that

$$\begin{align*}\operatorname{\mathrm{Ext}}_{ {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} }^n( {\cal O} , \omega) \end{align*}$$

is one-dimensional. This is Proposition 4.2.2 below. The compact case, that is $M = \overline {M}$ , is easier and Lemma 4.2.3 can be applied directly.

Proof. By Proposition 3.5.5, we have an exact sequence

and a finite resolution of the form

(14)

We get the long exact sequence

(all Ext-groups are computed in the category $ {[{ {\overline {M}}\textbf{-}\mathbf {FJ}\textbf{-}\mathbf{coh} }]} $ ). By Lemma 4.2.3 below, the dimension of $\operatorname {\mathrm {Ext}}^{n}( {\cal O} , \omega _{M^\vee })$ is one. Hence, it suffices to show that $\operatorname {\mathrm {Ext}}^{n-1}( {\cal O} , \mathcal {D} ) = \operatorname {\mathrm {Ext}}^{n}( {\cal O} , \mathcal {D} ) = 0$ . Splitting up the exact sequence (14) into short exact sequences, we see that this follows from $\operatorname {\mathrm {Ext}}^{i}( {\cal O} , \omega _{\overline {Y}}) = 0$ for $i \ge \dim (Y)$ and for $Y \not = M$ . We have fully faithful embeddings (cf. Corollary 3.4.10)

such that the image of $\omega _{M^\vee _Y} = (\Lambda ^n (\operatorname {\mathrm {Lie}}(G_Y)/\operatorname {\mathrm {Lie}}(Q_Y)))^*$ under the composition is $\omega _{\overline {Y}}$ .

Furthermore, we have

$$\begin{align*}{\cal O} = \iota_{0,!}\,\iota_{0}^*\, {\cal O}. \end{align*}$$

Hence

$$\begin{align*}\begin{array}{rcll} && \operatorname{\mathrm{Hom}}_{D^b( {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} )}(\iota_{0,!} \,\iota_{0}^*\, {\cal O} , \iota_{0,!}\, (0)_{Y,*}\, \omega_{M^\vee_Y}[i]) \\ &=& \operatorname{\mathrm{Hom}}_{D^b( {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}{\ge 0}\textbf{-}\mathbf{coh} }]} )}(\iota_{0}^*\, {\cal O} , (0)_{Y,*}\, \omega_{M^\vee_Y}[i] ) & \text{(fully faithfulness)} \\ & =& \operatorname{\mathrm{Hom}}_{D^b( {[{ {\left[M^\vee_Y / G_Y\right]}\textbf{-}\mathbf{coh} }]} )}( {\cal O} _{M^\vee_Y}, \omega_{M^\vee_Y}[i] ) & \text{(adjunction)} \end{array} .\end{align*}$$

Therefore, the proposition follows from the boundary vanishing condition (axiom B):

$$\begin{align*}H^i(\left[ M^\vee_Y / G_Y \right], \omega_{M^\vee_Y}) = 0 \text{ for } i \ge \dim(Y).\\[-37pt] \end{align*}$$

Proof. We have a fully faithful embedding (cf. Corollary 3.4.10)

$$\begin{align*}D^b( {[{ {\left[ M^\vee/G_M \right]}\textbf{-}\mathbf{coh} }]} ) \hookrightarrow D^b( {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} ). \end{align*}$$

The functor $R \operatorname {\mathrm {Hom}}( {\cal O} , -)$ is the same as the composition

$$\begin{align*}D^b( {[{ {\left[ M^\vee/G_M \right]}\textbf{-}\mathbf{coh} }]} ) \rightarrow D^b( {[{ {\left[\cdot / G_M\right]}\textbf{-}\mathbf{coh} }]} ) \rightarrow D^b( {[{ {\operatorname{\mathrm{spec}}(k)}\textbf{-}\mathbf{coh} }]} ), \end{align*}$$

where the first functor is the right derived functor of taking global sections and the second is the functor of $G_M$ -invariants. However, the last functor is exact (because $G_M$ is reductive) and therefore we have

$$\begin{align*}\operatorname{\mathrm{Ext}}_{ {[{ {\overline{M}}\textbf{-}\mathbf{FJ}\textbf{-}\mathbf{coh} }]} }^n( {\cal O} , \omega_{M^\vee}) = H^n(M^\vee, \omega_{M^\vee})^{G_M}. \end{align*}$$

Since $H^n(M^\vee , \omega _{M^\vee })$ is one-dimensional by Serre duality, and thus $G_M$ acts trivially because its centre does act trivially on $M^\vee $ , the lemma follows. Note that axiom (T), cf. Section 3.1.9, implies that $n=\dim (\overline {M})=\dim (M^\vee )$ .