$\\{(\mathcal{G}_{1,l},\varpi_{1,l})\\}_{l}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\\{(\mathcal{G}_{2,l},\varpi_{2,l})\\}_{l}$.
This concludes the injectivity of (211), and hence, completes the proof of
this lemma. ∎

##  4\. Indigenous bundles on smooth schemes

This section deals with indigenous bundles (including the case of finite
level) on a smooth scheme, generalizing the classical notion defined on a
Riemann surface (cf. [21]). As one of the main results of this section, we
prove that, for each level, $p$-flat Cartan geometries correspond bijectively
to dormant indigenous bundles (cf. Propositions 4.4.2, 4.5.4, and Corollary
4.5.5). In particular, the proofs of Theorems A and B will be completed in
this section. Also, the resulting correspondence enables us to prove several
properties on a smooth scheme having an $F^{\infty}$-$(G,P)$-structure (cf.
Propositions 4.6.1, 4.6.2). The last two subsections will be devoted to
studying $F^{N}$-$(G,P)$-structures in the case where $G$ is a product of
$\mathbb{G}_{m}$’s and $\mathbb{G}_{a}$.

Let $S$ be a scheme over $k$ and $X$ a smooth scheme over $S$.

###  4.1. The Kodaira-Spencer map

Let $\mathcal{E}_{H}$ be an $H$-bundle on $X$, and write
$\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$. The differential of the natural
inclusion $\mathcal{E}_{H}\hookrightarrow\mathcal{E}_{G}$ induces an
$\mathcal{O}_{X}$-linear injection
$\widetilde{\mathcal{T}}_{\mathcal{E}_{H}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$
(cf. § 7.1 for the definition of $\widetilde{\mathcal{T}}_{(-)/S}$). We shall
regard $\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}$ as an
$\mathcal{O}_{X}$-submodule of $\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}$
via this injection.

######  Definition 4.1.1.

Let
$\nabla:\mathcal{T}_{X/S}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}$
be an $S$-connection on the $G$-bundle $\mathcal{E}_{G}$. The
$\mathcal{O}_{X}$-linear composite

(212)
$\displaystyle\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}:\mathcal{T}_{X/S}\xrightarrow{\nabla}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}\twoheadrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}/\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}$

is called the Kodaira-Spencer map associated with $(\mathcal{E}_{H},\nabla)$.

######  Remark 4.1.2.

Let us keep the above notation. Differentiating the inclusion
$\mathcal{E}_{H}\hookrightarrow\mathcal{E}_{G}$ gives the following morphism
of short exact sequences:

(217)

This morphism induces an $\mathcal{O}_{X}$-linear isomorphism

(218)
$\displaystyle\mathfrak{g}_{\mathcal{E}_{G}}/\mathfrak{h}_{\mathcal{E}_{H}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}/\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}.$

By this isomorphism, the Kodaira-Spencer map
$\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$ may be regarded as an
$\mathcal{O}_{X}$-linear morphism
$\mathcal{T}_{X/S}\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}/\mathfrak{h}_{\mathcal{E}_{H}}$.

######  Example 4.1.3 (The cases of $G=\mathbb{G}_{m}$ and $\mathbb{G}_{a}$).

  * (i)

First, we shall consider the case where $G=\mathbb{G}_{m}$ and $H=\\{e\\}$.
Let us take a $1$-form $\omega$ on $X/S$. Recall that $\omega$ induces an
$S$-connection $\nabla_{\omega}$ on the trivial $\mathbb{G}_{m}$-bundle
$X\times\mathbb{G}_{m}$ (cf. (89)). Then, the Kodaira-Spencer map
$\mathrm{KS}_{(X\times\\{e\\},\nabla_{\omega})}$ associated with the pair
$(X\times\\{e\\},\nabla_{\omega})$ coincides with $\omega^{\triangleright}$
under the natural identification

(219)
$\displaystyle\mathcal{O}_{X}\cong\mathrm{Lie}(\mathbb{G}_{m})_{X\times\mathbb{G}_{m}}/\mathrm{Lie}(\\{e\\})_{X\times\\{e\\}}\stackrel{{\scriptstyle(\ref{Eoraplw2})}}{{\cong}}\widetilde{\mathcal{T}}_{X\times\mathbb{G}_{m}/S}/\widetilde{\mathcal{T}}_{X\times\\{e\\}/S}.$

Note that giving a $\mathbb{G}_{m}$-bundle equipped with both an
$S$-connection and an $\\{e\\}$-reduction is equivalent to giving an
$S$-connection on the trivial $\mathbb{G}_{m}$-bundle, i.e., $\nabla_{\omega}$
for some $\omega$ as above.

  * (ii)

Next, suppose that $G=\mathbb{G}_{a}$ and $H=\\{e\\}$. We shall identify
$\mathbb{G}_{a}$ as the subgroup of $\mathrm{GL}_{2}$ consisting of matrices
$\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}$ for some $a\in\mathbb{G}_{a}$. Each
$\mathbb{G}_{a}$-bundle may be thought of as a rank $2$ vector bundle
$\mathcal{V}$ equipped with a structure of extension
$0\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{V}\rightarrow\mathcal{O}_{X}\rightarrow
0$. Moreover, a choice of an $\\{e\\}$-reduction corresponds to a splitting
$\mathcal{V}\cong\mathcal{O}_{X}^{\oplus 2}$ of this extension. The vector
bundle $\mathcal{O}_{X}^{\oplus 2}$ equipped with the extension
$0\rightarrow\mathcal{O}_{X}\xrightarrow{a\mapsto(a,0)}\mathcal{O}_{X}^{\oplus
2}\xrightarrow{(a,b)\mapsto b}\mathcal{O}_{X}\rightarrow 0$ corresponds to the
trivial $\mathbb{G}_{a}$-bundle $X\times\mathbb{G}_{a}$.

For each $1$-form $\chi$ on $X/S$, denote by

(220) $\displaystyle\chi^{\dagger}:\mathcal{O}_{X}^{\oplus
2}\rightarrow\Omega_{X/S}\otimes_{\mathcal{O}_{X}}(\mathcal{O}_{X}^{\oplus
2})\left(=\Omega_{X/S}^{\oplus 2}\right)$

the $\mathcal{O}_{X}$-linear morphism defines as $\begin{pmatrix}0&\chi\\\
0&0\end{pmatrix}$. Then, the morphism

(221) $\displaystyle\nabla_{\chi}^{\dagger}:=d^{\oplus 2}+\chi^{\dagger}$

turns out to define an $S$-connection on the trivial $\mathbb{G}_{a}$-bundle.
The Kodaira-Spencer map
$\mathrm{KS}_{(X\times\\{e\\},\nabla_{\chi}^{\dagger})}$ associated with the
pair $(X\times\\{e\\},\nabla^{\dagger}_{\chi})$ coincides with
$\chi^{\triangleright}$ under the identification
$\mathcal{O}_{X}\cong\widetilde{\mathcal{T}}_{X\times\mathbb{G}_{a}/S}/\widetilde{\mathcal{T}}_{X\times\\{e\\}/S}$
similar to (219). Also, giving a $\mathbb{G}_{a}$-bundle equipped with both an
$S$-connection and an $\\{e\\}$-reduction is equivalent to giving an
$S$-connection on $\mathcal{O}_{X}^{\oplus 2}$ of the form
$\nabla_{\chi}^{\dagger}$ for some $\chi$ as above.

######  Definition 4.1.4.

  * (i)

An indigenous $(G,H)$-bundle on $X/S$ is a pair

(222) $\displaystyle\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$

consisting of an $H$-bundle $\mathcal{E}_{H}$ on $X$ and an $S$-connection
$\nabla$ on $\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$ such that the
Kodaira-Spencer map $\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$ is an
isomorphism.

  * (ii)

Let $\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ and
$\mathcal{E}^{\prime\spadesuit}:=(\mathcal{E}^{\prime}_{H},\nabla^{\prime})$
be indigenous $(G,H)$-bundles on $X/S$. An isomorphism of indigenous
$(G,H)$-bundles from $\mathcal{E}^{\spadesuit}$ to
$\mathcal{E}^{\prime\spadesuit}$ is defined as an isomorphism of $H$-bundles
$\mathcal{E}_{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}^{\prime}_{H}$
such that the induced isomorphism of $G$-bundles
$\mathcal{E}_{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}^{\prime}_{G}$
is compatible with the respective connections, i.e., $\nabla$ and
$\nabla^{\prime}$.

######  Definition 4.1.5.

Let $\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ be an indigenous
$(G,H)$-bundle on $X$. We shall say that $\mathcal{E}^{\spadesuit}$ is flat if
the curvature $\psi_{\nabla}$ (cf. (442)) of $\nabla$ vanishes.

Denote by

(223) $\displaystyle\mathrm{Ind}_{(G,H),X/S}\ \left(\text{resp.,}\
\mathrm{Ind}_{(G,H),X/S}^{\flat}\right)$

the set of isomorphism classes of indigenous $(G,H)$-bundles (resp., flat
indigenous $(G,H)$-bundles) on $X/S$.

######  Example 4.1.6 (Standard example).

Recall that $\mathcal{E}^{\mathrm{triv}}_{H}:=S\times G$ (cf. Example 3.4.3)
specifies an $H$-reduction on $P_{S}\times G$ via
$(\mathrm{id}_{S}\times\pi_{G},\mathrm{pr}_{2}):\mathcal{E}^{\mathrm{triv}}_{H}\hookrightarrow
P_{S}\times G$. In particular, there exists a natural isomorphism of
$G$-bundles
$\mathcal{E}^{\mathrm{triv}}_{G}\left(:=\mathcal{E}^{\mathrm{triv}}_{H}\times^{H}G\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}P_{S}\times
G$ extending this inclusion. Since the composite
$G\xrightarrow{(\mathrm{id}_{S}\times\pi_{G},\mathrm{pr}_{2})}P_{S}\times
G\xrightarrow{\mathrm{pr}_{2}}G$ coincides with the identity morphism, the
composite

(224)
$\displaystyle\widetilde{\mathcal{T}}_{\mathcal{E}_{H}^{\mathrm{triv}}/S}\hookrightarrow\widetilde{\mathcal{T}}_{P_{S}\times
G/S}\left(\stackrel{{\scriptstyle\tau}}{{\cong}}\mathcal{T}_{P_{S}/S}\oplus\mathcal{O}_{P_{S}}\otimes\mathfrak{g}\right)\xrightarrow{\mathrm{pr}_{2}}\mathcal{O}_{P_{S}}\otimes\mathfrak{g}$

is an isomorphism. This implies that the Kodaira-Spencer map
$\mathrm{KS}_{(\mathcal{E}_{H}^{\mathrm{triv}},\nabla_{0})}$ associated with
$(\mathcal{E}_{H}^{\mathrm{triv}},\nabla_{0})$ (cf. (89) for the definition of
$\nabla_{(-)}$) is an isomorphism. That is to say, the pair

(225)
$\displaystyle\mathcal{E}^{\mathrm{triv},\spadesuit}:=(\mathcal{E}^{\mathrm{triv}}_{H},\nabla_{0})$

forms a flat indigenous $(G,H)$-bundle on $P_{S}/S$. It is clear that
$\mathcal{E}^{\mathrm{triv},\spadesuit}$ is also dormant (cf. Definition 4.4.1
described later).

###  4.2. Pull-back and base-change of Indigenous $(G,H)$-bundles

Let $Y$ and $y$ be as in § 2.4. Also, let
$\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ be an indigenous
$(G,H)$-bundle on $X/S$. The base-change $y^{*}(\mathcal{E}_{H})$ (resp.,
$y^{*}(\mathcal{E}_{G})$) of $\mathcal{E}_{H}$ (resp.,
$\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$) has an étale morphism
$y^{*}(\mathcal{E}_{H})\rightarrow\mathcal{E}_{H}$ (resp.,
$y^{*}(\mathcal{E}_{G})\rightarrow\mathcal{E}_{G}$), whose differential yields
an isomorphism
$d\widetilde{y}_{H}:\widetilde{\mathcal{T}}_{y^{*}(\mathcal{E}_{H})/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}y^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S})$
(resp.,
$d\widetilde{y}_{G}:\widetilde{\mathcal{T}}_{y^{*}(\mathcal{E}_{G})/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}y^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S})$).
The pull-back $y^{*}(\nabla):y^{*}(\mathcal{T}_{X/S})\rightarrow
y^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S})$ of $\nabla$ specifies an
$S$-connection on $y^{*}(\mathcal{E}_{G})$ under the identifications
$\mathcal{T}_{Y/S}=y^{*}(\mathcal{T}_{X/S})$,
$\widetilde{\mathcal{T}}_{y^{*}(\mathcal{E}_{G})/S}=y^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S})$
given by $dy$, $d\widetilde{y}_{G}$ respectively. Since $d\widetilde{y}_{G}$
is restricted to $d\widetilde{y}_{H}$, we obtain the following commutative
square diagram:

(230)

This implies that the Kodaira-Spencer map
$\mathrm{KS}_{(y^{*}(\mathcal{E}_{H}),y^{*}(\nabla))}$ is an isomorphism, and
hence, the pair of pull-backs

(231) $\displaystyle
y^{*}(\mathcal{E}^{\spadesuit}):=(y^{*}(\mathcal{E}_{H}),y^{*}(\nabla))$

forms an indigenous $(G,H)$-bundle on $Y/S$. We shall refer
$y^{*}(\mathcal{E}^{\spadesuit})$ as the pull-back of
$(\mathcal{E}_{H},\nabla)$. Because of the equality
$y^{*}(\psi_{\nabla})=\psi_{y^{*}(\nabla)}$, we see that
$y^{*}(\mathcal{E}^{\spadesuit})$ is flat if $\mathcal{E}^{\spadesuit}$ is
flat. Hence, the assignment $\mathcal{E}^{\spadesuit}\mapsto
y^{*}(\mathcal{E}^{\spadesuit})$ defines maps

(232)
$\displaystyle\mathrm{Ind}_{(G,H),X/S}\rightarrow\mathrm{Ind}_{(G,H),Y/S},\hskip
14.22636pt\mathrm{Ind}_{(G,H),X/S}^{\flat}\rightarrow\mathrm{Ind}_{(G,H),Y/S}^{\flat}.$

The formation of a(n) (flat) indigenous $(G,H)$-bundle on $X/S$ has descent
with respect to the étale topology on $X$.

Next, let $s:S^{\prime}\rightarrow S$ and $X^{\prime}$ be as in § 2.4 and let
$\mathcal{E}^{\spadesuit}$ be as above. We shall use the notation $s^{*}(-)$
to denote the result of base-change by $s$. The base-change
$s^{*}(\mathcal{E}_{H})$ defines an $H$-reduction of $s^{*}(\mathcal{E}_{G})$,
and the differential of the projection
$s^{*}(\mathcal{E}_{G})\rightarrow\mathcal{E}_{G}$ yields an isomorphism
$d\widetilde{s}_{G}:\widetilde{\mathcal{T}}_{s^{*}(\mathcal{E}_{G})/S^{\prime}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}s^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S})$.
Just as in the above discussion, the base-change $s^{*}(\nabla)$ defines an
$S^{\prime}$-connection on $s^{*}(\mathcal{E}_{G})$ under the identification
$\widetilde{\mathcal{T}}_{s^{*}(\mathcal{E}_{G})/S^{\prime}}=s^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S})$
given by $d\widetilde{s}_{G}$. Also, the Kodaira-Spencer map associated with
$(s^{*}(\mathcal{E}_{H}),s^{*}(\nabla))$ may be identified with the
isomorphism $s^{*}(\mathrm{KS}_{(\mathcal{E}_{H},\nabla)})$. Thus, the pair of
base-changes

(233) $\displaystyle s^{*}(\mathcal{E}^{\spadesuit})\left(\text{or}\
S^{\prime}\times_{S}\mathcal{E}^{\spadesuit}\right):=(s^{*}(\mathcal{E}_{H}),s^{*}(\nabla))$

forms an indigenous $(G,H)$-bundle on $X^{\prime}/S^{\prime}$, called the
base-change of $\mathcal{E}^{\spadesuit}$. Since the equality
$s^{*}(\psi_{\nabla})=\psi_{s^{*}(\nabla)}$ holds,
$s^{*}(\mathcal{E}^{\spadesuit})$ is flat if $\mathcal{E}^{\spadesuit}$ is
flat. Thus, the resulting assignment $\mathcal{E}^{\spadesuit}\mapsto
s^{*}(\mathcal{E}^{\spadesuit})$ determines maps

(234)
$\displaystyle\mathrm{Ind}_{(G,H),X/S}\rightarrow\mathrm{Ind}_{(G,H),X^{\prime}/S^{\prime}},\hskip
14.22636pt\mathrm{Ind}_{(G,H),X/S}^{\flat}\rightarrow\mathrm{Ind}_{(G,H),X^{\prime}/S^{\prime}}^{\flat}.$

###  4.3. From $(G,H)$-Cartan geometries to Indigenous $(G,H)$-bundles

We shall construct an indigenous $(G,H)$-bundle by using a Cartan geometry
with model $(G,H)$.

Let $\mathcal{E}^{\clubsuit}:=(\mathcal{E}_{H},\omega)$ be a Cartan geometry
with model $(G,H)$ on $X/S$. Denote by $\pi:\mathcal{E}_{H}\rightarrow X$ the
structure morphism of $\mathcal{E}_{H}$ and by
$\jmath:\mathcal{E}_{H}\hookrightarrow\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$
the natural inclusion. The composite
$\jmath^{*}(\omega^{\triangleright}_{\mathcal{E}_{G}/X})^{-1}\circ\omega^{\triangleright}$
(cf. (78)) defines an $\mathcal{O}_{\mathcal{E}_{H}}$-linear isomorphism
$\mathcal{T}_{\mathcal{E}_{H}/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\jmath^{*}(\mathcal{T}_{\mathcal{E}_{G}/X})$.
The direct image
$\pi_{*}(\mathcal{T}_{\mathcal{E}_{H}/S})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\pi_{*}(\jmath^{*}(\mathcal{T}_{\mathcal{E}_{G}/X}))$
of this morphism preserves the $H$-actions, so it is restricted to an
isomorphism
$\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\pi_{*}(\jmath^{*}(\mathcal{T}_{\mathcal{E}_{G}/X}))^{H}$.
Since $\mathfrak{g}_{\mathcal{E}_{G}}$ is naturally isomorphic to
$\pi_{*}(\jmath^{*}(\mathcal{T}_{\mathcal{E}_{G}/X}))^{H}$, it may be regarded
as an isomorphism

(235)
$\displaystyle\widetilde{\omega}^{\triangleright}:\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left(\mathfrak{g}_{\mathcal{E}_{H}}=\right)\mathfrak{g}_{\mathcal{E}_{G}}.$

By the commutativity of (96), the restriction of
$\widetilde{\omega}^{\triangleright}$ to
$\mathfrak{h}_{\mathcal{E}_{H}}\left(\subseteq\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\right)$
coincides with the inclusion
$\mathfrak{h}_{\mathcal{E}_{H}}\hookrightarrow\mathfrak{g}_{\mathcal{E}_{G}}$.
Observe here that (217) induces an $\mathcal{O}_{X}$-linear isomorphism
$\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\times^{\mathfrak{h}_{\mathcal{E}_{H}}}\mathfrak{g}_{\mathcal{E}_{G}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}$.
Under the identification
$\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\times^{\mathfrak{h}_{\mathcal{E}_{H}}}\mathfrak{g}_{\mathcal{E}_{G}}=\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}$
given by this isomorphism, the morphism
$(\widetilde{\omega}^{\triangleright},\mathrm{id}_{\mathfrak{g}_{\mathcal{E}_{G}}}):\widetilde{\mathcal{T}}_{\mathcal{E}_{H}}\times^{\mathfrak{h}_{\mathcal{E}_{H}}}\mathfrak{g}_{\mathcal{E}_{G}}\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}$
determines a split surjection
$\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}\twoheadrightarrow\mathfrak{g}_{\mathcal{E}_{G}}$
of (441) for $\mathcal{E}_{G}$. The corresponding split injection

(236)
$\displaystyle\nabla_{\mathcal{E}^{\clubsuit}}:\mathcal{T}_{X/S}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}$

specifies an $S$-connection on the $G$-bundle $\mathcal{E}_{G}$.

The direct image
$\pi_{*}(\bigwedge^{2}\Omega_{\mathcal{E}_{H}/S}\otimes\mathfrak{g})\left(=\pi_{*}(\bigwedge^{2}\Omega_{\mathcal{E}_{H}/S})\otimes\mathfrak{g}\right)$
has a right $H$-action given by $(a\otimes v)\cdot
h:=d\mathrm{R}_{h}^{\vee}(a)\otimes\mathrm{Ad}_{G}(h^{-1})(v)$ for
$a\in\pi_{*}(\bigwedge^{2}\Omega_{\mathcal{E}_{H}/S})$, $v\in\mathfrak{g}$,
and $h\in H$. Because of the second condition in Definition 3.4.1, (i), the
curvature $\psi_{\omega}$ of the $1$-form $\nabla_{\omega}$, being an element
of
$\Gamma(\mathcal{E}_{H},\bigwedge^{2}\Omega_{\mathcal{E}_{H}/S}\otimes\mathfrak{g})\left(=\Gamma(X,\pi_{*}(\bigwedge^{2}\Omega_{\mathcal{E}_{H}/S}\otimes\mathfrak{g}))\right)$,
is invariant under this $H$-action. Hence, $\psi_{\omega}$ defines an element
of
$\Gamma(X,\pi_{*}(\bigwedge^{2}\Omega_{\mathcal{E}_{H}/S}\otimes\mathfrak{g})^{H})$.

######  Proposition 4.3.1.

The pair

(237)
$\displaystyle\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}:=(\mathcal{E}_{H},\nabla_{\mathcal{E}^{\clubsuit}})$

defines an indigenous $(G,H)$-bundle on $X/S$. Moreover,
$\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}$ is flat if and only if
$\mathcal{E}^{\clubsuit}$ is flat.

###### Proof.

For simplicity, we shall write $\nabla:=\nabla_{\mathcal{E}^{\clubsuit}}$.
First, let us consider the former assertion. By the definition of
$\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$, the following diagram forms a
morphism of short exact sequences:

(242)

Hence, since $\widetilde{\omega}^{\triangleright}$ is an isomorphism, the
Kodaira-Spencer map $\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$ is an
isomorphism. That is to say, the pair
$\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}:=(\mathcal{E}_{H},\nabla)$
specifies an indigenous $(G,H)$-bundle on $X/S$, which completes the proof of
the former assertion.

Next, let us consider the latter assertion. Note that the assignments
$\mathcal{V}\mapsto\pi_{*}(\mathcal{V})^{H}$ and
$\mathcal{U}\mapsto\pi^{*}(\mathcal{U})$ define an equivalence of categories

(243) $\displaystyle\left(\begin{matrix}\text{the category of $H$-equivariant
}\\\ \text{vector bundles on
$\mathcal{E}_{H}$}\end{matrix}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left(\begin{matrix}\text{the
category of}\\\ \text{vector bundles on $X$}\end{matrix}\right).$

In particular, since
$\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}=\pi_{*}(\mathcal{T}_{\mathcal{E}_{H}/S})^{H}$
and
$\mathfrak{g}_{\mathcal{E}_{G}}=\pi_{*}(\mathcal{O}_{\mathcal{E}_{H}}\otimes\mathfrak{g})^{H}$,
the natural morphisms

(244)
$\displaystyle\pi^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S})\rightarrow\mathcal{T}_{\mathcal{E}_{H}/S}\hskip
8.53581pt\text{and}\hskip
8.53581pt\pi^{*}(\mathfrak{g}_{\mathcal{E}_{G}})\rightarrow\mathcal{O}_{\mathcal{E}_{H}}\otimes\mathfrak{g}$

are isomorphisms. We shall define $\psi_{\widetilde{\omega}^{\triangleright}}$
to be the $\mathcal{O}_{X}$-linear morphism
$\bigwedge^{2}\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}$
determined by

(245)
$\displaystyle\psi_{\widetilde{\omega}^{\triangleright}}(\partial_{1},\partial_{2})=\partial_{1}(\widetilde{\omega}^{\triangleright}(\partial_{2}))-\partial_{2}(\widetilde{\omega}^{\triangleright}(\partial_{1}))+[\widetilde{\omega}^{\triangleright}(\partial_{1}),\widetilde{\omega}^{\triangleright}(\partial_{2})]-\widetilde{\omega}^{\triangleright}([\partial_{1},\partial_{2}])$

for any local sections
$\partial_{1},\partial_{2}\in\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}$.
Under the identifications
$\pi^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S})=\mathcal{T}_{\mathcal{E}_{H}/S}$,
$\pi^{*}(\mathfrak{g}_{\mathcal{E}_{G}})=\mathcal{O}_{\mathcal{E}_{H}}\otimes\mathfrak{g}$
given by (244), the equality
$\pi^{*}(\psi_{\widetilde{\omega}^{\triangleright}})=\psi_{\omega}$ holds (cf.
the last paragraph in the discussion preceding this proposition). Hence,
$\psi_{\widetilde{\omega}^{\triangleright}}=0$ if and only if
$\psi_{\omega}=0$. On the other hand, $\widetilde{\omega}^{\triangleright}$
can be expressed as $\widetilde{\omega}^{\triangleright}=\nabla\circ
d_{\mathcal{E}_{H}}-\mathrm{id}_{\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}}$.
If we take arbitrary local sections
$\partial_{1},\partial_{2}\in\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}$ and
write $\overline{\partial}_{1}:=d_{\mathcal{E}_{H}}(\partial_{1})$,
$\overline{\partial}_{2}:=d_{\mathcal{E}_{H}}(\partial_{2})\left(\in\mathcal{T}_{X/S}\right)$,
then (245) induces the following sequence of equalities:

(246) $\displaystyle\
\psi_{\widetilde{\omega}^{\triangleright}}(\partial_{1},\partial_{2})$
$\displaystyle=$ $\displaystyle\
[\partial_{1},\nabla(\overline{\partial}_{2})-\partial_{2}]-[\partial_{2},\nabla(\overline{\partial}_{1})-\partial_{1}]$
$\displaystyle\
+[\nabla(\overline{\partial}_{1})-\partial_{1},\nabla(\overline{\partial}_{2})-\partial_{2}]-\left(\nabla(d_{\mathcal{E}_{H}}([\partial_{1},\partial_{2}]))-[\partial_{1},\partial_{2}]\right)$
$\displaystyle=$ $\displaystyle\
\left(\cancel{[\partial_{1},\nabla(\overline{\partial}_{2})]}-\cancel{[\partial_{1},\partial_{2}]}\right)-\left(\cancel{[\partial_{2},\nabla(\overline{\partial}_{1})]}-\cancel{[\partial_{2},\partial_{1}]}\right)$
$\displaystyle\
+\left([\nabla(\overline{\partial}_{1}),\nabla(\overline{\partial}_{2})]-\cancel{[\nabla(\overline{\partial}_{1}),\partial_{2}]}-\cancel{[\partial_{1},\nabla(\overline{\partial}_{2})]}+\cancel{[\partial_{1},\partial_{2}]}\right)-\left(\nabla([\overline{\partial}_{1},\overline{\partial}_{2}])-\cancel{[\partial_{1},\partial_{2}]}\right)$
$\displaystyle=$ $\displaystyle\
\psi_{\nabla}(\overline{\partial}_{1},\overline{\partial}_{2}).$

Thus, since
$d_{\mathcal{E}_{H}}:\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\rightarrow\mathcal{T}_{X/S}$
is surjective, we see that $\psi_{\nabla}=0$ if and only if
$\psi_{\widetilde{\omega}^{\triangleright}}=0$, or equivalently,
$\psi_{\omega}=0$ by the above discussion. This completes the proof of the
latter assertion. ∎

######  Corollary 4.3.2.

The assignment
$\mathcal{E}^{\clubsuit}\mapsto\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}$
resulting from Proposition 4.3.1 defines a bijection of sets

(247)
$\displaystyle\mathrm{Car}_{(G,H),X/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ind}_{(G,H),X/S}\
\left(\text{resp.,}\
\mathrm{Car}_{(G,H),X/S}^{\flat}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ind}_{(G,H),X/S}^{\flat}\right).$

Moreover, the formation of this bijection is compatible with pull-back to
étale schemes over $X$, as well as with base-change to schemes over $S$.

###### Proof.

Since the resp’d assertion follows from the non-resp’d assertion and the
latter assertion of Proposition 4.3.1, it suffices to consider the non-resp’d
assertion. To prove this, we shall construct the inverse to the assignment
$\mathcal{E}^{\clubsuit}\mapsto\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}$.
Let $\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ be an indigenous
$(G,H)$-bundle on $X/S$. Denote by
$\widetilde{\omega}^{\triangleright}_{\mathcal{E}^{\spadesuit}}$ the composite
of the natural inclusion
$\widetilde{\mathcal{T}}_{\mathcal{E}_{H}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$
and the split surjection
$\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/S}\twoheadrightarrow\mathfrak{g}_{\mathcal{E}_{G}/S}$
of (441) corresponding to $\nabla$. This morphism fits into the following
morphism of short exact sequences:

(252)

The Kodaira-Spencer map $\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$ is an
isomorphism, so the five lemma shows that
$\widetilde{\omega}_{\mathcal{E}^{\spadesuit}}^{\triangleright}$ is an
isomorphism. Under the identifications given by (244), the pull-back of
$\widetilde{\omega}_{\mathcal{E}^{\spadesuit}}^{\triangleright}$ by $\pi$
defines an $\mathcal{O}_{\mathcal{E}_{H}}$-linear isomorphism
$\mathcal{T}_{\mathcal{E}_{H}/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{\mathcal{E}_{H}}\otimes\mathfrak{g}$.
If $\omega_{\mathcal{E}^{\spadesuit}}$ denotes the corresponding
$\mathfrak{g}$-valued $1$-form on $\mathcal{E}_{H}/S$, then the pair
$\mathcal{E}^{\spadesuit\Rightarrow\clubsuit}:=(\mathcal{E}_{H},\omega_{\mathcal{E}^{\spadesuit}})$
forms a Cartan geometry with model $(G,H)$ on $X/S$. (The commutativity of the
left-hand square in (252) implies, via pull-back by $\pi$, the first condition
in Definition 3.4.1, (i).) Moreover, the resulting assignment
$\mathcal{E}^{\spadesuit}\mapsto\mathcal{E}^{\spadesuit\Rightarrow\clubsuit}$
is verified to define the inverse to the assignment
$\mathcal{E}^{\clubsuit}\mapsto\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}$.
This implies the bijectivity of (247). The required compatibility of the
formation follows immediately from its construction, and hence, we finish the
proof of this corollary. ∎

###  4.4. Dormant indigenous $(G,H)$-bundles

Next, we make the definition of a dormant indigenous bundle. If the underlying
space is a curve, then this notion was already defined and has been studied in
previous works (cf., e.g., [40]).

######  Definition 4.4.1.

We shall say that an indigenous $(G,H)$-bundle
$\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ is dormant if it is flat
and the $p$-curvature ${{}^{p}}\psi_{\nabla}$ of $\nabla$ (cf. (458)) vanishes
identically on $X$.

Denote by

(253) $\displaystyle\mathrm{Ind}_{(G,H),X/S}^{{}^{\mathrm{Zzz...}}}$

the subset of $\mathrm{Ind}_{(G,H),X}^{\flat}$ consisting of dormant
indigenous $(G,H)$-bundles. Then, we obtain the remaining portion of Theorem A
described below. (The map $\zeta^{\heartsuit\Rightarrow\spadesuit}$, which is
one of the required bijections in that theorem, is defined as the composite of
$\zeta^{\clubsuit\Rightarrow\spadesuit}$ constructed below and
$\zeta^{\heartsuit\Rightarrow\clubsuit}$.)

######  Proposition 4.4.2.

The bijection
$\mathrm{Car}_{(G,H),X/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ind}_{(G,H),X/S}$
resulting from Corollary 4.3.2 is restricted to a bijection of sets

(254)
$\displaystyle\zeta^{\clubsuit\Rightarrow\spadesuit}:\mathrm{Car}^{p\text{-}\mathrm{flat}}_{(G,H),X/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ind}_{(G,H),X/S}^{{}^{\mathrm{Zzz...}}}.$

Moreover, the formation of this bijection commutes with pull-back to étale
schemes over $X$, as well as with base-change to schemes over $S$.

###### Proof.

We shall consider the former assertion. Let
$\mathcal{E}^{\clubsuit}:=(\mathcal{E}_{H},\omega)$ be a Cartan geometry with
model $(G,H)$ on $X/S$. Denote by $\pi:\mathcal{E}_{H}\rightarrow X$ the
structure morphism of $\mathcal{E}_{H}$ and by
$\mathcal{E}^{\clubsuit\Rightarrow\spadesuit}:=(\mathcal{E}_{H},\nabla)$ the
indigenous $(G,H)$-bundle corresponding to $\mathcal{E}^{\clubsuit}$ via
(247), i.e., $\nabla:=\nabla_{\mathcal{E}^{\clubsuit}}$. Under the
identifications
$\pi^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S})=\mathcal{T}_{\mathcal{E}_{H}/S}$,
$\pi^{*}(\mathfrak{g}_{\mathcal{E}_{G}})=\mathcal{O}_{\mathcal{E}_{H}}\otimes\mathfrak{g}$
given by (244), the $p$-curvature ${{}^{p}}\psi_{\omega}$ of $\omega$ may be
regarded as an
$\mathcal{O}_{\mathcal{E}_{H}}\left(=\pi^{*}(\mathcal{O}_{X})\right)$-linear
morphism
$\left(F_{\mathcal{E}_{H}}^{*}(\pi^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}))=\right)\pi^{*}(F^{*}_{X}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}))\rightarrow\pi^{*}(\mathfrak{g}_{\mathcal{E}_{G}})$.
This morphism comes from an $\mathcal{O}_{X}$-linear morphism
${{}^{p}}\psi_{\widetilde{\omega}^{\triangleright}}:F^{*}_{X}(\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S})\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}$
via pull-back by $\pi$. If
$\widetilde{\omega}^{\triangleright}:\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}_{\mathcal{E}_{G}}$
denotes the isomorphism constructed from $\mathcal{E}^{\clubsuit}$ as in
(235), then ${{}^{p}}\psi_{\widetilde{\omega}^{\triangleright}}$ is defined by

(255)
$\displaystyle{{}^{p}}\psi_{\widetilde{\omega}^{\triangleright}}(F^{-1}_{X}(\partial))=\widetilde{\omega}^{\triangleright}(\partial)^{[p]}-\widetilde{\omega}^{\triangleright}(\partial^{[p]})+\sum_{i=1}^{p-1}\frac{s_{i}(\partial,\widetilde{\omega}^{\triangleright}(\partial))}{i}$

for any local section $\partial\in\mathcal{T}_{X/S}$. Let us take an arbitrary
local section $\partial\in\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}$ and
write
$\overline{\partial}:=d_{\mathcal{E}_{H}}(\partial)\in\mathcal{T}_{X/S}$.
Since $\widetilde{\omega}^{\triangleright}=\nabla\circ
d_{\mathcal{E}_{H}}-\mathrm{id}_{\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}}$,
we obtain the following sequence of equalities:

(256) $\displaystyle\
{{}^{p}}\psi_{\widetilde{\omega}^{\triangleright}}(F^{-1}_{X}(\partial))$
$\displaystyle=$ $\displaystyle\ (\nabla\circ
d_{\mathcal{E}_{H}}-\mathrm{id})(\partial)^{[p]}-(\nabla\circ
d_{\mathcal{E}_{H}}-\mathrm{id})(\partial^{[p]})+\sum_{i=1}^{p-1}\frac{s_{i}(\partial,(\nabla\circ
d_{\mathcal{E}_{H}}-\mathrm{id})(\partial))}{i}$ $\displaystyle=$
$\displaystyle\
(\nabla(\overline{\partial})-\partial)^{[p]}-(\nabla(\overline{\partial}^{[p]})-\partial^{[p]})+\sum_{i=1}^{p-1}\frac{s_{i}(\partial,\nabla(\overline{\partial})-\partial)}{i}$
$\displaystyle=$ $\displaystyle\
\cancel{(-1)^{p}\cdot\partial^{[p]}}+\nabla(\overline{\partial})^{[p]}+\cancel{\sum_{i=1}^{p-1}\frac{s_{i}(-\partial,\nabla(\overline{\partial}))}{i}}-\nabla(\overline{\partial}^{[p]})+\cancel{\partial^{[p]}}+\cancel{\sum_{i=1}^{p-1}\frac{s_{i}(\partial,\nabla(\overline{\partial})-\partial)}{i}}$
$\displaystyle=$ $\displaystyle\
{{}^{p}}\psi_{\nabla}(F^{-1}_{X}(\overline{\partial})),$

where the last equality follows from Lemma 4.4.3 described below. Hence, the
surjectivity of
$d_{\mathcal{E}_{H}}:\widetilde{\mathcal{T}}_{\mathcal{E}_{H}/S}\rightarrow\mathcal{T}_{X/S}$
implies that ${{}^{p}}\psi_{\nabla}=0$ if and only if
${{}^{p}}\psi_{\widetilde{\omega}^{\triangleright}}=0$, or equivalently,
${{}^{p}}\psi_{\omega}=0$ by the above discussion. This completes the proof of
the former assertion. The latter assertion follows immediately from its
construction. ∎

The following lemma was used in the proof of the above proposition.

######  Lemma 4.4.3.

Let $\mathfrak{a}$ be a Lie algebra over a $k$-algebra $R$. Then, for any
elements $v,u\in\mathfrak{a}$, the following equality holds:

(257)
$\displaystyle\sum_{i=1}^{p-1}\frac{s_{i}(v,u-v)}{i}=-\sum_{i=1}^{p-1}\frac{s_{i}(-v,u)}{i}.$

###### Proof.

By the definition of $s_{i}$, the following sequence of equalities holds:

(258) $\displaystyle\sum_{i=1}^{p-1}s_{i}(v,u-v)t^{i-1}=$ $\displaystyle\
\mathrm{ad}(vt+u-v)^{p-1}(v)$ $\displaystyle=$ $\displaystyle\
\mathrm{ad}(v(t-1)+u)^{p-1}(v)$ $\displaystyle=$ $\displaystyle\
\sum_{j=1}^{p-1}s_{j}(v,u)(t-1)^{j-1}$ $\displaystyle=$ $\displaystyle\
\sum_{i=1}^{p-1}\left(\sum_{j=i}^{p-1}(-1)^{j-i}\cdot\binom{j-1}{i-1}\cdot
s_{j}(v,u)\right)t^{i-1}.$

This implies
$s_{i}(v,u-v)=\sum\limits_{j=i}^{p-1}(-1)^{j-i}\cdot\binom{j-1}{i-1}\cdot
s_{j}(v,u)$ for $i=1,\cdots,p-1$. Thus, we have

(259) $\displaystyle\sum_{i=1}^{p-1}\frac{s_{i}(v,u-v)}{i}=$ $\displaystyle\
\sum_{i=1}^{p-1}\sum\limits_{j=i}^{p-1}\frac{1}{i}\cdot(-1)^{j-i}\cdot\binom{j-1}{i-1}\cdot
s_{j}(v,u)$ $\displaystyle=$ $\displaystyle\
\sum_{j=1}^{p-1}\frac{s_{j}(v,u)}{j}\cdot\left(\sum_{i=1}^{j}(-1)^{j-i}\cdot\binom{j}{i}\right)$
$\displaystyle=$ $\displaystyle\
\sum_{j=1}^{p-1}\frac{s_{j}(v,u)}{j}\cdot(-1)^{j+1}$ $\displaystyle=$
$\displaystyle\ -\sum_{j=1}^{p-1}\frac{s_{j}(-v,u)}{j}.$

This completes the proof of this lemma. ∎

###  4.5. Indigenous $(G,H)$-bundles of higher level

We generalize both the notion of an indigenous $(G,H)$-bundle and the
bijection $\zeta^{\clubsuit\Rightarrow\spadesuit}$ to higher level. In the
rest of this section, suppose that $G$ is affine. Let $N$ be an element of
$\mathbb{Z}_{>0}\sqcup\\{\infty\\}$. (Whenever we deal with $(N-1)$-PD
stratifications on a $G$-bundle for $N\in\mathbb{Z}_{>0}$, we always suppose
that the scheme $S$ is equipped with an $(N-1)$-PD ideal
$(\mathfrak{a},\mathfrak{b},\gamma)$, as in § 7.4, such that $\gamma$ extends
to $X$.)

######  Definition 4.5.1.

  * (i)

An indigenous $(G,H)$-bundle of level $N$ on $X/S$ is a pair

(260)
$\displaystyle\mathcal{E}^{\spadesuit}_{N}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$

consisting of an $H$-bundle $\mathcal{E}_{H}$ on $X$ and an $(N-1)$-PD
stratification $\varepsilon^{\blacklozenge}$ (cf. Definition 7.5.1) on
$\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$ such that if $\nabla$ denotes
the flat $S$-connection on $\mathcal{E}_{G}$ induced by the $0$-PD
stratification $\varepsilon^{\blacklozenge}|^{\langle 0\rangle}$, i.e.,
$\nabla:=\nabla_{\varepsilon^{\blacklozenge}|^{\langle 0\rangle}}$ (cf. (513)
and (525)), then $(\mathcal{E}_{H},\nabla)$ forms an indigenous $(G,H)$-bundle
on $X/S$.

  * (ii)

Let
$\mathcal{E}^{\spadesuit}_{N}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$
and
$\mathcal{E}^{\prime\spadesuit}_{N}:=(\mathcal{E}^{\prime}_{H},\varepsilon^{\prime\blacklozenge})$
be indigenous $(G,H)$-bundles of level $N$ on $X/S$. An isomorphism of
indigenous $(G,H)$-bundles of level $N$ from $\mathcal{E}^{\spadesuit}_{N}$ to
$\mathcal{E}^{\prime\spadesuit}_{N}$ is an isomorphism
$\mathcal{E}_{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}^{\prime}_{H}$
of $H$-bundles such that the induced isomorphism of $G$-bundles
$\mathcal{E}_{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}^{\prime}_{G}$
is compatible with the respective $(N-1)$-PD stratifications, i.e.,
$\varepsilon^{\blacklozenge}$ and $\varepsilon^{\prime\blacklozenge}$.

Given an indigenous $(G,H)$-bundle $\mathcal{E}_{N}^{\blacklozenge}$ of level
$N$, we can define, in an obvious way, the pull-back of
$\mathcal{E}_{N}^{\blacklozenge}$ to an étale $X$-scheme, as well as with the
base-change of $\mathcal{E}_{N}^{\blacklozenge}$ to an $S$-scheme. (But we
leave the details to the reader.)

######  Definition 4.5.2.

Let
$\mathcal{E}^{\spadesuit}_{N}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$
be an indigenous $(G,H)$-bundle of level $N$. If $N\in\mathbb{Z}_{>0}$ (i.e.,
$N\neq\infty$), then we shall say that $\mathcal{E}^{\spadesuit}_{N}$ is
dormant if the $p$-$(N-1)$-curvature
${{}^{p}}\psi_{\varepsilon^{\blacklozenge}}$ of $\varepsilon^{\blacklozenge}$
(cf. Definition 7.7.1) vanishes identically. Also, if $N=\infty$, then we
shall refer to any such $\mathcal{E}^{\spadesuit}_{N}$ as being dormant, for
convenience.

Denote by

(261) $\displaystyle\mathrm{Ind}_{(G,H),X/S,N}^{{}^{\mathrm{Zzz...}}}$

the set of isomorphism classes of dormant indigenous $(G,H)$-bundles of level
$N$ on $X/S$. It follows from Proposition 7.8.1 that the notion of dormant
indigenous $(G,H)$-bundle is equivalent to that of level $1$. That is to say,
we have a natural identification

(262)
$\displaystyle\mathrm{Ind}_{(G,H),X/S,1}^{{}^{\mathrm{Zzz...}}}=\mathrm{Ind}_{(G,H),X/S}^{{}^{\mathrm{Zzz...}}}.$

Let $N^{\prime}$ be another element of $\mathbb{Z}_{>0}\sqcup\\{\infty\\}$
with $N\leq N^{\prime}$ and and
$\mathcal{E}^{\spadesuit}_{N}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$
a(n) (dormant) indigenous $(G,H)$-bundle of level $N^{\prime}$. Then, the pair

(263) $\displaystyle\mathcal{E}^{\spadesuit}_{N}|^{\langle
N\rangle}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge}|^{(N-1)})$

(cf. (513)) specifies a(n) (dormant) indigenous $(G,H)$-bundle of level $N$,
which is called the $N$-th truncation of $\mathcal{E}^{\spadesuit}$. The
resulting assignments
$\mathcal{E}^{\spadesuit}\mapsto\mathcal{E}^{\spadesuit}|^{\langle N\rangle}$
for various pairs $(N,N^{\prime})$ with $N\leq N^{\prime}$ give a projective
system

(264)
$\displaystyle\mathrm{Ind}_{(G,H),X/S,\infty}^{{}^{\mathrm{Zzz...}}}\rightarrow\cdots\rightarrow\mathrm{Ind}_{(G,H),X/S,N}^{{}^{\mathrm{Zzz...}}}\rightarrow\cdots\rightarrow\mathrm{Ind}_{(G,H),X/S,2}^{{}^{\mathrm{Zzz...}}}\rightarrow\mathrm{Ind}_{(G,H),X/S,1}^{{}^{\mathrm{Zzz...}}}.$

######  Example 4.5.3 (Standard example).

Let $N\in\mathbb{Z}_{>0}$. Then, the trivial $G$-bundle $P_{S}^{(N)}\times G$
on $P_{S}^{(N)}$ determines an $(N-1)$-PD stratification
$\varepsilon^{\mathrm{triv},\blacklozenge}_{N}$ on $P_{S}\times G$ via (521).
Then, the pair

(265)
$\displaystyle\mathcal{E}_{N}^{\mathrm{triv},\spadesuit}:=(\mathcal{E}_{H}^{\mathrm{triv}},\varepsilon^{\mathrm{triv},\blacklozenge}_{N})$

(cf. (225)) forms a dormant indigenous $(G,P)$-bundle of level $N$ on
$P_{S}/S$. Since the formation of
$\varepsilon^{\mathrm{triv},\blacklozenge}_{N}$ commutes with truncation to
lower levels, the collection
$\\{\varepsilon^{\mathrm{triv},\blacklozenge}_{N}\\}_{N}$ specifies an
$\infty$-PD stratification
$\varepsilon_{\infty}^{\mathrm{triv},\blacklozenge}$. Thus, we obtain a
dormant indigenous $(G,H)$-bundle
$\mathcal{E}_{\infty}^{\mathrm{triv},\spadesuit}:=(\mathcal{E}_{H}^{\mathrm{triv}},\varepsilon^{\mathrm{triv},\blacklozenge}_{\infty})$
of level $\infty$. It is immediately verified that
$\mathcal{E}_{N}^{\mathrm{triv},\spadesuit}$ corresponds to the standard
example of $F^{N}$-Cartan geometry $\mathcal{E}^{\mathrm{triv},\clubsuit}_{N}$
(cf. (202)) via the bijection asserted in Proposition 4.5.4 later.

The remaining portions of Theorem B are completed by the following Proposition
4.5.4 and Corollary 4.5.5. (The map
$\zeta_{N}^{\heartsuit\Rightarrow\spadesuit}$, which is one of the required
bijections in that theorem, is defined as the composite of
$\zeta_{N}^{\clubsuit\Rightarrow\spadesuit}$ below and the bijection
$\zeta^{\heartsuit\Rightarrow\clubsuit}_{N}$ asserted in Propositions 3.10.5
and 3.11.2.)

######  Proposition 4.5.4.

(Recall that $G$ is assumed to be affine.) For any
$N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$, there exists a canonical bijection of
sets

(266)
$\displaystyle\zeta_{N}^{\clubsuit\Rightarrow\spadesuit}:F^{N}\text{-}\mathrm{Car}_{(G,H),X/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ind}_{(G,H),X/S,N}^{{}^{\mathrm{Zzz...}}},$

and the formation of this bijection commutes with truncation to lower levels,
pull-back to étale $X$-schemes, and base-change to $S$-schemes.

###### Proof.

If $N\in\mathbb{Z}_{>0}$ and
$\mathcal{E}_{N}^{\clubsuit}:=(\mathcal{E}_{H},\omega,\mathcal{G},\kappa)$ is
an $F^{N}$-Cartan geometry with model $(G,H)$, then the pair
$\mathcal{E}_{N}^{\clubsuit\Rightarrow\spadesuit}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$,
where $\varepsilon^{\blacklozenge}$ denotes the $(N-1)$-PD stratification on
$\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$ corresponding to $\mathcal{G}$
via (521), forms a dormant indigenous $(G,H)$-bundle of level $N$; the
resulting assignment
$\mathcal{E}_{N}^{\clubsuit}\mapsto\mathcal{E}_{N}^{\clubsuit\Rightarrow\spadesuit}$
defines the required bijection for $N\in\mathbb{Z}_{>0}$ because of
Proposition 4.4.2 and Proposition 7.7.3, (i). Moreover, since this bijection
commutes with truncation to lower levels (cf. Proposition 7.7.3, (ii)), the
case of $N=\infty$ follows from Corollary 7.7.5. ∎

######  Corollary 4.5.5.

Suppose that $G$ is affine, $S=\mathrm{Spec}(k)$, and $X$ is proper over $k$.
Then, the composite

(267)
$\displaystyle\zeta_{\infty}^{\heartsuit\Rightarrow\spadesuit}:=\zeta_{\infty}^{\clubsuit\Rightarrow\spadesuit}\circ\zeta_{\infty}^{\heartsuit\Rightarrow\clubsuit}:F^{\infty}\text{-}\mathrm{Ehr}_{(G,P),X/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ind}_{(G,H),X/S,\infty}^{{}^{\mathrm{Zzz...}}}$

(cf. (209) for the definition of
$\zeta_{\infty}^{\heartsuit\Rightarrow\clubsuit}$) is bijective. In
particular, the map of sets

(268)
$\displaystyle\mathrm{Ind}_{(G,H),X/S,\infty}^{{}^{\mathrm{Zzz...}}}\rightarrow\varprojlim_{N\geq\in\mathbb{Z}_{>0}}\mathrm{Ind}_{(G,H),X/S,N}^{{}^{\mathrm{Zzz...}}}$

induced by (264) is bijective.

###### Proof.

The assertion follows from Propositions 3.11.2 and 4.5.4. ∎

######  Example 4.5.6.

We shall briefly recall the study of [53], in which indigenous $(G,H)$-bundles
for $(G,H)=(\mathrm{PGL}_{n+1},\mathrm{PGL}_{n+1}^{\circledcirc})$ and
$(\mathrm{PGL}_{n+1},\mathrm{PGL}_{n+1}^{\circledcirc})$ are discussed in
terms of vector bundles. Suppose that $X$ is a smooth projective scheme over
$S=\mathrm{Spec}(k)$ of dimension $n$, and denote by $\omega_{X}$ the
canonical line bundle on $X$. Then, we obtain the composite isomorphism

(269)
$\displaystyle\mathrm{det}(\mathcal{D}_{X/k,1}^{(N-1)})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{det}(\mathcal{D}_{X/k,0}^{(N-1)})\otimes_{\mathcal{O}_{X}}\mathrm{det}(\mathcal{D}_{X/k,1}^{(N-1)}/\mathcal{D}_{X/k,0}^{(N-1)})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{det}(\mathcal{T}_{X/k})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\omega_{X}^{\vee}$

(cf. (476) for the definition of $\mathcal{D}_{X/k,(-)}^{(N-1)}$).

  * (i)

First, let us consider the case where
$(G,H)=(\mathrm{PGL}_{n+1},\mathrm{PGL}_{n+1}^{\circledcirc})$ (cf. Example
2.3.3). Suppose further that $p\nmid(n+1)$ and $X$ admits an $F^{N}$-theta
characteristic (cf. [53, § 3, Definition 3.5.1]), i.e., a pair
$(\varTheta,\nabla^{0})$ consisting of a line bundle $\varTheta$ on $X$ and a
$\mathcal{D}_{X/k}^{(N-1)}$-module structure $\nabla^{0}$ on
$\omega_{X}^{\vee}\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes(n+1)}$ with
vanishing $p$-$(N-1)$-curvature (cf. (516)). According to [53, § 3, Theorem
3.6.4, (i)], each element of
$\mathrm{Ind}_{(\mathrm{PGL}_{n+1},\mathrm{PGL}_{n+1}^{\circledcirc}),X/k,N}^{{}^{\mathrm{Zzz...}}}$
may be expressed as a $\mathcal{D}_{X/k}^{(N-1)}$-module structure

(270)
$\displaystyle\nabla:\mathcal{D}_{X/k}^{(N-1)}\otimes_{\mathcal{O}_{X}}(\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta)\rightarrow\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta$

on the rank $(n+1)$ vector bundle
$\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta$ with vanishing
$p$-$(N-1)$-curvature satisfying the following conditions:

    * (a)

The following composite defines an automorphism of
$\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta$ (cf. [53, § 2,
Proposition 2.3.3]):

(271)
$\displaystyle\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta\hookrightarrow\mathcal{D}_{X/k}^{(N-1)}\otimes_{\mathcal{O}_{X}}(\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta)\xrightarrow{\nabla}\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta,$

where the first arrow arises from the natural inclusions
$\mathcal{D}_{X/k,1}^{(N-1)}\hookrightarrow\mathcal{D}_{X/k}^{(N-1)}$ and
$\varTheta\left(=\mathcal{D}_{X/k,0}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta\right)\hookrightarrow\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta$.

    * (b)

The determinant of $\nabla$ corresponds to $\nabla^{0}$ via the composite
isomorphism

(272)
$\displaystyle\mathrm{det}(\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{det}(\mathcal{D}_{X/k,1}^{(N-1)})\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes(n+1)}\xrightarrow{(\ref{Ewq1})\otimes\mathrm{id}}\omega_{X}^{\vee}\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes(n+1)}.$

If $X$ is a curve, then we can take $\varTheta$ as a theta characteristic in
the usual sense and $\nabla^{0}$ as the trivial connection
$\nabla_{0}\left(=d\right)$ under a fixed identification
$\omega_{X}^{\vee}\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes
2}=\mathcal{O}_{X}$. In this situation, a connection $\nabla$ as above is
nothing but what we call a dormant $\mathrm{SL}_{2}$-oper.

  * (ii)

Next, let us recall the case of
$(G,H)=(\mathrm{Aff}_{n},\mathrm{Aff}_{n}^{\circledcirc})$. According to [53,
§ 3, Theorem 3.6.4, (i)], each element of
$\mathrm{Ind}_{(\mathrm{Aff}_{n},\mathrm{Aff}_{n}^{\circledcirc}),X/k,N}^{{}^{\mathrm{Zzz...}}}$
may be expressed as a pair

(273) $\displaystyle(\nabla,\delta)$

consisting of a $\mathcal{D}_{X/k}^{(N-1)}$-module structure $\nabla$ on the
vector bundle $\mathcal{D}_{X/k,1}^{(N-1)}$ with vanishing
$p$-$(N-1)$-curvature and a left inverse morphism
$\delta:\mathcal{D}_{X/k}^{(N-1)}\twoheadrightarrow\mathcal{O}_{X}$ of the
natural inclusion $\mathcal{O}_{X}\hookrightarrow\mathcal{D}_{X/k,1}^{(N-1)}$
satisfying the following conditions:

    * (c)

The following composite defines an automorphism of
$\mathcal{D}_{X/k,1}^{(N-1)}$:

(274)
$\displaystyle\mathcal{D}_{X/k,1}^{(N-1)}\left(=\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}\right)\hookrightarrow\mathcal{D}_{X/k}^{(N-1)}\otimes_{\mathcal{O}_{X}}\mathcal{D}_{X/k,1}^{(N-1)}\xrightarrow{\nabla}\mathcal{D}_{X/k,1}^{(N-1)},$

where the first arrow arises from the natural inclusions
$\mathcal{D}_{X/k,1}^{(N-1)}\hookrightarrow\mathcal{D}_{X/k}^{(N-1)}$ and
$\mathcal{O}_{X}\left(=\mathcal{D}_{X/k,0}^{(N-1)}\right)\hookrightarrow\mathcal{D}_{X/k,1}^{(N-1)}$.

    * (d)

The subbundle $\mathrm{Ker}(\delta)$ of $\mathcal{D}_{X/k,1}^{(N-1)}$ is
closed under the $\mathcal{D}_{X/k}^{(N-1)}$-action $\nabla$ and the
$\mathcal{D}_{X/k}^{(N-1)}$-module structure on $\mathcal{O}_{X}$ induced from
$\nabla$ via $\delta$ coincides with the trivial one.

Now, suppose that $p\nmid(n+1)$ and there exists an $F^{N}$-theta
characteristic $(\varTheta,\nabla^{0})$ on $X$. Let us take a pair
$(\nabla,\delta)$ as above. Since
$\mathrm{det}((\mathcal{D}_{X/k,1}^{(N-1)})^{\vee})\cong\omega_{X}$ (cf.
(269)), we have the $\mathcal{D}_{X/k}^{(N-1)}$-module structure
$\nabla_{\omega_{X}}$ on $\omega_{X}$ induced by (the determinant of the dual
of) $\nabla$. Then, it follows from [53, § 3, Lemma 3.6.3] that there exists
uniquely a $\mathcal{D}_{X/k}^{(N-1)}$-module structure $\nabla_{\varTheta}$
on $\varTheta$ such that its $(n+1)$-st tensor product coincides, via
$\varTheta^{\otimes(n+1)}\cong\omega_{X}\otimes_{\mathcal{O}_{X}}(\omega_{X}^{\vee}\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes(n+1)})$,
with the tensor product of $\nabla_{\omega_{X}}$ and $\nabla^{0}$ (cf. [53, §
2, (70)]). Then, the tensor product of $\nabla$ and $\nabla_{\varTheta}$
defines a $\mathcal{D}_{X/k}^{(N-1)}$-module structure $\nabla^{\prime}$ on
the vector bundle
$\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\varTheta$, satisfying
the two conditions (a) and (b) in (i). One may verify that the resulting
assignment $\nabla\mapsto\nabla^{\prime}$ coincides with (55) under the
identification
$F^{N}\text{-}\mathrm{Ehr}_{(G,H),X/S}=\mathrm{Ind}_{(G,H),X/S,N}^{{}^{\mathrm{Zzz...}}}$
given by the bijection
$\zeta_{N}^{\heartsuit\Rightarrow\spadesuit}:=\zeta_{N}^{\clubsuit\Rightarrow\spadesuit}\circ\zeta_{N}^{\heartsuit\Rightarrow\clubsuit}$.

Let us keep the assumption in § 7.4 and take an $F^{N}$-$(G,P)$-structure
$\mathcal{S}^{\heartsuit}$ on $X/S$ with $N\in\mathbb{Z}_{>0}$. Then, we
obtain the indigenous $(G,H)$-bundle
$\mathcal{E}^{\heartsuit}_{N}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$
of level $N$ corresponding to $\mathcal{S}^{\heartsuit}$ via
$\zeta_{N}^{\heartsuit\Rightarrow\spadesuit}$. Then, the $(N-1)$-PD stratified
$G$-bundle $(\mathcal{E}_{G},\varepsilon^{\blacklozenge})$, where
$\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$, detemines an $(N-1)$-crystal of
$G$-bundles $\mathcal{E}^{\diamondsuit}$ on $X/S$ (cf. Proposition 7.6.2).

######  Definition 4.5.7.

We shall refer to $\mathcal{E}^{\diamondsuit}$ as the monodromy crystal
associated with $\mathcal{S}^{\heartsuit}$.

###  4.6. Basic properties on $F^{\infty}$-$(G,P)$-structures

By using the result of Theorem B, we prove several properties concerning
smooth varieties admitting an $F^{\infty}$-$(G,P)$-structure.

Recall that a rational curve in a $k$-scheme $X$ is a non-constant
$k$-morphism $f:\mathbb{P}^{1}\rightarrow X$. We shall say that a rational
curve $f:\mathbb{P}^{1}\rightarrow X$ is free if $f^{*}(\mathcal{T}_{X})$ is
spanned by its global sections. Also, a $k$-scheme is called convex if every
rational curve in it is free.

######  Proposition 4.6.1.

Suppose that $G$ is affine. Then, any proper smooth $k$-scheme admitting an
$F^{\infty}$-$(G,P)$-structure is convex.

###### Proof.

Let $X$ be a proper smooth $k$-scheme admitting an
$F^{\infty}$-$(G,P)$-structure $\mathcal{S}^{\heartsuit}$. Denote by
$\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\varepsilon^{\blacklozenge})$ the
indigenous $(G,H)$-bundle of level $\infty$ corresponding to
$\mathcal{S}^{\heartsuit}$ via
$\zeta_{\infty}^{\heartsuit\Rightarrow\spadesuit}$ asserted in Theorem B and
by $\mathcal{G}_{\infty}:=\\{(\mathcal{G}_{l},\varpi_{l})\\}_{l}$ the
$F$-divided $G$-bundle (cf. Definition 7.7.4) corresponding to the $\infty$-PD
stratified $G$-bundle $(\mathcal{E}_{G},\varepsilon^{\blacklozenge})$ via
(524). Now, let us take an arbitrary rational curve
$f:\mathbb{P}^{1}\rightarrow X$ in $X$. By pulling-back $\mathcal{G}_{\infty}$
by $f$, we obtain an $F$-divided $G$-bundle $f^{*}(\mathcal{G}_{\infty})$ on
$\mathbb{P}^{1}$. Recall from [18, § 2, Theorem 2.2] that there are no
nontrivial $F$-divided vector bundles on $\mathbb{P}^{1}$. Hence, for each
finite-dimensional representation $V$ of $G$ over $k$, the associated
$F$-divided vector bundle $f^{*}(\mathcal{G}_{\infty})\times^{G}V$ may be
trivialized. By [6, § 3, Lemma 3.5, (ii)], we see that the $F$-divided
$G$-bundle $f^{*}(\mathcal{G}_{\infty})$ itself may be trivialized. Let us fix
an isomorphism
$f^{*}(\mathcal{E}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{P}^{1}\times
G$ preserving the $F$-divided structures, where $\mathbb{P}^{1}\times G$ is
equipped with the trivial $F$-divided structure. It gives an $H$-equivariant
composite

(275) $\displaystyle f^{*}(\mathcal{E}_{H})\hookrightarrow
f^{*}(\mathcal{E}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{P}^{1}\times
G\xrightarrow{\mathrm{pr}_{2}}G.$

This morphism induces, via taking the quotients with respect to the
$H$-actions, a morphism
$h:\mathbb{P}^{1}\left(:=f^{*}(\mathcal{E}_{H})/H\right)\rightarrow
P\left(:=G/H\right)$. Since the commutative diagram

(280)

is cartesian, we obtain an isomorphism of $H$-bundles
$f^{*}(\mathcal{E}_{H})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}h^{*}(G)\left(=\mathbb{P}^{1}\times_{h,P,\pi_{G}}G\right)$.
Thus, we have

(281) $\displaystyle f^{*}(\mathcal{T}_{X})$ $\displaystyle\cong
f^{*}(\mathfrak{g}_{\mathcal{E}_{G}}/\mathfrak{h}_{\mathcal{E}_{H}})$
$\displaystyle\cong
f^{*}(\mathcal{E}_{H}\times^{H}(\mathfrak{g}/\mathfrak{h}))$
$\displaystyle\cong
f^{*}(\mathcal{E}_{H})\times^{H}(\mathfrak{g}/\mathfrak{h})$
$\displaystyle\cong h^{*}(G)\times^{H}(\mathfrak{g}/\mathfrak{h})$
$\displaystyle\cong
h^{*}(\mathcal{E}_{H}^{\mathrm{triv}}\times^{H}(\mathfrak{g}/\mathfrak{h}))$
$\displaystyle\cong h^{*}(\mathcal{T}_{P}),$

where the first and last “$\cong$” follow from the Kodaira-Spencer maps
$\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$ and
$\mathrm{KS}_{(\mathcal{E}_{H}^{\mathrm{triv}},\nabla_{0})}$ (cf. Example
4.1.6) respectively. Observe here that $\mathcal{T}_{P}$, as well as its pull-
back $h^{*}(\mathcal{T}_{P})$, is spanned by the global sections because the
following composite is surjective:

(282)
$\displaystyle\mathcal{O}_{P}\otimes\mathfrak{g}\left(\cong\pi_{G*}(\mathcal{O}_{G}\otimes\mathfrak{g})^{H}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\pi_{G*}(\mathcal{T}_{G})^{H}\xrightarrow{d_{\mathcal{E}_{H}^{\mathrm{triv}}}}\mathcal{T}_{P},$

where the first arrow arises from $\omega_{G}^{\triangleright}$. Hence, by
(281), $f^{*}(\mathcal{T}_{X})$ turns out to be spanned by the global
sections, i.e., $f$ is free. This completes the proof of the assertion. ∎

Next, a connected $k$-scheme $X$ is called simply connected if the étale
fundamental group of $X$ is trivial. Then, we obtain the following assertion,
generalizing [53, Theorem C, (ii)].

######  Proposition 4.6.2.

Let $X$ be a simply connected, projective, and smooth scheme over $k$. Suppose
that $G$ is affine. Then, the following assertions hold:

  * (i)

Suppose that $X/k$ admits an $F^{\infty}$-$(G,P)$-structure. Then, there
exists a finite étale covering $X\rightarrow P$ over $k$. In particular, if,
moreover, $P$ is simply connected, then $X$ is isomorphic to $P$.

  * (ii)

Suppose that $P$ is affine and not isomorphic to $\mathrm{Spec}(k)$. Then,
there are no $F^{\infty}$-$(G,P)$-structures on $X/k$.

###### Proof.

We shall consider the former assertion of (i). Let
$\mathcal{S}^{\heartsuit}:=\\{\mathcal{S}_{N}^{\heartsuit}\\}_{N}$ be an
$F^{\infty}$-$(G,P)$-structure on $X/k$. Denote by
$\mathcal{G}_{\infty}:=\\{(\mathcal{G}_{l},\varpi_{l})\\}_{l\in\mathbb{Z}_{\geq
0}}$ the $F$-divided $G$-bundle induced by $\mathcal{S}^{\heartsuit}$, i.e.,
the underlying $F$-divided $G$-bundle of the $F^{\infty}$-Cartan geometry
corresponding to $\mathcal{S}^{\heartsuit}$. Once one chooses a morphism of
algebraic $k$-groups $G\rightarrow\mathrm{GL}_{n}$ ($n\geq 1$),
$\mathcal{G}_{\infty}$ induces an $F$-divided vector bundle
$\mathcal{V}_{\infty}$ via change of structure group by this morphism. But,
according to [14, Theorem 1.1], the simple connectedness of $X$ implies that
there are no nontrivial $F$-divided vector bundles on $X$, so
$\mathcal{V}_{\infty}$ must be trivial. Hence, by [6, § 3.4, Lemma 3.5, (ii)],
the $G$-bundle $\mathcal{G}_{1}$ is trivial, or equivalently, there exists a
global section of $\mathcal{S}_{1}^{\heartsuit}$. (Note that the result in
loc. cit. is asserted under the assumption that $X$ is separably rationally
connected. But, this assumption was not used in its proof.) This global
section specifies an étale $k$-morphism $f:X\rightarrow P$; it is finite since
$X$ is projective. This completes the proof of the former assertion (i).
Moreover, both the latter assertion of (i) and assertion (ii) follow
immediately from this fact. ∎

###  4.7. $F^{N}$-$(\mathbb{G}_{m}^{\times r_{m}}\times\mathbb{G}_{a}^{\times
r_{a}},\\{e\\})$-structures

This subsection is devoted to studying the case where $G$ is the product of
finite copies of $\mathbb{G}_{m}$ and $\mathbb{G}_{a}$ (and $H=\\{e\\}$).

#### 4.7.1.

Let $N$ be an element of $\mathbb{Z}_{>0}$. First, we discuss $(N-1)$-PD
stratified $\mathbb{G}_{m}$-bundles equipped with an $\\{e\\}$-reduction.
Denote by $C$ the Cartier operator
$C:F_{X/S*}(\mathrm{Ker}(\nabla_{0}^{(2)}))\rightarrow\Omega_{X^{(1)}/S}$ (cf.
(444) for the definition of $(-)^{(2)}$). Also, denote by

(283) $\displaystyle\Gamma(X,\Omega_{X/S})^{C-1}$

the submodule of
$\Gamma(X,\mathrm{Ker}(\nabla_{0}^{(2)}))\left(\subseteq\Gamma(X,\Omega_{X/S})\right)$
consisting of closed $1$-forms $\omega$ with
$C(\omega)=F^{*}_{S}(\omega)\left(:=(\mathrm{id}_{X}\times
F_{S})^{*}(\omega)\right)$. Recall from [28, § 7, Corollary 7.1.3] that, for a
$1$-form $\omega$ on $X/S$, the $S$-connection $\nabla_{\omega}:=d+\omega$ is
$p$-flat if and only if $\omega\in\Gamma(X,\Omega_{X/S})^{C-1}$.

Now, let us consider a triple

(284) $\displaystyle\omega^{+}:=(\omega,\mathcal{L},\eta),$

where $\omega$ denotes a $1$-form on $X/S$, $\mathcal{L}$ denotes a line
bundle on $X^{(N)}$, and $\eta$ denotes an isomorphism of flat line bundles
$(\mathcal{O}_{X},\nabla_{\omega})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(F^{(N)}_{X/S}(\mathcal{L}),\nabla^{\mathrm{can}(N)}_{\mathcal{L}})$
(cf. (518)). The $p$-flatness of $\nabla_{\mathcal{L}}^{\mathrm{can}(N)}$
implies that of $\nabla_{\omega}$. Hence, the above discussion shows that
$\omega$ automatically belongs to $\Gamma(X,\Omega_{X/S})^{C-1}$. Given two
such triples $\omega^{+}_{i}:=(\omega_{i},\mathcal{L}_{i},\eta_{i})$
($i=1,2$), we define an isomorphism
$\omega_{1}^{+}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\omega_{2}^{+}$ to
be an isomorphism
$\iota:\mathcal{L}_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}_{2}$
with $F^{(N)}_{X/S}(\iota)\circ\eta_{1}=\eta_{2}$. Since the morphism
$\mathcal{O}_{X^{(N)}}^{\times}\rightarrow
F_{X/S*}^{(N)}(\mathcal{O}_{X}^{\times})$ induced by $F_{X/S}^{(N)}$ is
injective, an isomorphism between $\omega_{1}^{+}$ and $\omega_{2}^{+}$ is, if
it exists, uniquely determined. We shall denote by

(285) $\displaystyle B_{m,X/S}^{(N)}$

the set consisting of isomorphism classes of triples
$(\omega,\mathcal{L},\eta)$ as above. By Proposition 7.7.3, (i), and the last
comment in Example 4.1.3, (i), the set $B_{m,X/S}^{(N)}$ is naturally
identified with the set of isomorphism classes of $(N-1)$-PD stratified
$\mathbb{G}_{m}$-bundles on $X/S$ with vanishing $p$-$(N-1)$-curvature whose
underlying $\mathbb{G}_{m}$-bundle is trivial. If $N=1$, then we obtain the
canonical bijection

(286) $\displaystyle
B_{m,X/S}^{(1)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Coker}\left(\Gamma(X,\mathcal{O}_{X}^{\times})\xrightarrow{u\mapsto
d\mathrm{log}(u)}\Gamma(X,\Omega_{X/S})^{C-1}\right)$

given by $(\omega,\mathcal{L},\eta)\mapsto\omega$.

If $N^{\prime}$ is another positive integer with $N\leq N^{\prime}$ and
$\omega^{+}:=(\omega,\mathcal{L},\eta)\in B_{m,X/S}^{(N^{\prime})}$, then the
$N$-th truncation of $\omega^{+}$ is defined as the element

(287) $\displaystyle\omega^{+}|^{\langle
N\rangle}:=(\omega,F^{(N^{\prime}-N)*}_{X^{(N)}/S}(\mathcal{L}),\eta)$

of $B_{m,X/S}^{(N)}$. Truncations to lower levels gives rise to a projective
system

(288) $\displaystyle\cdots\rightarrow
B_{m,X/S}^{(N)}\rightarrow\cdots\rightarrow B_{m,X/S}^{(2)}\rightarrow
B_{m,X/S}^{(1)}.$

#### 4.7.2.

Next, we shall consider $(N-1)$-PD stratified $\mathbb{G}_{a}$-bundles
equipped with an $\\{e\\}$-reduction. Recall from the discussion in Example
4.1.3 that $\mathbb{G}_{a}$ is identified with the subgroup of
$\mathrm{GL}_{2}$ consisting of matrices of the form $\begin{pmatrix}1&a\\\
0&1\end{pmatrix}$ for $a\in\mathbb{G}_{a}$. Denote by

(289) $\displaystyle\Gamma(X,\Omega_{X/S})^{C}$

the submodule of
$\Gamma(X,\mathrm{Ker}(\nabla_{0}^{(2)}))\left(\subseteq\Gamma(X,\Omega_{X/S})\right)$
consisting of closed $1$-forms $\chi$ with $C(\chi)=0$. Here, let us observe
the following assertion.

######  Lemma 4.7.1.

Let $\chi$ be a closed $1$-form on $X/S$, i.e., an element of
$\Gamma(X,\mathrm{Ker}(\nabla_{0}^{(2)}))$. Then, the $p$-curvature
${{}^{p}}\psi_{\nabla_{\chi}^{\dagger}}$ of $\nabla^{\dagger}_{\chi}$ (cf.
(221)) is given by

(290)
$\displaystyle{{}^{p}}\psi_{\nabla^{\dagger}_{\chi}}(F_{X}^{-1}(\partial))=-\begin{pmatrix}0&F_{X/S}^{-1}(\langle
F_{S}^{*}(\partial),C(\chi)\rangle)\\\ 0&0\end{pmatrix}$

for any local section $\partial\in\mathcal{T}_{X/S}$. In particular,
$\nabla^{\dagger}_{\chi}$ has vanishing $p$-curvature if and only if
$\chi\in\Gamma(X,\Omega_{X/S})^{C}$.

###### Proof.

Let us take a local section $\partial\in\mathcal{T}_{X/S}$. Then, we have the
following sequence of equalities:

(291)
$\displaystyle{{}^{p}}\psi_{\nabla^{\dagger}_{\chi}}(F_{X}^{-1}(\partial))$
$\displaystyle=\nabla(\partial)^{p}-\nabla(\partial^{p})$
$\displaystyle=\left(\partial^{\oplus
2}+\begin{pmatrix}0&\langle\partial,\chi\rangle\\\
0&0\end{pmatrix}\right)^{p}-(\partial^{p})^{\oplus
2}-\begin{pmatrix}0&\langle\partial^{p},\chi\rangle\\\ 0&0\end{pmatrix}$
$\displaystyle=\left((\partial^{p})^{\oplus
2}+\begin{pmatrix}0&\partial^{p-1}(\langle\partial,\chi\rangle)\\\
0&0\end{pmatrix}\right)-(\partial^{p})^{\oplus
2}-\begin{pmatrix}0&\langle\partial^{p},\chi\rangle\\\ 0&0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&\partial^{p-1}(\langle\partial,\chi\rangle)-\langle\partial^{p},\chi\rangle\\\
0&0\end{pmatrix}$ $\displaystyle=-\begin{pmatrix}0&F_{X/S}^{-1}(\langle
F_{S}^{*}(\partial),C(\chi)\rangle)\\\ 0&0\end{pmatrix},$

where the third equality follows from Jacobson’s formula and the last equality
follows from [28, § 7, (7.1.2.6)]. This completes the proof of this lemma. ∎

Now, we shall consider a triple

(292) $\displaystyle\chi^{+}:=(\chi,\mathcal{V},\lambda)$

consisting of a $1$-form $\chi$ on $X/S$, a rank $2$ vector bundle
$\mathcal{V}$ on $X^{(N)}$ equipped with a structure of extension
$0\rightarrow\mathcal{O}_{X^{(N)}}\rightarrow\mathcal{V}\rightarrow\mathcal{O}_{X^{(N)}}\rightarrow
0$, and an isomorphism of flat bundles $\lambda:(\mathcal{O}_{X}^{\oplus
2},\nabla_{\chi}^{\dagger})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(F_{X/S}^{(N)*}(\mathcal{V}),\nabla^{\mathrm{can}(N)}_{\mathcal{V}})$
preserving the extension structures. The $p$-flatness of
$\nabla_{\mathcal{V}}^{\mathrm{can}(N)}$ implies that of
$\nabla_{\omega}^{\dagger}$, so the above lemma shows that $\chi$ belongs to
$\Gamma(X,\Omega_{X/S})^{C}$. An isomorphism between such triples can be
defined naturally and is, if it exists, uniquely determined for the same
reason as the case of $(G,H)=(\mathbb{G}_{m},\\{e\\})$. Denote by

(293) $\displaystyle B_{a,X/S}^{(N)}$

the set of isomorphism classes of triples $(\chi,\mathcal{V},\lambda)$ as
above. By Proposition 7.7.3, (i), and the last comment in Example 4.1.3, (ii),
this set is naturally identified with the set of isomorphism classes of
$(N-1)$-PD stratified $\mathbb{G}_{a}$-bundles on $X/S$ with vanishing
$p$-$(N-1)$-curvature whose underlying $\mathbb{G}_{a}$-bundle is trivial. If
$N=1$, then we obtain the canonical bijection

(294) $\displaystyle
B_{a,X/S}^{(1)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Coker}\left(\Gamma(X,\mathcal{O}_{X})\xrightarrow{u\mapsto
du}\Gamma(X,\Omega_{X/S})\right)$

given by $(\chi,\mathcal{V},\lambda)\mapsto\chi$.

Just as in the case of $B_{m,X/S}^{(N)}$, the $N$-th truncation
$\chi^{+}|^{\langle N\rangle}$ of $\chi^{+}\in B_{a,X/S}^{(N^{\prime})}$ with
$N\leq N^{\prime}$ can be defined. In particular, truncations to various lower
levels yield a projective system consisting of $B_{a,X/S}^{(N)}$’s.

#### 4.7.3.

Finally, we combine the above discussions to study $(N-1)$-PD stratified
bundles whose structure group is the product of finite copies of
$\mathbb{G}_{m}$ and $\mathbb{G}_{a}$. Let $s$ and $t$ be nonnegative integers
with $s+t=n\left(:=\mathrm{dim}(X/S)\right)$. Write $G:=\mathbb{G}_{m}^{\times
s}\times\mathbb{G}_{a}^{\times t}$. The product of sets
$(B_{m,X/S}^{(N)})^{\times s}\times(B_{a,X/S}^{(N)})^{\times t}$ is in
bijection with the set of isomorphism classes of $(N-1)$-PD stratified
$G$-bundles with vanishing $p$-$(N-1)$-curvature equipped with an
$\\{e\\}$-reduction (i.e., whose underlying $G$-bundles are trivial). Denote
by

(295) $\displaystyle B_{X/S}^{(N),s,t}$

the elements of $(B_{m,X/S}^{(N)})^{\times s}\times(B_{a,X/S}^{(N)})^{\times
t}$ represented by collections
$(\omega^{+}_{1},\cdots,\omega_{s}^{+},\chi_{1}^{+},\cdots,\chi_{t}^{+})$,
where $\omega_{i}^{+}:=(\omega_{i},\mathcal{L}_{i},\eta_{i})$ and
$\chi_{j}^{+}:=(\chi_{j},\mathcal{V}_{j},\lambda_{j})$ ($i=1,\cdots,s$ and
$j=1,\cdots,t$), such that the global sections
$\omega_{1},\cdots,\omega_{s},\chi_{1},\cdots,\chi_{t}$ forms a basis of the
$\Gamma(X,\mathcal{O}_{X})$-module $\Gamma(X,\Omega_{X/S})$. Truncations to
lower levels defined before give rise to a projective system consisting of
$B_{X/S}^{(N),s,t}$’s for various $N$. Hence, we obtain the set

(296) $\displaystyle
B_{X/S}^{(\infty),s,t}:=\varprojlim_{N\in\mathbb{Z}_{>0}}B_{X/S}^{(N),s,t}.$

######  Proposition 4.7.2.

Let us keep the above notation.

  * (i)

For any $N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$, there exists a canonical
bijection of sets

(297) $\displaystyle
F^{N}\text{-}\mathrm{Ehr}_{(G,G),X/S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}B_{X/S}^{(N),s,t}.$

In particular, $X/S$ admits no $F^{N}$-$(G,G)$-structures unless
$\Gamma(X,\Omega_{X/S})\cong\Gamma(X,\mathcal{O}_{X})^{\oplus n}$.

  * (ii)

The following two conditions are equivalent to each other:

    * (a)

There exists an $F^{1}$-$(G,G)$-structure on $X/S$;

    * (b)

There exists a basis
$\\{\omega_{1},\cdots,\omega_{s},\chi_{1},\cdots,\chi_{t}\\}$ of
$\Gamma(X,\Omega_{X/S})$ such that each $\omega_{i}$ ($i=1,\cdots,s$) belongs
to $\Gamma(X,\Omega_{X/S})^{C-1}$ and each $\chi_{j}$ ($j=1,\cdots,t$) belongs
to $\Gamma(X,\Omega_{X/S})^{C}$.

Moreover, if $S=\mathrm{Spec}(k)$ and $X$ is proper over $k$, then these
conditions are equivalent to the following condition:

    * (c)

The following equalities hold:

(298) $\displaystyle n=\mathrm{dim}_{k}(\Gamma(X,\Omega_{X/k})),\hskip
8.53581pts=\mathrm{dim}_{\mathbb{F}_{p}}(H^{1}_{\mathrm{fl}}(X,\mu_{p})),\hskip
8.53581ptt=\mathrm{dim}_{k}(H^{1}_{\mathrm{fl}}(X,\alpha_{p})),$

where $H^{1}_{\mathrm{fl}}$ denotes the $1$-st flat cohomology.

###### Proof.

Let us consider assertion (i). The problem is reduced to the case of
$N\in\mathbb{Z}_{>0}$ because the assertion for $N=\infty$ follows immediately
from this case together with (40) and (296). First, we shall construct a map
of sets $F^{N}\text{-}\mathrm{Ehr}_{(G,G),X/S}\rightarrow B_{X/S}^{(N),s,t}$.
Take an $F^{N}$-$(G,G)$-structure $\mathcal{S}^{\heartsuit}$ on $X/S$, and
denote by $\mathcal{E}^{\spadesuit}_{N}$ the dormant indigenous
$(G,\\{e\\})$-bundle of level $N$ corresponding to $\mathcal{S}^{\heartsuit}$
(cf. Theorem B). As mentioned above, the data $\mathcal{E}^{\spadesuit}_{N}$
gives a collection of data

(299)
$\displaystyle(\omega_{1}^{+},\cdots,\omega_{s}^{+},\chi_{1}^{+},\cdots,\chi^{+}_{t}),$

where each $\omega^{+}_{i}:=(\omega_{i},\mathcal{L}_{i},\eta_{i})$
($i=1,\cdots,s$) denotes an element of $B_{m,X/S}^{(N)}$ and each
$\chi^{+}_{j}:=(\chi_{j},\mathcal{V}_{j},\lambda_{j})$ ($j=1,\cdots,t$)
denotes an element of $B_{a,X/S}^{(N)}$. Under the natural identifications
$\mathrm{Lie}(G)\cong\mathrm{Lie}(\mathbb{G}_{m})^{\oplus
s}\oplus\mathrm{Lie}(\mathbb{G}_{a})^{\oplus t}\cong k^{\oplus(s+t)}$, the
vector bundle $\mathrm{Lie}(G)_{X\times
G}/\mathrm{Lie}(\\{e\\})_{X\times\\{e\\}}$ may be identified with
$\mathcal{O}_{X}^{\oplus(s+t)}$. In particular, if $\nabla$ denotes the
$S$-connection on the trivial $G$-bundle defining
$\mathcal{E}^{\spadesuit}_{N}|^{\langle 1\rangle}$ (cf. (262)), then the
Kodaira-Spencer map $\mathrm{KS}_{(X\times\\{e\\},\nabla)}$ may be regarded as
an $\mathcal{O}_{X}$-linear morphism
$\mathcal{T}_{X/S}\rightarrow\mathcal{O}_{X}^{\oplus(s+t)}$, or equivalently,
an element of $\Gamma(X,\Omega_{X/S})^{\oplus(s+t)}$ (cf. Remark 4.1.2). By
taking account of the discussions in Example 4.1.3, (i) and (ii), we see that
$\mathrm{KS}_{(X\times\\{e\\},\nabla)}$ coincides with
$(\omega_{1},\cdots,\omega_{s},\chi_{1},\cdots,\chi_{t})$. Since
$\mathrm{KS}_{(X\times\\{e\\},\nabla)}$ is an isomorphism, the global sections
$\omega_{1},\cdots,\omega_{s},\chi_{1},\cdots,\chi_{t}$ form a basis of
$\Gamma(X,\Omega_{X/S})$. Thus, the collection of data (299) specifies an
element of $B_{X/S}^{(N),s,t}$. The resulting assignment
$\mathcal{S}^{\heartsuit}\mapsto(\omega_{1}^{+},\cdots,\omega_{s}^{+},\chi_{1}^{+},\cdots,\chi^{+}_{t})$
defines a map of sets $F^{N}\text{-}\mathrm{Ehr}_{(G,G),X/S}\rightarrow
B_{X/S}^{(N),s,t}$. Moreover, by the above discussion, this map is verified to
be bijective, so this completes the proof of assertion (i).

Next, let us consider assertion (ii). The equivalence (a) $\Leftrightarrow$
(b) follows immediately from assertion (i) together with (286) and (294). The
equivalence (b) $\Leftrightarrow$ (c) under the assumption mentioned in the
statement follows from the fact that $\Gamma(X,\Omega_{X/k})^{C-1}$,
$\Gamma(X,\Omega_{X/k})^{C}$ are naturally isomorphic to the flat cohomology
groups $H^{1}_{\mathrm{fl}}(X,\mu_{p})$, $H^{1}_{\mathrm{fl}}(X,\alpha_{p})$
respectively (cf. [35, Chap. III, § 4, Proposition 4.14]). (Note that the
natural $k$-linear morphism
$k\otimes_{\mathbb{F}_{p}}\Gamma(X,\Omega_{X/k})^{C-1}\rightarrow\Gamma(X,\Omega_{X/k})$
is verified to be injective.) This completes the proof of the assertion. ∎

In the following examples, suppose that $N$ is a positive integer.

######  Example 4.7.3.

Let us consider $F^{N}$-$(\mathbb{G}_{m},\mathbb{G}_{m})$-structures on the
$k$-curve $\mathbb{G}_{m}:=\mathrm{Spec}(k[T^{\pm 1}])$. For each
$u\in\Gamma(\mathbb{G}_{m},\mathcal{O}^{\times}_{\mathbb{G}_{m}})\left(=\\{a\cdot
T^{n}\,|\,n\in\mathbb{Z},a\in k^{\times}\\}\right)$, we shall write
$\mathrm{mult}_{u}$ for the automorphism of $\mathcal{O}_{\mathbb{G}_{m}}$
given by multiplication by $u$. The pull-back
$\mathrm{mult}_{u}^{*}(\nabla_{0})$ of the trivial connection $\nabla_{0}$ on
$\mathcal{O}_{\mathbb{G}_{m}}$ via $\mathrm{mult}_{u}$ is the $k$-connection
expressed as $\mathrm{mult}_{u}^{*}(\nabla_{0})=\nabla_{d\mathrm{log}(u)}$,
where $d\mathrm{log}(u):=u^{-1}\cdot du$. Hence, since any line bundle on
$\mathbb{G}_{m}^{(N)}$ is trivial, each element of
$B_{m,\mathbb{G}_{m}/k}^{(N)}$ may be represented by
$(d\mathrm{log}(u),\mathcal{O}_{\mathbb{G}_{m}^{(N)}},\mathrm{mult}_{u})$ for
some $u\in\Gamma(\mathbb{G}_{m},\mathcal{O}^{\times}_{\mathbb{G}_{m}})$. Such
a triple belongs to $B_{\mathbb{G}_{m}/k}^{(N),1,0}\left(\subseteq
B_{m,\mathbb{G}_{m}/k}^{(N)}\right)$ if and only if $d\mathrm{log}(u)$ is
nowhere vanishing, i.e., $u\in\\{a\cdot T^{n}\,|\,n\in\mathbb{Z},(p,n)=1,a\in
k^{\times}\\}$. Also, an isomorphism from
$(d\mathrm{log}(u),\mathcal{O}_{\mathbb{G}_{m}^{(N)}},\mathrm{mult}_{u})$ to
$(d\mathrm{log}(v),\mathcal{O}_{\mathbb{G}_{m}^{(N)}},\mathrm{mult}_{v})$ is
given by the automorphism of $\mathcal{O}_{\mathbb{G}_{m}^{(N)}}$ defined as
multiplication by an element
$w\in\Gamma(\mathbb{G}_{m}^{(N)},\mathcal{O}_{\mathbb{G}_{m}^{(N)}}^{\times})$
$\left(=\\{a\cdot T^{n\cdot p^{N}}\,|\,n\in\mathbb{Z},a\in
k^{\times}\\}\right)$ satisfying $u=w\cdot v$. This implies that each
isomorphism class in $B_{\mathbb{G}_{m}/k}^{(N),1,0}$ may be represented by a
triple
$(d\mathrm{log}(u),\mathcal{O}_{\mathbb{G}_{m}^{(N)}},\mathrm{mult}_{u})$ for
$u=T^{m}$ with $1\leq m\leq p^{N}-1$ and $(p,m)=1$. Consequently, the
assignment
$m\mapsto(d\mathrm{log}(T^{m}),\mathcal{O}_{\mathbb{G}_{m}^{(N)}},\mathrm{mult}_{T^{m}})$
defines a canonical bijection

(300) $\displaystyle\left\\{m\in\mathbb{Z}\,|\,1\leq m\leq
p^{N}-1,(p,m)=1\right\\}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}B_{\mathbb{G}_{m}/k}^{(N),1,0}.$

In particular, $\mathbb{G}_{m}/k$ admits an
$F^{N}$-$(\mathbb{G}_{m},\mathbb{G}_{m})$-structure for any $N$.

######  Example 4.7.4.

Next, we shall consider $F^{N}$-$(\mathbb{G}_{a},\mathbb{G}_{a})$-structures
on the affine line $\mathbb{A}^{1}:=\mathrm{Spec}(k[T])$. For any
$u\in\Gamma(\mathbb{A}^{1},\mathcal{O}_{\mathbb{A}^{1}})$, we shall write
$\mathrm{add}_{u}$ for the automorphism of
$\mathcal{O}_{\mathbb{A}^{1}}^{\oplus 2}$ given by multiplication by
$\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}$. The pull-back
$\mathrm{add}^{*}_{u}(\nabla_{0})$ of the trivial connection $\nabla_{0}$ on
$\mathcal{O}_{\mathbb{A}^{1}}$ via $\mathrm{add}_{u}$ is the $k$-connection
expressed as $\mathrm{add}_{u}^{*}(\nabla_{0})=\nabla_{du}^{\dagger}$. Hence,
since any rank $2$ vector bundle on $\mathbb{A}^{1(N)}$ is trivial, each
element of $B_{a,\mathbb{A}^{1}/k}^{(N)}$ may be represented by
$(du,\mathcal{O}^{\oplus 2}_{\mathbb{A}^{1(N)}},\mathrm{add}_{u})$ for
$u\in\Gamma(\mathbb{A}^{1},\mathcal{O}_{\mathbb{A}^{1}})$. Such a triple
belongs to $B_{\mathbb{A}^{1}/k}^{(N),0,1}\left(\subseteq
B_{a,\mathbb{A}^{1}/k}^{(N)}\right)$ if and only if $du$ is nowhere vanishing,
i.e., $u\in\left\\{a\cdot T+f(T)\,|\,a\in k^{\times},f(T)\in
k[T^{p}]\right\\}$. Also, an isomorphism from $(du,\mathcal{O}^{\oplus
2}_{\mathbb{A}^{1(N)}},\mathrm{add}_{u})$ to $(dv,\mathcal{O}^{\oplus
2}_{\mathbb{A}^{1(N)}},\mathrm{add}_{v})$ is given by the automorphism of
$\mathcal{O}_{\mathbb{A}^{1(N)}}^{\oplus 2}$ defined as multiplication by
$\begin{pmatrix}1&w\\\ 0&1\end{pmatrix}$ for some
$w\in\Gamma(\mathbb{A}^{1(N)},\mathcal{O}_{\mathbb{A}^{1(N)}})\left(=k[T^{p^{N}}]\right)$
satisfying $u=v+w$. This implies that each isomorphism class in
$B_{\mathbb{A}^{1}/k}^{(N),0,1}$ may be represented by a triple
$(du,\mathcal{O}^{\oplus 2}_{\mathbb{A}^{1(N)}},\mathrm{add}_{u})$ for
$u=a\cdot T+\sum_{i\geq 0}a_{i}\cdot T^{i}\in k[T]$ such that $a\in
k^{\times}$ and $a_{i}\neq 0$ only if $i\in p\mathbb{Z}\setminus
p^{N}\mathbb{Z}$. Consequently, the assignment
$u\mapsto(du,\mathcal{O}_{\mathbb{A}^{1(N)}},\mathrm{add}_{u})$ defines a
canonical bijection

(301) $\displaystyle\left\\{a\cdot T+\sum_{i\geq 0}a_{i}\cdot T^{i}\in
k[T]\,\Bigg{|}\,a\in k^{\times}\ \text{and}\ a_{i}\neq 0\ \text{only if}\ i\in
p\mathbb{Z}\setminus
p^{N}\mathbb{Z}\right\\}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}B_{\mathbb{A}^{1}/k}^{(N),0,1}.$

In particular, the affine line $\mathbb{A}^{1}/k$ admits an
$F^{N}$-$(\mathbb{G}_{a},\mathbb{G}_{a})$-structure for any $N$.

###  4.8. $F^{N}$-$(\mathbb{G}_{m}^{\times s}\times\mathbb{G}_{a}^{\times
t},\\{e\\})$-structures on Abelian varieties

Let $X$ be an Abelian variety over $k$ of dimension $n$. For each positive
integer $l$, denote by $l_{X}$ the endomorphism of $X$ given by multiplication
by $l$. Let $\widehat{X}$ denote the dual Abelian variety of $X$. Recall that
the $p$-rank of $X$ is the nonnegative integer
$\mu(X):=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{Hom}(\mu_{p},X))\left(=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{Hom}(\mu_{p},\mathrm{Ker}(p_{X})))\right)$,
which coincides with the value
$\mathrm{dim}_{\mathbb{F}_{p}}(\mathbb{Z}/p\mathbb{Z},X)$. Also, the
$a$-number of $X$ is defined as
$\alpha(X):=\mathrm{dim}_{k}(\mathrm{Hom}(\alpha_{p},X))\left(=\mathrm{dim}_{k}(\mathrm{Hom}(\alpha_{p},\mathrm{Ker}(p_{X})))\right)$.
These values satisfy the inequalities $0\leq\mu(X)\leq g$ and
$1\leq\alpha(X)+\mu(X)\leq g$.

Regarding $F^{N}$-$(\mathbb{G}_{m}^{\times s}\times\mathbb{G}_{a}^{\times
t},\\{e\\})$-structures for $N=1$, Proposition 4.7.2 implies the following
assertion.

######  Proposition 4.8.1.

Let us keep the above notation. Also, let $(s,t)$ be a pair of nonnegative
integers with $n=s+t$. Then, the Abelian variety $X$ admits an
$F^{1}$-$(\mathbb{G}_{m}^{\times s}\times\mathbb{G}_{a}^{\times
t},\\{e\\})$-structure if and only if $s=\mu(\widehat{X})$ and
$t=\alpha(\widehat{X})$.

###### Proof.

Observe that

(302)
$\displaystyle\mathrm{dim}_{\mathbb{F}_{p}}(H^{1}_{\mathrm{fl}}(X,\mu_{p}))$
$\displaystyle=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{Hom}(\varprojlim_{l}\mathrm{Ker}(l_{X}),\mu_{p}))$
$\displaystyle=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{Hom}(\mathrm{Ker}(p_{X}),\mu_{p}))$
$\displaystyle=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{Hom}(\widehat{\mu_{p}},\widehat{\mathrm{Ker}(p_{X})}))$
$\displaystyle=\mathrm{dim}_{\mathbb{F}_{p}}(\mathbb{Z}/p\mathbb{Z},\mathrm{Ker}(p_{\widehat{X}}))$
$\displaystyle=\mu(\widehat{X}),$

where $\widehat{(-)}$ denotes Cartier dual and the first equality follows from
[35, Chap. III, § 4, Corollary 4.19, (a)]. By a similar discussion together
with $\widehat{\alpha_{p}}=\alpha_{p}$, we have
$\mathrm{dim}_{k}(H^{1}_{\mathrm{fl}}(X,\alpha_{p}))=\alpha(\widehat{X})$.
Thus, the assertion follows from the equivalence (a) $\Leftrightarrow$ (c)
asserted in Proposition 4.7.2, (ii). ∎

Recall that the Abelian variety $X$ is called ordinary (resp., superspecial)
if $\mu(X)=g$ (resp., $\alpha(X)=g$). It is verified that $X$ is ordinary
(resp., superspecial) if and only if the dual $\widehat{X}$ is ordinary
(resp., superspecial). The following assertion concerns the cases where
$G=\mathbb{G}_{m}^{\times n}$ and $G=\mathbb{G}_{a}^{\times n}$.

######  Proposition 4.8.2.

  * (i)

Let $N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$. Then, $X$ admits an
$F^{N}$-$(\mathbb{G}_{m}^{\times n},\mathbb{G}_{m}^{\times n})$-structure if
and only if $X$ is ordinary. Moreover, when $X$ is ordinary, the map of sets

(303) $\displaystyle F^{N}\text{-}\mathrm{Ehr}_{(\mathbb{G}_{m}^{\times
n},\mathbb{G}_{m}^{\times n}),X/k}/\mathfrak{S}_{n}\rightarrow
F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{n},\mathbb{A}^{n}),X/k}$

defined in Example 2.5.2 is injective, and moreover bijective if $N=\infty$.

  * (ii)

There exists an $F^{1}$-$(\mathbb{G}_{a}^{\times n},\mathbb{G}_{a}^{\times
n})$-structure on $X/k$ if and only if $X$ is superspecial. Moreover, for any
$N\geq 2$, there are no $F^{N}$-$(\mathbb{G}_{a}^{\times
n},\mathbb{G}_{a}^{\times n})$-structures on any Abelian variety.

###### Proof.

Let us consider the former assertion of (i). The “only if” part of the
required equivalence is a direct consequence of Proposition 4.8.1 for
$(s,t)=(n,0)$ together with the equivalence “$X$ is ordinary”
$\Leftrightarrow$ “$\widehat{X}$ is ordinary”. To prove the “if” part, let us
suppose that $X$, or equivalently, $\widehat{X}$ is ordinary. Also, for a
positive integer $N$, suppose that we are given an
$F^{N}$-$(\mathbb{G}_{m}^{\times n},\mathbb{G}_{m}^{\times n})$-structure on
$X/k$; it specifies an element $(\omega_{1}^{+},\cdots,\omega_{n}^{+})$ of
$B_{X/k}^{(N),n,0}$, where
$\omega_{i}^{+}:=(\omega_{i},\mathcal{L}_{i},\eta_{i})$ ($i=1,\cdots,n$), via
(297). Since the map $\widehat{X^{(N+1)}}(k)\rightarrow\widehat{X^{(N)}}(k)$
arising from pull-back by $F_{X^{(N)}/k}:X^{(N)}\rightarrow X^{(N+1)}$ is
surjective, there exists a collection of line bundles
$\mathcal{L}_{1}^{\nabla},\cdots,\mathcal{L}_{n}^{\nabla}$ on $X^{(N+1)}$ with
$F^{*}_{X^{(N)}/k}(\mathcal{L}_{i}^{\nabla})\cong\mathcal{L}_{i}$
($i=1,\cdots,n$). The collection
$(\widetilde{\omega}_{1}^{+},\cdots,\widetilde{\omega}_{n}^{+})$ consisting of
triples
$\widetilde{\omega}_{i}^{+}:=(\omega_{i},\mathcal{L}_{i}^{\nabla},\eta_{i})$
($i=1,\cdots,n$) defines an element of $B_{X/k}^{(N+1),n,0}$ whose $N$-th
truncation coincides with $(\omega_{1}^{+},\cdots,\omega_{n}^{+})$. This
proves the surjectivity of the truncation map $B_{X/k}^{(N+1),n,0}\rightarrow
B_{X/k}^{(N),n,0}$. Observe here that $B_{X/k}^{(1),n,0}$ is nonempty because
of the result of Proposition 4.8.1 and the ordinariness assumption on $X$.
Hence, we can apply the above argument inductively to conclude that
$B_{X/k}^{(N),n,0}\neq\emptyset$ for any
$N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$. That is to say, $X/k$ admits an
$F^{N}$-$(\mathbb{G}_{m}^{\times n},\mathbb{G}_{m}^{\times n})$-structure on
$X/k$ via (297). This completes the former assertion of (i).

Next, let us consider the latter assertion of (i) under the assumption that
$X$ is ordinary. Suppose here that $N\in\mathbb{Z}_{>0}$, and let us take an
element $(\omega_{1}^{+},\cdots,\omega_{n}^{+})$ of $B_{X/k}^{(N),n,0}$, where
$\omega_{i}^{+}:=(\omega_{i},\mathcal{L}_{i},\eta_{i})$. Write
$\nabla:=\nabla_{0}\oplus\bigoplus_{i=1}^{n}\nabla^{\mathrm{can}(N-1)}_{\mathcal{L}_{i}}$,
which specifies a $\mathcal{D}_{X/k}^{(N-1)}$-module structure on
$\mathcal{O}_{X}^{\oplus(n+1)}$ via
$\mathrm{id}_{\mathcal{O}_{X}}\oplus\bigoplus_{i=1}^{n}\eta_{i}$. Since
$\omega_{1},\cdots,\omega_{n}$ forms a basis of $\Gamma(X,\Omega_{X})$, the
following composite turns out to be an isomorphism:

(304)
$\displaystyle\mathcal{D}_{X/k,1}^{(N-1)}\left(=\mathcal{D}_{X/k,1}^{(N-1)}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}\right)\hookrightarrow\mathcal{D}_{X/k}^{(N-1)}\otimes_{\mathcal{O}_{X}}\mathcal{O}^{\oplus(n+1)}_{X}\xrightarrow{\nabla}\mathcal{O}^{\oplus(n+1)}_{X},$

where the first arrow arises from both the natural inclusion
$\mathcal{D}_{X/k,1}^{(N-1)}\hookrightarrow\mathcal{D}_{X/k}^{(N-1)}$ and the
diagonal embedding
$\mathcal{O}_{X}\hookrightarrow\mathcal{O}_{X}^{\oplus(n+1)}$. Under the
identification $\mathcal{D}_{X/k,1}^{(N-1)}=\mathcal{O}^{\oplus(n+1)}_{X}$
given by this isomorphism, we shall regard $\nabla$ as a
$\mathcal{D}_{X/k}^{(N-1)}$-module structure on $\mathcal{D}_{X/k,1}^{(N)}$.
Also, we regard the $(n+1)$-st projection
$\delta:=\mathrm{pr}_{n+1}:\mathcal{O}_{X}^{\oplus(n+1)}\rightarrow\mathcal{O}_{X}$
as a left inverse morphism
$\mathcal{D}_{X/k}^{(N-1)}\twoheadrightarrow\mathcal{O}_{X}$ of the natural
inclusion
$\mathcal{O}_{X}\left(:=\mathcal{D}_{X/k,0}^{(N-1)}\right)\hookrightarrow\mathcal{D}_{X/k,1}^{(N-1)}$.
By construction, the pair $(\nabla,\delta)$ satisfies the conditions (c) and
(d) described in Example 4.5.6. That is to say, $(\nabla,\delta)$ specifies an
elmement of
$F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{n},\mathbb{A}^{n}),X/k}$. If
$B_{X/k}^{(N),n,0}/\mathfrak{S}_{n}$ denotes the quotient set of
$B_{X/k}^{(N),n,0}$ by the $\mathfrak{S}_{n}$-action determined by permutation
of factors, then the resulting assignment
$(\omega_{1}^{+},\cdots,\omega_{n}^{+})\mapsto(\nabla,\delta)$ induces a map
of sets $\varsigma_{N}:B_{X/k}^{(N),n,0}/\mathfrak{S}_{n}\rightarrow
F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{n},\mathbb{A}^{n}),X/k}$. One may
verify that $\varsigma_{N}$ coincides with (303) under the identification
$B_{X/k}^{(N),n,0}/\mathfrak{S}_{n}=F^{N}\text{-}\mathrm{Ehr}_{(\mathbb{G}_{m}^{\times
n},\mathbb{G}_{m}^{\times n}),X/k}/\mathfrak{S}_{n}$ induced by (297). Then,
the required assertion for $N\in\mathbb{Z}_{>0}$ follows from the injectivity
of $\varsigma_{N}$ resulting from [53, § 6, Theorem 6.4.1]. Moreover, the case
of $N=\infty$ follows from the bijectivity of
$\varprojlim_{N\in\mathbb{Z}_{>0}}\varsigma_{N}$ asserted in the same theorem.

Finally, we shall consider assertion (ii). The former assertion follows
directly from Proposition 4.8.1 for $(s,t)=(0,n)$ together with the
equivalence “$X$ is superspecial” $\Leftrightarrow$ “$\widehat{X}$ is
superspecial”. To prove the latter assertion, it suffices, by the former
assertion, to consider the case where $X$ is superspecial. Suppose, on the
contrary, that there exists an $F^{N}$-$(\mathbb{G}_{a}^{\times
n},\mathbb{G}_{a}^{\times n})$-structure $\mathcal{S}^{\heartsuit}$ on $X/k$
for some $N\geq 2$; it specifies, via truncation to level $2$, an element
$(\chi^{+}_{1},\cdots,\chi_{n}^{+})$, where
$\chi_{j}^{+}:=(\chi_{j},\mathcal{W}_{j},\lambda_{j})$ ($j=1,\cdots,n$), of
$B_{X/k}^{(2),0,n}$ via (297). Let us fix $j\in\\{1,\cdots,n\\}$ and write
$\mathcal{V}_{j}:=F_{X^{(1)}/k}^{*}(\mathcal{W}_{j})$. Also, denote by $v$
(resp., $w$) the element of $H^{1}(X^{(1)},\mathcal{O}_{X^{(1)}})$ (resp.,
$H^{1}(X^{(2)},\mathcal{O}_{X^{(2)}})$) determined by the extension structure
of $\mathcal{V}_{j}$ induced by that of $\mathcal{W}_{j}$ (resp., the
extension structure of $\mathcal{W}_{j}$) under the natural identification
$H^{1}(X^{(1)},\mathcal{O}_{X^{(1)}})\cong\mathrm{Ext}^{1}_{\mathcal{O}_{X^{(1)}}}(\mathcal{O}_{X^{(1)}},\mathcal{O}_{X^{(1)}})$
(resp.,
$H^{1}(X^{(2)},\mathcal{O}_{X^{(2)}})\cong\mathrm{Ext}^{1}_{\mathcal{O}_{X^{(2)}}}(\mathcal{O}_{X^{(2)}},\mathcal{O}_{X^{(2)}})$).
Since the $1$-form $\chi_{j}$ is nozero, there are no split injections
$\mathcal{O}_{X}\hookrightarrow\mathcal{O}_{X}^{\oplus 2}$ of the trivial
extension $0\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}^{\oplus
2}\rightarrow\mathcal{O}_{X}\rightarrow 0$ whose image is closed under
$\nabla_{\chi_{j}}^{\dagger}$. It follows that the extension structure of
$\mathcal{V}_{j}$ is nontrivial, or equivalently, $v\neq 0$. On the other
hand, since $X$, as well as $X^{(1)}$, is superspecial, the morphism
$F^{*}_{X^{(1)}/k}:H^{1}(X^{(2)},\mathcal{O}_{X^{(2)}})\rightarrow
H^{1}(X^{(1)},\mathcal{O}_{X^{(1)}})$ induced by $F_{X^{(1)}/k}$ is zero. This
implies $0=F^{*}_{X^{(1)}/k}(w)=v$, so we obtain a contradiction. This
completes the proof of the latter assertion. ∎

##  5\. Classification for algebraic curves

The purpose of this section is to examine the classification problem of
Frobenius-Ehresmann structures on smooth proper curves. Recall a result by B.
Laurent (cf. [32, Theorem 1.1, (1)]), asserting that the list of homogenous
curves, i.e., homogenous space of dimension $1$, may be described as follows:

  *     * Case (a):

$G=\mathrm{PGL}_{2}$, $H=\mathrm{PGL}_{2}^{\circledcirc}$;

    * Case (b):

$G=\mathrm{Aff}_{1}$, $H=\mathrm{Aff}_{1}^{\circledcirc}$;

    * Case (c):

$G=\mathbb{G}_{m}$, $H=\\{e\\}$;

    * Case (d):

$G=\mathbb{G}_{a}$, $H=\\{e\\}$;

    * Case (e):

$G$ is an elliptic curve over $k$ and $H=\\{e\\}$.

In what follows, we determine which curves have Frobenius-Ehresmann structures
with a fixed model $(G,H)$ according to this classification. Let $X$ be a
smooth proper curve over $k$ of genus $g\geq 0$.

###  5.1. Case (a):

The first case is the pair of the projective linear group $G=\mathrm{PGL}_{2}$
and its subgroup $H=\mathrm{PGL}^{\circledcirc}_{2}$. In several literatures,
$F^{N}$-$(\mathrm{PGL}_{2},\mathbb{P}^{1})$-structures on curves have been
studied under the name of Frobenius-projective structure (cf. [23], [53]). If
$N=1$, then the corresponding dormant indigenous
$(\mathrm{PGL}_{2},\mathrm{PGL}_{2}^{\circledcirc})$-bundles are called
dormant indigenous bundles or dormant $\mathrm{PGL}_{2}$-opers (cf. [40],
[51], [54]). Moreover, these objects are closely related to second order
differential operators having a full set of roots functions (cf. [54, Chap. 4,
§ 4.10, Proposition 4.65]). Here, let us pick up a previous result of [53,
Theorem E], concerning the existence of
$F^{N}$-$(\mathrm{PGL}_{2},\mathbb{P}^{1})$-structures.

######  Proposition 5.1.1.

For any $N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$, $X$ admits an
$F^{N}$-$(\mathrm{PGL}_{2},\mathbb{P}^{1})$-structure. Moreover, if $g>1$,
$N\in\mathbb{Z}_{>0}$, and $\Theta$ is a theta characteristic of $X$ (i.e., a
line bundle on $X$ equipped with an isomorphism $\Theta^{\otimes
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega_{X}$), then there exists
a canonical bijection of sets

(305) $\displaystyle
F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{PGL}_{2},\mathbb{P}^{1}),X/k}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{Q}_{N,\Theta}^{2,\mathrm{triv}}(k),$

where $\mathcal{Q}_{N,\Theta}^{2,\mathrm{triv}}$ denotes the Quot-scheme over
$k$ classifying $\mathcal{O}_{X^{(N)}}$-submodules $\mathcal{F}$ of
$F_{X/k}^{(N)*}(\Theta^{\vee})$ with $\mathrm{rank}(\mathcal{F})=2$ and
$\mathrm{det}(\mathcal{F})\cong\mathcal{O}_{X^{(N)}}$.

There are some results related to the counting problem of
$F^{N}$-$(\mathrm{PGL}_{2},\mathbb{P}^{1})$-structures. S. Mochizuki showed
that there are only finitely many
$F^{1}$-$(\mathrm{PGL}_{2},\mathbb{P}^{1})$-structures on $X$ (cf. [40, Chap.
II, § 2.3, Theorem 2.8]). Moreover, the main result of [51, Theorem A]
explicitly computes the number of such structures by means of the genus “$g$”
of $X$ and the characteristic “$p$” of the base field $k$. To be precise, if
$X$ is sufficiently general in the moduli stack of smooth proper genus $g$
curves, then we have the equality

(306) $\displaystyle
F\text{-}\mathrm{Ehr}_{(\mathrm{PGL}_{2},\mathbb{P}^{1}),X/k}=\frac{p^{g-1}}{2^{2g-1}}\cdot\sum_{\theta=1}^{p-1}\frac{1}{\sin^{2g-2}\big{(}\frac{\pi\cdot\theta}{p}\big{)}}.$

This formula is proved by using (305) for $N=1$ and applying a formula
computing the degree of certain Quot-schemes (i.e., a special case of the
Vafa-Intriligator formula). However, for large $N$’s, the author does not know
much about the finiteness of
$F^{N}$-$(\mathrm{PGL}_{2},\mathbb{P}^{1})$-structures at the time of writing
the present paper. So far, the finiteness of the set
$F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{PGL}_{2},\mathbb{P}^{1}),X/k}$ with
$N\geq 2$ has been shown only for genus $2$ curves (cf. [53, § 7, Theorem
7.4.2]). By taking account of (306) and its proof, we expect that the
following conjectural formula will hold for any large $N$; this conjecture may
be thought of as a generalization of Joshi’s conjecture for the rank $2$ case
(cf. [26, Conjecture 8.1]).

######  Conjecture 5.1.2.

Suppose that $X$ is a sufficiently general smooth proper curve of genus $g$
over $k$. Then, for each $N\in\mathbb{Z}_{>0}$, the cardinality of the set
$F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{PGL}_{2},\mathbb{P}^{1}),X/k}$ is given
by the following formula:

(307) $\displaystyle
F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{PGL}_{2},\mathbb{P}^{1}),X/k}=\frac{p^{n(g-1)}}{2^{2g-1}}\cdot\sum_{\theta=1}^{p^{n}-1}\frac{1}{\sin^{2g-2}\big{(}\frac{\pi\cdot\theta}{p^{n}}\big{)}}.$

###  5.2. Case (b):

Next, we briefly review some results of the case where $G=\mathrm{Aff}_{1}$
and $H=\mathrm{Aff}_{1}^{\circledcirc}$. Recall that
$F^{N}$-$(\mathrm{Aff}_{1},\mathrm{Aff}_{1}^{\circledcirc})$-structures on
curves have been studied in [24] and [53] under the name of Frobenius-affine
structure. If $N=1$, then these objects are equivalent to what we call dormant
(generic) Miura $\mathrm{PGL}_{2}$-opers or Tango structures (cf. [41], [52],
[55]).

The existence of such an object is very restrictive because if there is a
Tango curve (= a curve admitting a Tango structure) of genus $g$, then $p$
must divide $g-1$. M. Raynaud provided an explicit example of a Tango curve
(cf. [46] or [41, § 1.1, Example 1.3]). Also, Tango curves can be constructed
by means of solutions to the Bethe ansatz equations for Gaudin model over
finite fields (cf. [55]). According to [52, Theorem B], the moduli stack
parametrizing Tango curves of genus $g$ forms, if it is nonempty, an
equidimensional smooth Deligne-Mumford stack of dimension $2\cdot
g-2+\frac{2\cdot g-2}{p}$. Regarding higher-level cases, Y. Hoshi proved (cf.
[25, Theorem 3]) that if an integer $g>1$ satisfies $p^{N}\mid(g-1)$ for a
positive integer $N$, then there exists a smooth proper curve of genus $g$
admitting an $F^{N}$-$(\mathrm{Aff}_{1},\mathbb{A}^{1})$-structure. The
following two propositions describe some results proved previously (cf. [24],
[53, Theorem D and § 7, Propositions 7.5.1, 7.5.2]).

######  Proposition 5.2.1.

  * (i)

For $N=1$, $X/k$ admits an
$F^{1}$-$(\mathrm{Aff}_{1},\mathbb{A}^{1})$-structure if and only if $X$ is
either a genus $1$ curve or a Tango curve.

  * (ii)

For an integer $N$ with $N\geq 2$, $X/k$ admits an
$F^{N}$-$(\mathrm{Aff}_{1},\mathbb{A}^{1})$-structure if and only if $X$ is
either an ordinary genus $1$ curve or a Tango curve with $p^{N}\mid g-1$.

  * (iii)

For $N=\infty$, $X/k$ admits an
$F^{\infty}$-$(\mathrm{Aff}_{1},\mathbb{A}^{1})$-structure if and only if $X$
is an ordinary genus $1$ curve.

Here, for each ring $R$ and a free $R$-module $M$ of rank $1$, we shall denote
by $\mathbb{G}_{m}(R;M)$ the set of generators in $M$.

######  Proposition 5.2.2.

Suppose that $X$ is an ordinary genus $1$ curve, and denote by $T_{p}X$ the
$p$-adic Tate module of $X$. Then, there exists a bijection of sets

(308) $\displaystyle
F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{1},\mathbb{A}^{1}),X/k}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{G}_{m}(R_{N};T_{p}X\otimes_{\mathbb{Z}_{p}}R_{N}),$

where $R_{N}:=\mathbb{Z}/p^{N}\mathbb{Z}$ if $N\in\mathbb{Z}_{>0}$ and
$R_{N}:=\mathbb{Z}_{p}$ if $N=\infty$. In particular, if
$N\in\mathbb{Z}_{>0}$, then the set
$F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{1},\mathbb{A}^{1}),X/k}$ is finite
and its cardinality can be given by the following formula:

(309)
$\displaystyle\sharp(F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{1},\mathbb{A}^{1}),X/k})=p^{N-1}(p-1).$

###  5.3. Cases (c) and (d):

The classification problems for the cases (c) and (d) were already finished in
the discussion of § 4.8. The following assertions are the results for curves
obtained there.

######  Proposition 5.3.1.

Let $N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$. Then, $X$ admits an
$F^{N}$-$(\mathbb{G}_{m},\mathbb{G}_{m})$-structure if and only if $X$ is an
ordinary genus $1$ curve. If $X$ is an ordinary genus $1$ curve, then the map
of sets

(310) $\displaystyle
F^{N}\text{-}\mathrm{Ehr}_{(\mathbb{G}_{m},\mathbb{G}_{m}),X/k}\rightarrow
F^{N}\text{-}\mathrm{Ehr}_{(\mathrm{Aff}_{1},\mathbb{A}^{1}),X/k}$

defined in Example 2.5.2 is bijective.

###### Proof.

The assertion follow from Proposition 4.8.2, (i), for $n=1$. Note that the
surjectivity of (310) for $N\in\mathbb{Z}_{>0}$ can be verified from the
discussion in the proof of Proposition 4.8.2, (i). ∎

######  Proposition 5.3.2.

  * (i)

There exists an $F^{1}$-$(\mathbb{G}_{a},\mathbb{G}_{a})$-structure on $X/k$
if and only if $X$ is a supersingular genus $1$ curve. Moreover, for any
$N\geq 2$, there are no $F^{N}$-$(\mathbb{G}_{a},\mathbb{G}_{a})$-structures
on $X/k$.

  * (ii)

Suppose that $X$ is a supersingular genus $1$ curve. Then, there exists a
canonical bijection

(311) $\displaystyle
F\text{-}\mathrm{Ehr}_{(\mathbb{G}_{a},\mathbb{G}_{a}),X/k}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{G}_{m}(k;\Gamma(X,\Omega_{X/k}))\left(=\Gamma(X,\Omega_{X/k})\setminus\\{0\\}\right).$

###### Proof.

The assertions follow from (294), the latter assertion of Proposition 4.7.2,
(i), and Proposition 4.8.2, (ii). ∎

###  5.4. Case (e):

Finally, let us consider the case where $G$ is an elliptic curve and
$H=\\{e\\}$.

######  Proposition 5.4.1.

For each $N\in\mathbb{Z}_{>0}\sqcup\\{\infty\\}$, there exists an
$F^{N}$-$(G,G)$-structure on $X/k$ if and only if $X$ is of genus $1$ and
isogenous to $G$. In particular, there are no $F^{N}$-$(G,G)$-structures on
any smooth proper curve of genus $>1$.

###### Proof.

The latter assertion is a direct consequence of the former assertion, so we
only discuss the former assertion. First, let us consider the “if” part of the
required equivalence. Suppose that $X$ is of genus $1$ and there exists an
isogeny morphism $X\rightarrow G$. This morphism decomposes as
$X\xrightarrow{F_{X/k}^{(m)}}X^{(m)}\xrightarrow{w}G$ for some $m\geq 0$,
where $w$ denotes a finite étale $k$-covering. Then, the pull-back
$w^{*}(\mathcal{S}^{\mathrm{triv},\heartsuit}_{N})$ of the standard
$F^{N}$-$(G,G)$-structure $\mathcal{S}^{\mathrm{triv},\heartsuit}_{N}$ on
$G/k$ (cf. (41)) forms an $F^{N}$-$(G,G)$-structure on $X^{(m)}$. Moreover, it
induces an $F^{N}$-$(G,G)$-structure on $X/k$ via base-change by
$(F_{k}^{-1})^{m}:\mathrm{Spec}(k)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Spec}(k)$.
This completes the proof of the “if” part.

Next, we shall consider the “only if” part. To prove this, it suffices to
consider the case of $N=1$. Suppose that $X$ is not isogenous to $G$ and
admits an $F^{1}$-$(G,G)$-structure $\mathcal{S}^{\heartsuit}$. Denote by
$\mathcal{E}^{\spadesuit}:=(X\times\\{e\\},\nabla)$ the dormant indigenous
$(G,\\{e\\})$-bundle corresponding to $\mathcal{S}^{\heartsuit}$. Under the
natural identification $\mathrm{Lie}(G)_{X\times
G}/\mathrm{Lie}(\\{e\\})_{X\times\\{e\\}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left(\mathcal{O}_{X}\otimes\mathrm{Lie}(G)=\right)\mathcal{O}_{X}$,
the Kodaira-Spencer map $\mathrm{KS}_{(X\times\\{e\\},\nabla)}$ may be
regarded as an $\mathcal{O}_{X}$-linear morphism
$\mathcal{T}_{X}\rightarrow\mathcal{O}_{X}$. Since
$\mathrm{KS}_{(X\times\\{e\\},\nabla)}$ is assumed to be an isomorphism, the
equality $\mathrm{deg}(\mathcal{T}_{X})=0$ holds, in particular, $X$ must be
of genus $1$. Next, denote by $\mathcal{G}$ the (right) $G$-bundle on
$X^{(1)}$ corresponding to the left $G_{X/S}^{(1)}$-torsor
$\mathcal{S}^{\heartsuit}$. Let us fix an isomorphism of $G$-bundles
$\alpha:X\times
G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{*}_{X/k}(\mathcal{G})$. Note
that there exist a smooth proper curve $Y$ and a finite étale covering
$y:Y\rightarrow X$ such that $\mathcal{G}$ can be trivialized after pull-back
by the finite étale covering $y^{(1)}:Y^{(1)}\rightarrow X^{(1)}$ induced by
$y$ (cf. [37, Chap II, § 5, Lemma 5.5]). That is to say, we can find an
isomorphism
$\beta:Y^{(1)}\times_{X^{(1)}}\mathcal{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}Y^{(1)}\times
G$. Both $\alpha$ and $\beta$ yield the composite

(312) $\displaystyle\gamma:Y\xrightarrow{a\mapsto(a,e)}Y\times
G\xrightarrow{y^{*}(\alpha)}y^{*}(F^{*}_{X/k}(\mathcal{G}))\left(=F^{*}_{Y/k}(y^{(1)*}(\mathcal{G}))\right)\xrightarrow{F^{*}_{Y/k}(\beta)}Y\times
G\xrightarrow{\mathrm{pr}_{2}}G.$

The curve $Y$ is not isogenous to $G$ because $X$ is assumed not to be
isogenous to $G$. Hence, the image of $\gamma$ consists exactly of one point.
It follows that its differential
$d\gamma:\mathcal{T}_{Y}\rightarrow\gamma^{*}(\mathcal{T}_{G})\left(=\mathcal{O}_{Y}\otimes\mathrm{Lie}(G)\right)$
is the zero map. This contradicts the fact that $d\gamma$ coincides with the
pull-back of the isomorphism $\mathrm{KS}_{(X\times\\{e\\},\nabla)}$ by $y$.
This completes the proof of the “only if” part. ∎

##  6\. Deformation theory of (dormant) indigenous bundles

The topic of this section is the deformation theory of (dormant) indigenous
$(G,H)$-bundles. After proving basic facts on various deformation functors, we
formulate and prove the Ehresmann-Weil-Thurston principle for Frobenius-
Ehresmann structures (cf. Theorem 6.9.1), which is the main result of this
section.

Denote by $\mathfrak{S}\mathfrak{e}\mathfrak{t}$ the category of (small) sets
and by $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ the category of local
Artinian $k$-algebras with residue field $k$. We shall write
$k_{\epsilon}:=k[\epsilon]/(\epsilon^{2})$, which is an object of
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$. For each
$R\in\mathrm{Ob}(\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k})$, we write
$\mathfrak{m}_{R}$ for the maximal ideal of $R$. Throughout this section, we
fix a smooth scheme $X$ of finite type over $k$. Note that each morphism of
sheaves $\nabla:\mathcal{K}^{0}\rightarrow\mathcal{K}^{1}$ on a scheme may be
regarded as a complex concentrated at degrees $0$ and $1$; we shall denote
this complex by $\mathcal{K}^{\bullet}[\nabla]$. Also, for each sheaf
$\mathcal{F}$, we shall write $\mathcal{F}[0]$ for the complex defined to be
$\mathcal{F}$ concentrated at degree $0$.

###  6.1. Deformation functors

By a functor on $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ (or sometimes just
a functor when there is no fear of confusion), we shall mean a covariant
functor
$D:\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}\rightarrow\mathfrak{S}\mathfrak{e}\mathfrak{t}$
such that $D(k)$ is a singleton. Also, following [16, § Definition 2.8], we
refer to a functor $D$ on $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ as a
functor with good deformation theory (or simply, a functor with gdt) if it
satisfies the following two of the Schlessinger conditions:

  * (H1)

For any morphism $\sigma^{\prime}:R^{\prime}\rightarrow R$ and any surjective
morphism $\sigma^{\prime\prime}:R^{\prime\prime}\twoheadrightarrow R$ both of
which belong to $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, the map

(313) $\displaystyle D(R^{\prime}\times_{R}R^{\prime\prime})\rightarrow
D(R^{\prime})\times_{D(R)}D(R^{\prime\prime})$

induced naturally by $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ is
surjective. (This condition remains equivalent even if $\sigma^{\prime\prime}$
is moreover assumed to be a principal small extension.)

  * (H2)

With the above notation, (313) becomes bijective when $R=k$. (This condition
remains equivalent even if $R^{\prime\prime}$ is moreover assumed to be
$k_{\epsilon}$.)

Let
$D:\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}\rightarrow\mathfrak{S}\mathfrak{e}\mathfrak{t}$
be a deformation functor with gdt and write $TD:=D(k_{\epsilon})$. Then,
according to [16, § 2, Lemma 2.12], the following properties hold:

  * •

$TD$ has a canonical structure of $k$-vector space;

  * •

For any small extension in $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ of the
form $e:=(0\rightarrow M\rightarrow R\rightarrow k\rightarrow 0)$, there
exists a canonical bijection $M\otimes
TD\stackrel{{\scriptstyle\sim}}{{\rightarrow}}D(R)$;

  * •

For any small extension $e:=(0\rightarrow M\rightarrow
R\xrightarrow{\sigma}R_{0}\rightarrow 0)$ in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, there exists an $M\otimes
TD$-action on $D(R)$ which is restricted to a transitive action on the fibers
of $D(\sigma):D(R)\rightarrow D(R_{0})$. Moreover, the formation of this
action is compatible, in an evident sense, with morphisms of small extensions.

Here, we recall several notions concerning deformation functors used in our
discussion. A morphism of functors $\gamma:D\rightarrow D^{\prime}$ is called
formally smooth if for every small extension $e:=(0\rightarrow M\rightarrow
R\xrightarrow{\sigma}R_{0}\rightarrow 0)$ in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, the map of sets $D(R)\rightarrow
D(R_{0})\times_{D^{\prime}(R_{0})}D^{\prime}(R)$ induced by $D(\sigma)$ and
$\gamma(R)$ is surjective. If $\gamma:D\rightarrow D^{\prime}$ is a morphism
between functors with gdt, then the map
$d\gamma:=\gamma(k_{\epsilon}):TD\rightarrow TD^{\prime}$ induced by it
becomes a $k$-linear morphism of vector spaces. We shall refer to $d\gamma$ as
the differential of $\gamma$.

Next, an obstruction theory for a functor $D$ is defined as a collection
$(O,\\{o_{e}\\}_{e})$ consisting of a $k$-vector space $O$ and a map
$o_{e}:D(R_{0})\rightarrow M\otimes O$ for every small extension
$e:=(0\rightarrow M\rightarrow R\rightarrow R_{0}\rightarrow 0)$ such that an
element $a\in D(R_{0})$ satisfies $o_{e}(a)=0$ if and only if $a$ lifts to
$D(R)$. The $k$-vector space $O$ is called an obstruction space of $D$. We
shall often abuse notation by just writing $O$ for $(O,\\{o_{e}\\}_{e})$.

To every complete local $k$-algebra $R$ with residue field $k$, we can
associate a functor with gdt
$h_{R}:\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}\rightarrow\mathfrak{S}\mathfrak{e}\mathfrak{t}$
defined by $h_{R}(R^{\prime}):=\mathrm{Hom}_{k}(R,R^{\prime})$. Recall that a
functor on $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ is called pro-
representable if it is isomorphic to $h_{R}$ for such an $R$.

######  Definition 6.1.1.

  * (i)

A deformation triple is a collection

(314) $\displaystyle\mathbb{D}:=(D,T,O),$

consisting of a functor $D$ with gdt, a $k$-vector space $T$ equipped with an
identification $T\stackrel{{\scriptstyle\sim}}{{\rightarrow}}TD$, and an
obstruction theory $O$ for $D$.

  * (ii)

Let $\mathbb{D}:=(D,T,O)$ and
$\mathbb{D}^{\prime}:=(D^{\prime},T^{\prime},O^{\prime})$ be deformation
triples. A morphism of deformation triples from $\mathbb{D}$ to
$\mathbb{D}^{\prime}$ is defined as a triple

(315) $\displaystyle(\gamma_{D},\gamma_{T},\gamma_{O}),$

where

    * –

$\gamma_{D}$ denotes a morphism of functors $\gamma_{D}:D\rightarrow
D^{\prime}$;

    * –

$\gamma_{T}$ denotes a $k$-linear morphism $T\rightarrow T^{\prime}$ such that
for any small extension $e:=(0\rightarrow M\rightarrow
R\xrightarrow{\sigma}R_{0}\rightarrow 0)$ in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, the $M\otimes T$-action on $D(R)$
is compatible with the $M\otimes T^{\prime}$-action on $D^{\prime}(R)$ via
$\gamma_{T}$ and $\gamma_{D}$;

    * –

$\gamma_{O}$ denotes a $k$-linear morphism $O\rightarrow O^{\prime}$ such that
for any $e$ as above, the following square diagram is commutative:

(320)

######  Definition 6.1.2.

Let $D$ be a functor with gdt. We shall say that $D$ is rigid if for any small
extension $e:=(0\rightarrow M\rightarrow R\xrightarrow{\sigma}R_{0}\rightarrow
0)$ in $\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, the transitive $M\otimes
TD$-action on every fiber of $D(\sigma):D(R)\rightarrow D(R_{0})$ is free,
i.e., the fibers form $M\otimes TD$-torsors.

######  Remark 6.1.3.

Let
$D:\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}\rightarrow\mathfrak{S}\mathfrak{e}\mathfrak{t}$
be a functor with gdt.

  * (i)

It follows from [48, § 2, Theorem 2.11] that $D$ is pro-representable if and
only if $D$ is rigid and satisfies $\mathrm{dim}_{k}(TD)<\infty$.

  * (ii)

Suppose that $D$ is associated with a deformation category $\mathcal{D}$ in
the sense of [49, § 2, Definition 2.16]. For each
$R\in\mathrm{Ob}(\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k})$ and
$\xi\in\mathcal{D}(R)$, we shall write $\mathrm{Aut}_{R}(\xi)$ for the
automorphism of $\xi$ over the identity morphism of $R$. Then, the rigidity of
$D$ holds if, for any small extension $0\rightarrow M\rightarrow
R\xrightarrow{\sigma}R_{0}\rightarrow 0$ in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, the group homomorphism
$\mathrm{Aut}_{R}(\xi)\rightarrow\mathrm{Aut}_{R_{0}}(\sigma^{*}(\xi))$ is
surjective.

######  Definition 6.1.4.

Let $\gamma:D\rightarrow D^{\prime}$ be a morphism of functors with gdt. We
shall say that $\gamma$ is a semi-isomorphism if it is formally smooth and its
differential $d\gamma:TD\rightarrow TD^{\prime}$ is bijective. (Recall that a
pro-representable hull for a functor $D$ is a semi-isomorphism
$h_{R}\rightarrow D$, where $R$ is a completes local $k$-algebra $R$ with
residue field $k$.)

######  Proposition 6.1.5.

Let $\gamma:D\rightarrow D^{\prime}$ be a morphism of functors with gdt.

  * (i)

Suppose that $D^{\prime}$ is rigid. Then, $\gamma$ is a semi-isomorphism if
and only if it is an isomorphism.

  * (ii)

Suppose that $\gamma$ is a semi-isomorphism and there exists a pro-
representable hull $\nu:h_{R}\rightarrow D$ for $D$. Then, the composite
$\gamma\circ\nu:h_{R}\rightarrow D^{\prime}$ is a pro-representable hull for
$D^{\prime}$.

  * (iii)

Suppose that there exist pro-representable hulls $\nu:h_{R}\rightarrow D$,
$\nu^{\prime}:h_{R^{\prime}}\rightarrow D^{\prime}$ for $D$, $D^{\prime}$
respectively. Then, the following conditions are equivalent to each other:

    * –

The morphism $\gamma$ is a semi-isomorphism.

    * –

There exists an isomorphism
$\sigma:R\stackrel{{\scriptstyle\sim}}{{\rightarrow}}R^{\prime}$ of complete
local $k$-algebras such that the induced isomorphism
$h_{\sigma}:h_{R}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}h_{R^{\prime}}$
makes the following diagram commute:

(325)

###### Proof.

The assertions follow immediately from various definitions involved together
with routine arguments of deformation functors. ∎

######  Remark 6.1.6.

Let $\gamma:D\rightarrow D^{\prime}$ be a morphism of functors. Then, without
the rigidity assumption on $D^{\prime}$, the morphism $\gamma$ can be a semi-
isomorphism, yet $\gamma$ need not be an isomorphism. It is because of this
subtle difference that a semiuniversal family of a deformation functor can
fail to be universal (see [42, § 3, Example 12] for an example).

######  Proposition 6.1.7.

Let
$(\gamma_{D},\gamma_{T},\gamma_{O}):(D,T,O)\rightarrow(D^{\prime},T^{\prime},O^{\prime})$
be a morphism of deformation triples. Suppose that $\gamma_{T}:T\rightarrow
T^{\prime}$ is an isomorphism and $\gamma_{O}:O\rightarrow O^{\prime}$ is
injective. Then, $\gamma_{D}:D\rightarrow D^{\prime}$ is a semi-isomorphism.

###### Proof.

The formal smoothness of $\gamma_{D}$ can be proved by a routine diagram-
chasing argument using the commutative diagram

(330)

defined for every small extension $e:=(0\rightarrow M\rightarrow
R\xrightarrow{\sigma}R_{0}\rightarrow 0)$ in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$. ∎

###  6.2. Deformation of smooth schemes

For $R\in\mathrm{Ob}(\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k})$, a
deformation of $X$ over $R$ is a smooth scheme $X_{R}$ over $R$ together with
a $k$-isomorphism
$X\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k\times_{R}X_{R}$. The notion of
an isomorphism between two such deformations can be defined in an evident
manner. The trivial deformation of $X$ over $R$ is the smooth $R$-scheme

(331) $\displaystyle X_{R}^{\circ}:=R\times X$

together with the natural isomorphism
$X\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k\times_{R}(R\times X)$. The
assignment from each
$R\in\mathrm{Ob}(\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k})$ to the set
$\mathrm{Def}_{X}(R)$ of isomorphism classes of deformations of $X$ over $R$
defines a functor with gdt

(332)
$\displaystyle\mathrm{Def}_{X}:\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}\rightarrow\mathfrak{S}\mathfrak{e}\mathfrak{t}.$

For each $i\geq 0$, we shall write

(333) $\displaystyle H^{i}:=H^{i}(X,\mathcal{T}_{X}).$

Then, it is well-known that $(\mathrm{Def}_{X},H^{1},H^{2})$ forms a
deformation triple, and the group of infinitesimal automorphisms of $X$ is
isomorphic to $H^{0}$.

###  6.3. Deformation of indigenous $(G,H)$-bundles

We shall fix an indigenous $(G,H)$-bundle
$\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ on $X/k$.

######  Definition 6.3.1.

  * (i)

Let $R\in\mathrm{Ob}(\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k})$. A
deformation of $(X,\mathcal{E}^{\spadesuit})$ over $R$ is a collection of data

(334) $\displaystyle(X_{R},\mathcal{E}_{R}^{\spadesuit},\varrho),$

where $X_{R}$ denotes a deformation of $X$ over $R$,
$\mathcal{E}_{R}^{\spadesuit}$ denotes an indigenous $(G,H)$-bundle on
$X_{R}/R$, and $\varrho$ denotes an isomorphism of indigenous $(G,H)$-bundles
$\mathcal{E}^{\spadesuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k\times_{R}\mathcal{E}_{R}^{\spadesuit}$.
A deformation of $\mathcal{E}^{\spadesuit}$ over $R$ is defined as a pair

(335) $\displaystyle(\mathcal{E}_{R}^{\spadesuit},\varrho)$

so that $(X_{R}^{\circ},\mathcal{E}_{R}^{\spadesuit},\varrho)$ forms a
deformation of $(X,\mathcal{E}^{\spadesuit})$ over $R$. We shall often abuse
notation by just writing $(X_{R},\mathcal{E}_{R}^{\spadesuit})$ (resp.,
$\mathcal{E}_{R}^{\spadesuit}$) for
$(X_{R},\mathcal{E}_{R}^{\spadesuit},\varrho)$ (resp.,
$(\mathcal{E}_{R}^{\spadesuit},\varrho)$).

  * (ii)

Let $\sigma:R^{\prime}\rightarrow R$ be a morphism in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ and denote by $\sigma^{\sharp}$
the morphism $\mathrm{Spec}(R)\rightarrow\mathrm{Spec}(R^{\prime})$ induced by
$\sigma$. Also, let $(X_{R},\mathcal{E}_{R}^{\spadesuit},\varrho)$ (resp.,
$(X_{R^{\prime}},\mathcal{E}_{R^{\prime}}^{\spadesuit},\varrho^{\prime})$) be
a deformation of $(X,\mathcal{E}^{\spadesuit})$ over $R$ (resp.,
$R^{\prime}$). A morphism over $\sigma$ from
$(X_{R},\mathcal{E}_{R}^{\spadesuit},\varrho)$ to
$(X_{R^{\prime}},\mathcal{E}^{\spadesuit}_{R^{\prime}},\varrho^{\prime})$ is a
pair

(336) $\displaystyle(\xi_{X},\xi_{\mathcal{E}})$

consisting of a morphism $\xi_{X}:X_{R}\rightarrow X_{R^{\prime}}$ over
$\sigma^{\sharp}$ and an isomorphism of indigenous $(G,H)$-bundles
$\xi_{\mathcal{E}}:\mathcal{E}_{R}^{\spadesuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\xi^{*}_{X}(\mathcal{E}_{R^{\prime}}^{\spadesuit})$
whose reduction modulo $\mathfrak{m}_{R}$ coincides with the identity morphism
of $(X,\mathcal{E}^{\spadesuit})$ via $\varrho$ and $\varrho^{\prime}$. If
$\sigma=\mathrm{id}_{R}$, then we shall refer to such a pair as a morphism
over $R$. Finally, we can define the notion of a morphism between deformations
of $\mathcal{E}^{\spadesuit}$ in an evident manner.

For each $R\in\mathrm{Ob}(\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k})$, we
shall write

(337)
$\displaystyle\mathrm{Def}_{\mathcal{E}^{\spadesuit}}(R):=\left(\begin{matrix}\text{the
set of isomorphism classes}\\\ \text{of deformations of
$\mathcal{E}^{\spadesuit}$ over $R$}\end{matrix}\right)\hskip 34.1433pt$
$\displaystyle\left(\text{resp.,}\
\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R):=\left(\begin{matrix}\text{the
set of isomorphism classes}\\\ \text{of deformations of
$(X,\mathcal{E}^{\spadesuit})$ over $R$}\end{matrix}\right)\right).$

If $\sigma:R^{\prime}\rightarrow R$ is a morphism in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$, then changing the base by the
induced morphism $\mathrm{Spec}(R)\rightarrow\mathrm{Spec}(R^{\prime})$ gives
a map of sets
$\sigma^{*}:\mathrm{Def}_{\mathcal{E}^{\spadesuit}}(R^{\prime})\rightarrow\mathrm{Def}_{\mathcal{E}^{\spadesuit}}(R)$
(resp.,
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R^{\prime})\rightarrow\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R)$).
Thus, the assignments $R\mapsto\mathrm{Def}_{\mathcal{E}^{\spadesuit}}(R)$
(resp., $R\mapsto\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R)$) and
$\sigma\mapsto\sigma^{*}$ determine a functor

(338) $\displaystyle\mathrm{Def}_{\mathcal{E}^{\spadesuit}}\
\left(\text{resp.,}\
\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}\right):\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}\rightarrow\mathfrak{S}\mathfrak{e}\mathfrak{t}.$

One may verify from various definitions involved that this functor specifies a
functor with gdt. The assignment
$\mathcal{E}^{\spadesuit}_{R}\mapsto(X_{R}^{\circ},\mathcal{E}_{R}^{\spadesuit})$
(resp., $(X_{R},\mathcal{E}_{R}^{\spadesuit})\mapsto X_{R}$) determines a
morphism of functors

(339)
$\displaystyle\gamma_{\mathcal{E}^{\spadesuit}}:\mathrm{Def}_{\mathcal{E}^{\spadesuit}}\rightarrow\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}\
\left(\text{resp.,}\
\delta_{\mathcal{E}^{\spadesuit}}:\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}\rightarrow\mathrm{Def}_{X}\right).$

###  6.4. Cohomological descriptions of
$\mathrm{Def}_{\mathcal{E}^{\spadesuit}}$ and
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}$

In what follows, let us describe the tangent and obstruction spaces of the
deformation functors introduced above in terms of cohomology groups. The
corresponding discussion for holomorphic Cartan geometries can be found in [9]
(cf. Remark 6.4.5).

Let $\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{H},\nabla)$ be as above and write
$\mathcal{E}_{G}:=\mathcal{E}_{H}\times^{H}G$. Denote by
$\nabla^{\mathrm{ad}}_{G}$ the $k$-connection
$\mathfrak{g}_{\mathcal{E}_{G}}\rightarrow\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}}$
on the adjoint bundle $\mathfrak{g}_{\mathcal{E}_{G}}$ induced by $\nabla$ via
change of structure group by
$\mathrm{Ad}_{G}:G\rightarrow\mathrm{GL}(\mathfrak{g})$. Denote by

(340)
$\displaystyle\nabla^{\mathrm{ad}}_{H}:\mathfrak{h}_{\mathcal{E}_{H}}\rightarrow\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}}$

the morphism obtained by restricting the domain of $\nabla_{G}^{\mathrm{ad}}$
to
$\mathfrak{h}_{\mathcal{E}_{H}}\left(\subseteq\mathfrak{g}_{\mathcal{E}_{G}}\right)$.
For any local sections
$v\in\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}\left(:=\widetilde{\mathcal{T}}_{\mathcal{E}_{G}/k}\right)$
and $\partial\in\mathcal{T}_{X}$, we shall write

(341)
$\displaystyle\widetilde{\nabla}^{\mathrm{ad}}_{\partial}(v):=[\nabla(\partial),v]-\nabla([\partial,d_{\mathcal{E}_{G}}(v)])\in\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}.$

######  Lemma 6.4.1.

The local section $\widetilde{\nabla}^{\mathrm{ad}}_{\partial}(v)$ lies in
$\mathfrak{g}_{\mathcal{E}_{G}}\left(\subseteq\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}\right)$.

###### Proof.

Note that $d_{\mathcal{E}_{G}}$ preserves the Lie bracket operations. Hence,
for any local sections $v\in\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$ and
$\partial\in\mathcal{T}_{X}$, the following sequence of equalities holds:

(342) $\displaystyle
d_{\mathcal{E}_{G}}(\widetilde{\nabla}_{\partial}^{\mathrm{ad}}(v))$
$\displaystyle=d_{\mathcal{E}_{G}}([\nabla(\partial),v]-\nabla([\partial,d_{\mathcal{E}_{G}}(v)]))$
$\displaystyle=[d_{\mathcal{E}_{G}}\circ\nabla(\partial),d_{\mathcal{E}_{G}}(v)]-d_{\mathcal{E}_{G}}\circ\nabla([\partial,d_{\mathcal{E}_{G}}(v)])$
$\displaystyle=[\partial,d_{\mathcal{E}_{G}}(v)]-[\partial,d_{\mathcal{E}_{G}}(v)]$
$\displaystyle=0.$

This means that the image of $\widetilde{\nabla}^{\mathrm{ad}}_{\partial}(v)$
lies in
$\mathrm{Ker}(d_{\mathcal{E}_{G}})\left(=\mathfrak{g}_{\mathcal{E}_{G}}\right)$.
∎

Let $\Box$ denote either $G$ or $H$. By the above lemma, the assignment
$v\mapsto\widetilde{\nabla}^{\mathrm{ad}}_{(-)}(v)$ defines a $k$-linear
morphism

(343)
$\displaystyle\widetilde{\nabla}_{\Box}^{\mathrm{ad}}:\widetilde{\mathcal{T}}_{\mathcal{E}_{\Box}}\rightarrow\left(\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{T}_{X},\mathfrak{g}_{\mathcal{E}_{G}})=\right)\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}}.$

This morphism fits into the following morphism of short exact sequences:

(348)

In other words, we have the following short exact sequence of complexes:

(349) $\displaystyle
0\longrightarrow\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\Box}]\longrightarrow\mathcal{K}^{\bullet}[\widetilde{\nabla}^{\mathrm{ad}}_{\Box}]\longrightarrow\mathcal{T}_{X}[0]\longrightarrow
0.$

For each $i\geq 0$, we shall write

(350)
$\displaystyle\mathbb{H}^{i}_{\Box}:=\mathbb{H}^{i}(X,\mathcal{K}^{\bullet}[\nabla_{\Box}^{\mathrm{ad}}]),\hskip
14.22636pt\widetilde{\mathbb{H}}_{\Box}^{i}:=\mathbb{H}^{i}(X,\mathcal{K}^{\bullet}[\widetilde{\nabla}_{\Box}^{\mathrm{ad}}]).$

By applying the hypercohomology functor $\mathbb{H}^{i}(X,-)$ ($i\geq 0$) to
(349), we obtain morphisms

(351)
$\displaystyle\mathbb{H}^{i}_{\Box}\xrightarrow{\gamma^{i}_{\Box}}\widetilde{\mathbb{H}}_{\Box}^{i}\xrightarrow{\delta^{i}_{\Box}}H^{i}.$

If $X$ is proper over $k$, then the $k$-vector spaces appearing in (351) are
all finite-dimensional.

######  Lemma 6.4.2.

Let us consider the $\mathcal{O}_{X}$-linear endomorphism $\eta$ of
$\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$ given by $v\mapsto v-(\nabla\circ
d_{\mathcal{E}_{G}})(v)$ for each local section
$v\in\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$. Then, this endomorphism is
restricted to a morphism (resp., an isomorphism)

(352)
$\displaystyle\eta_{G}:\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}\
\left(\text{resp.,}\
\eta_{H}:\widetilde{\mathcal{T}}_{\mathcal{E}_{H}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}_{\mathcal{E}_{G}}\right).$

Moreover, for $\Box\in\\{G,H\\}$, the following diagram is commutative:

(357)

In particular, the isomorphism
$\eta^{\bullet}_{H}:\mathcal{K}^{\bullet}[\widetilde{\nabla}^{\mathrm{ad}}_{H}]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{G}]$
determined by the pair of morphisms
$(\eta_{H},\mathrm{id}_{\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}}})$
induces, for each $i\geq 0$, an isomorphism of $k$-vector spaces
$\widetilde{\mathbb{H}}_{H}^{i}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{i}_{G}$.

###### Proof.

First, let us prove the former assertion. To this end, we only consider the
case of $\eta_{H}$ because the other case is entirely similar. If we take a
local section $v\in\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$, then we have

(358) $\displaystyle d_{\mathcal{E}_{G}}(\eta(v))$
$\displaystyle=d_{\mathcal{E}_{G}}(v-(\nabla\circ d_{\mathcal{E}_{G}})(v))$
$\displaystyle=d_{\mathcal{E}_{G}}(v)-(d_{\mathcal{E}_{G}}\circ\nabla)(d_{\mathcal{E}_{G}}(v))$
$\displaystyle=d_{\mathcal{E}_{G}}(v)-d_{\mathcal{E}_{G}}(v)$
$\displaystyle=0.$

This implies that
$\eta(v)\in\mathrm{Ker}(d_{\mathcal{E}_{G}})=\mathfrak{g}_{\mathcal{E}_{G}}$,
so the image of $\eta$ is contained in $\mathfrak{g}_{\mathcal{E}_{G}}$. By
restricting the domain and codomain of $\eta$, we obtain an
$\mathcal{O}_{X}$-linear morphism
$\eta_{H}:\widetilde{\mathcal{T}}_{\mathcal{E}_{H}}\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}$.
One may verify that $\eta_{H}$ fits into the following morphism of short exact
sequences:

(363)

Since the Kodaira-Spencer map $\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$ is an
isomorphism, $\eta_{H}$ turns out to be an isomorphism by the five lemma.

Next, we shall prove the latter assertion, i.e., the commutativity of (357).
Clearly, it suffices to consider the case of $\eta_{G}$. Denote by
$\langle-,-\rangle$ the pairing
$\mathcal{T}_{X}\times(\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}})\rightarrow\mathfrak{g}_{\mathcal{E}_{G}}$
arising from the natural pairing
$\mathcal{T}_{X}\times\Omega_{X}\rightarrow\mathcal{O}_{X}$. Then, for each
local section $v\in\widetilde{\mathcal{T}}_{\mathcal{E}_{G}}$, we have

(364)
$\displaystyle\langle\partial,(\nabla_{G}^{\mathrm{ad}}\circ\eta_{G})(v)\rangle$
$\displaystyle=\langle\partial,\nabla_{G}^{\mathrm{ad}}(v-(\nabla\circ
d_{\mathcal{E}_{H}})(v))\rangle$
$\displaystyle=[\nabla(\partial),v-(\nabla\circ d_{\mathcal{E}_{H}})(v)]$
$\displaystyle=[\nabla(\partial),v]-[\nabla(\partial),(\nabla\circ
d_{\mathcal{E}_{G}})(v)]$
$\displaystyle=[\nabla(\partial),v]-\nabla([\partial,d_{\mathcal{E}_{G}}(v)])$
$\displaystyle=\langle\partial,\widetilde{\nabla}^{\mathrm{ad}}_{G}(v)\rangle.$

This implies
$\nabla_{G}^{\mathrm{ad}}\circ\eta_{G}=\widetilde{\nabla}_{G}^{\mathrm{ad}}$,
completing the proof of this lemma. ∎

######  Proposition 6.4.3.

  * (i)

The collection
$(\mathrm{Def}_{\mathcal{E}^{\spadesuit}},\mathbb{H}^{1}_{H},\mathbb{H}^{2}_{H})$
forms a deformation triple. Moreover, the group of infinitesimal automorphisms
of $\mathcal{E}^{\spadesuit}$ (i.e., the group of automorphisms of
$k_{\epsilon}\times_{k}\mathcal{E}^{\spadesuit}$ inducing the identity
morphism of $\mathcal{E}^{\spadesuit}$ via reduction modulo $(\epsilon)$) is
isomorphic to $\mathbb{H}^{0}_{H}$.

  * (ii)

The collection
$(\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})},\widetilde{\mathbb{H}}^{1}_{H},\widetilde{\mathbb{H}}^{2}_{H})$
($\cong(\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})},\mathbb{H}^{1}_{G},\mathbb{H}^{2}_{G})$
by Lemma 6.4.2) forms a deformation triple. Moreover, the group of
infinitesimal automorphisms of $(X,\mathcal{E}^{\spadesuit})$ is isomorphic to
$\widetilde{\mathbb{H}}_{H}^{0}$ ($\cong\mathbb{H}^{0}_{G}$).

  * (iii)

Suppose that $X$ is proper over $k$. Then, there exist pro-representable hulls
for $\mathrm{Def}_{\mathcal{E}^{\spadesuit}}$ and
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}$.

###### Proof.

We will omit the proof of assertion (i) because it is entirely similar to the
proof of assertion (ii).

First, we shall consider the tangent space of
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}$. Let $e:=(0\rightarrow
M\rightarrow R\xrightarrow{\sigma}R_{0}\rightarrow 0)$ be a small extension in
$\mathfrak{A}\mathfrak{r}\mathfrak{t}_{/k}$ and
$(X_{R},\mathcal{E}^{\spadesuit}_{R})$ (where
$\mathcal{E}_{R}^{\spadesuit}:=(\mathcal{E}_{R,H},\nabla_{R})$) a deformation
of $(X,\mathcal{E}^{\spadesuit})$ over $R$. Also, let us take an element $v$
of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\mathrm{id}_{M}\otimes\widetilde{\nabla}_{H}^{\mathrm{ad}}])\left(\cong
M\otimes\widetilde{\mathbb{H}}^{1}_{H}\right)$, where
$\mathrm{id}_{M}\otimes\widetilde{\nabla}_{H}^{\mathrm{ad}}$ denotes the
tensor product $M\otimes\widetilde{\mathcal{T}}_{\mathcal{E}_{H}}\rightarrow
M\otimes\mathfrak{g}_{\mathcal{E}_{G}}$ of $\mathrm{id}_{M}$ and
$\widetilde{\nabla}_{H}^{\mathrm{ad}}$. Note here that
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\mathrm{id}_{M}\otimes\widetilde{\nabla}_{H}^{\mathrm{ad}}])$
can be realized as the total cohomology of the Čech double complex of
$\mathcal{K}^{\bullet}[\mathrm{id}_{M}\otimes\widetilde{\nabla}_{H}^{\mathrm{ad}}]$
associated with an affine covering. That is to say, we can find an affine open
covering $\mathcal{U}:=\\{U_{\alpha}\\}_{\alpha\in I}$ (where $I$ is a finite
index set) of $X$ and $v$ may be represented by a collection of data

(365) $\displaystyle
v=(\\{\widetilde{\partial}_{\alpha\beta}\\}_{(\alpha,\beta)\in
I_{2}},\\{\delta_{\alpha}\\}_{\alpha\in I}),$

where

  * •

$I_{2}$ denotes the subset of $I\times I$ consisting of pairs $(\alpha,\beta)$
with $U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}\neq\emptyset$;

  * •

$\\{\widetilde{\partial}_{\alpha\beta}\\}_{(\alpha,\beta)\in I_{2}}$ is a Čech
$1$-cocycle in
$\check{C}^{1}(\mathcal{U},M\otimes\widetilde{\mathcal{T}}_{\mathcal{E}_{H}})$,
where
$\widetilde{\partial}_{\alpha\beta}\in\Gamma(U_{\alpha\beta},M\otimes\widetilde{\mathcal{T}}_{\mathcal{E}_{H}})$;

  * •

$\\{\delta_{\alpha}\\}_{\alpha\in I}$ is a Čech $0$-cochain in
$\check{C}^{0}(\mathcal{U},M\otimes(\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}}))$,
where
$\delta_{\alpha}\in\Gamma(U_{\alpha},M\otimes(\Omega_{X}\otimes\mathfrak{g}_{\mathcal{E}_{G}}))=\mathrm{Hom}_{\mathcal{O}_{U_{\alpha}}}(\mathcal{T}_{X}|_{U_{\alpha}},M\otimes\mathfrak{g}_{\mathcal{E}_{G}}|_{U_{\alpha}})$,

such that $\\{\widetilde{\partial}_{\alpha\beta}\\}_{(\alpha,\beta)\in I_{2}}$
and $\\{\delta_{\alpha}\\}_{\alpha\in I}$ agree under
$\mathrm{id}_{M}\otimes\widetilde{\nabla}_{H}^{\mathrm{ad}}$ and the Čech
coboundary map. For each $(\alpha,\beta)\in I_{2}$, we set
$\partial_{\alpha\beta}:=d_{\mathcal{E}_{H}}(\widetilde{\partial}_{\alpha\beta})$.
We shall write $U_{R,\alpha}:=X_{R}|_{U_{\alpha}}$ for each $\alpha\in I$ and
write $U_{R,\alpha\beta}:=X_{R}|_{U_{\alpha\beta}}$ for each
$(\alpha,\beta)\in I_{2}$. The element $\partial_{\alpha\beta}$ (resp.,
$\widetilde{\partial}_{\alpha\beta}$) may be regarded as a $k$-linear morphism
$\partial_{\alpha\beta}:\mathcal{O}_{U_{\alpha\beta}}\rightarrow
M\otimes\mathcal{O}_{U_{\alpha\beta}}$ (resp.,
$\widetilde{\partial}_{\alpha\beta}:\mathcal{O}_{\mathcal{E}_{H}|_{U_{\alpha\beta}}}\rightarrow
M\otimes\mathcal{O}_{\mathcal{E}_{H}|_{U_{\alpha\beta}}}$) satisfying the
Leibniz rule. Then,
$\mathrm{id}_{\mathcal{O}_{U_{R,\alpha\beta}}}+\partial_{\alpha\beta}$ (resp.,
$\mathrm{id}_{\mathcal{O}_{\mathcal{E}_{R,H}|_{U_{R,\alpha\beta}}}}+\widetilde{\partial}_{\alpha\beta}$)
defines a well-defined automorphism $\partial^{\sharp}_{\alpha\beta}$ (resp.,
$\widetilde{\partial}^{\sharp}_{\alpha\beta}$) of $U_{R,\alpha\beta}$ (resp.,
$\mathcal{E}_{R,H}|_{U_{R,\alpha\beta}}$). The automorphism
$\widetilde{\partial}^{\sharp}_{\alpha\beta}$ is $H$-equivariant and the
following square diagram is commutative and cartesian:

(370)

By means of $\partial_{\alpha\beta}^{\sharp}$’s ($(\alpha,\beta)\in I_{2}$),
we can glue together $U_{R,\alpha}$’s ($\alpha\in I$) to obtain a smooth
scheme $X^{v}_{R}$ over $R$. Moreover, the pairs
$(\mathcal{E}_{R,H}|_{U_{R,\alpha}},\nabla|_{U_{R,\alpha}}+\delta_{\alpha})$
may be glued together by means of
$\widetilde{\partial}^{\sharp}_{\alpha\beta}$’s. Thus, we obtain an $H$-bundle
$\mathcal{E}_{R,H}^{v}$ on $X_{R}^{v}$ and an $R$-connection $\nabla^{v}_{R}$
on $\mathcal{E}_{R,G}^{v}:=\mathcal{E}_{R,H}^{v}\times^{H}G$. The Kodaira-
Spencer map $\mathrm{KS}_{(\mathcal{E}_{R,H}^{v},\nabla^{v}_{R})}$ is an
isomorphism because its reduction modulo $\mathfrak{m}_{R}$ coincides with the
isomorphism $\mathrm{KS}_{(\mathcal{E}_{H},\nabla)}$. It follows that the
resulting pair
$\mathcal{E}_{R}^{\spadesuit,v}:=(\mathcal{E}_{R,H}^{v},\nabla^{v}_{R})$ forms
an indigenous $(G,H)$-bundle on $X^{v}_{R}$ and the pair
$(X_{R}^{v},\mathcal{E}_{R}^{\spadesuit,v})$ specifies an element of
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R)$. The assignment
$((X_{R},\mathcal{E}_{R}^{\spadesuit}),v)\mapsto(X_{R}^{v},\mathcal{E}_{R}^{\spadesuit,v})$
defines a transitive action of $M\otimes\widetilde{\mathbb{H}}_{H}^{1}$ on the
fibers of
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(\sigma):\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R)\rightarrow\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R_{0})$.
(To verify the transitivity of this action, we may need the fact that
$M\otimes\widetilde{\mathbb{H}}_{H}^{1}$ is canonically isomorphic to the
$1$-st hypercohomology group of the complex of étale sheaves associated with
$\mathcal{K}^{\bullet}[\mathrm{id}_{M}\otimes\widetilde{\nabla}^{\mathrm{ad}}_{H}]$,
each of whose elements may be represented by a Čech $1$-cocycle with respect
to an étale covering; this is because $H$-bundles are locally trivial in the
étale topology, but not necessarily in the Zariski topology.) This action is,
by construction, verified to be free if $R=k$. Consequently,
$\widetilde{\mathbb{H}}_{H}^{1}$ is isomorphic to the tangent space of
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}$.

Next, we shall consider the assertion about an obstruction space. Let $e$ be
as above and $a$ an element of
$\mathrm{Def}_{(X,\mathcal{E}^{\spadesuit})}(R_{0})$. Choose a deformation
$(X_{R_{0}},\mathcal{E}_{R_{0}}^{\spadesuit})$ (where
$\mathcal{E}^{\spadesuit}_{R_{0}}:=(\mathcal{E}_{R_{0},H},\nabla_{R_{0}})$) of
$(X,\mathcal{E}^{\spadesuit})$ representing $a$. Observe that there exists an
affine open covering $\mathcal{V}:=\\{V_{\alpha}\\}_{\alpha\in J}$ of $X$ such
that, for each $\alpha\in J$, the $R_{0}$-scheme
$V_{R_{0},\alpha}:=X_{R_{0}}|_{V_{\alpha}}$ has a deformation $V_{R,\alpha}$
over $R$ and
$(\mathcal{E}_{R_{0},H}|_{V_{\alpha}},\nabla_{R_{0}}|_{V_{\alpha}})$ may be
lifted to a pair
$\mathcal{E}_{R,\alpha}^{\spadesuit}:=(\mathcal{E}_{R,H,\alpha},\nabla_{R,\alpha})$
defined over $V_{R,\alpha}$. After possibly replacing $\mathcal{V}$ by its
refinement, we can find a collection
$\\{\eta_{\alpha\beta}\\}_{(\alpha,\beta)\in J_{2}}$, where each
$\eta_{\alpha\beta}$ denotes an $H$-equivariant isomorphism
$\mathcal{E}_{R,H,\alpha}|_{V_{\alpha\beta}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}_{R,H,\beta}|_{V_{\alpha\beta}}$
inducing the identity morphism of $\mathcal{E}_{R_{0},H}|_{V_{\alpha\beta}}$
via reduction modulo $M$. There exists an element