$\pi:\underset{(P,\,b_{i}^{(m)})\in\mathfrak{S}}{\sqcup}\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}\rightarrow
T$.

Conversely, let $s$ be a point of $\text{Bun}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$
for some $(P,\,b_{i}^{(m)})\in\mathfrak{S}$. By definition, this means that
$\mathcal{V}|_{C_{s}}$ has a subbundle $\mathcal{W}$ over $C_{s}$ of the form

$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$

such that $HP(\mathcal{F}^{(0)})=P$. But we know that
$(P,\,b_{i}^{(m)})\in\mathfrak{S}$. We can compute the slope of $\mathcal{W}$
from the parabolic Quot data $(P,\,b_{i}^{(m)})$, and we see that
$\mathcal{W}$ must be a $P$-destabilizing subbundle of $\mathcal{V}|_{C_{s}}$.
Therefore $HN(\mathcal{V}|_{C_{s}})\nleq P$. Lemma 2.30 implies that
$HN(\mathcal{V}|_{C_{s}})=HN(\mathcal{V}|_{C_{\pi(s)}})$, so we must also have
$HN(\mathcal{V}|_{C_{\pi(s)}})\nleq P$. We conclude that $\pi(s)\in T^{\nleq
P}$. ∎

###  5.2 Openness of strata

The following proposition will be used to prove the main theorem of this
section.

######  Proposition 5.6.

Let $R$ be a discrete valuation ring over $k$. Let $\eta$ (resp. $s$) denote
the generic (resp. special) point of $\text{Spec}(R)$. Let $\mathcal{V}$ be a
parabolic vector bundle of rank $n$ over $C_{R}$. Let $P$ be a parabolic HN
datum. If $HN\left(\mathcal{V}|_{C_{s}})\right)\leq P$, then
$HN\left(\mathcal{V}|_{C_{\eta}}\right)\leq P$.

Before proving Proposition 5.6, let’s state a lemma that we will use in the
proof.

######  Lemma 5.7.

Let $R$ be a $k$-algebra. Let $\mathcal{E}$ be a vector bundle on $C_{R}$.
Suppose that $\mathcal{F}$ is a coherent subsheaf of $\mathcal{E}$ such that
the quotient $\mathcal{E}\,/\,\mathcal{F}$ is $R$-flat. Then, $\mathcal{F}$ is
a vector bundle on $C_{R}$.

###### Proof.

We want to show that $\mathcal{F}$ is $C_{R}$-flat. There is a short exact
sequence
$0\longrightarrow\mathcal{F}\longrightarrow\mathcal{E}\longrightarrow\mathcal{E}\,/\,\mathcal{F}\longrightarrow
0$. We can use the Tor long exact sequence at each stalk to conclude that
$\mathcal{F}$ is $R$-flat. By Lemma 4.11 with $X=C_{R}$ and
$B=\text{Spec}\,R$, it suffices to check flatness at each fiber. Choose a
point $t\in\text{Spec}\,R$. Proving flatness of $\mathcal{F}|_{C_{t}}$ amounts
to showing that $\mathcal{F}|_{C_{t}}$ is torsion-free, because $C_{t}$ is a
regular curve. Since $\mathcal{E}\,/\,\mathcal{F}$ is $R$-flat, the fiber
$\mathcal{F}|_{C_{t}}$ remains a subsheaf of $\mathcal{E}|_{C_{t}}$. Since
$\mathcal{E}|_{C_{t}}$ is torsion-free, we conclude that
$\mathcal{F}|_{C_{t}}$ is torsion-free. ∎

###### Proof of Proposition 5.6.

For the purpose of this proof, it will be convenient to fix the following
notation. For every sheaf $\mathcal{F}$ over $C_{R}$, we will write
${}^{\eta}\mathcal{F}\vcentcolon=\mathcal{F}|_{C_{\eta}}$ and
${}^{s}\mathcal{F}\vcentcolon=\mathcal{F}|_{C_{s}}$. We will use the same
notation for parabolic vector bundles.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C_{R}$. Suppose that
$HN\left({}^{\eta}\mathcal{V}\right)\nleq P$, where

${}^{\eta}\mathcal{V}=\left[\;{}^{\eta}\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,^{\eta}\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,^{\eta}\mathcal{E}_{i}^{(N_{i})}=\,^{\eta}\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$

By Lemma 5.2, there exists a $P$-destabilizing parabolic subbundle
${}^{\eta}\mathcal{W}\subset\,^{\eta}\mathcal{V}$. Set

${}^{\eta}\mathcal{W}=\left[\;{}^{\eta}\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,^{\eta}\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,^{\eta}\mathcal{F}_{i}^{(N_{i})}=\,^{\eta}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$

Let
${}^{\eta}\mathcal{Q}^{(m)}_{i}\vcentcolon=\,^{\eta}\mathcal{E}_{i}^{(m)}\,/\,^{\eta}\mathcal{F}_{i}^{(m)}$,
and set
${}^{\eta}\mathcal{L}_{i}^{(m)}\vcentcolon=\,^{\eta}\mathcal{Q}^{(0)}(x_{i})\,/\,^{\eta}\mathcal{Q}^{(m)}_{i}$.
Let $j$ denote the open immersion $j:C_{\eta}\hookrightarrow C_{R}$. Define
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\text{Im}\left(\;\mathcal{E}_{i}^{(m)}\xrightarrow{unit}j_{*}\,^{\eta}\mathcal{E}_{i}^{(m)}\longrightarrow
j_{*}\,^{\eta}\mathcal{Q}_{i}^{(m)}\;\right)$. Also, define
$\mathcal{F}_{i}^{(m)}\vcentcolon=\text{Ker}\left(\;\mathcal{E}_{i}^{(m)}\twoheadrightarrow\mathcal{Q}_{i}^{(m)}\;\right)$.
We have the following commutative diagram with exact columns:

${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0)}}$${\mathcal{Q}_{i}^{(1)}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2)}}$${\mathcal{Q}_{i}^{(N_{i}-1)}}$${\mathcal{Q}^{(0)}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0)}}$${\mathcal{F}_{i}^{(1)}}$${\cdots}$${\mathcal{F}_{i}^{(Ni-2)}}$${\mathcal{F}_{i}^{(N_{i}-1)}}$${\mathcal{F}^{(0)}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$
Figure 3: Diagram 3

Define
$\mathcal{L}_{i}^{(m)}\vcentcolon=\mathcal{Q}^{(0)}(x_{i})\,/\,\mathcal{Q}_{i}^{(m)}$.
Since $j$ is affine, the pushforward functor $j_{*}$ is exact on quasicoherent
sheaves. This implies that $\mathcal{L}_{i}^{(m)}\subset
j_{*}\,^{\eta}\mathcal{L}_{i}^{(m)}$, and hence $\mathcal{L}_{i}^{(m)}$ is
$R$-torsion free. Therefore $\mathcal{L}_{i}^{(m)}$ is $R$-flat, because $R$
is a discrete valuation ring.

We can use the short exact sequence,

$0\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\longrightarrow\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\longrightarrow\mathcal{L}_{i}^{(m)}\longrightarrow
0$

and the flatness of $\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}$ to
see that $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}$ is $R$-flat for
all $i$, $m$. Now we can use the short exact sequence

$0\longrightarrow\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m-1)}\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\longrightarrow
0$

to conclude that $\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}$ is $R$\-
flat for all $i$ and $m$. A similar argument shows that
$\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}$ is $R$ flat. Therefore,
Diagram 3 remains exact and with horizontal arrows given by inclusions after
any base-change with respect to $R$.

Since $\mathcal{Q}_{i}^{(m)}\subset j_{*}\,^{\eta}\mathcal{Q}_{i}^{(m)}$, we
have that $\mathcal{Q}_{i}^{(m)}$ is $R$-torsion free. This implies that
$\mathcal{Q}_{i}^{(m)}$ is $R$-flat, because $R$ is a discrete valuation ring.
By construction we have the short exact sequence

$0\longrightarrow\mathcal{F}_{i}^{(m)}\longrightarrow\mathcal{E}_{i}^{(m)}\longrightarrow\mathcal{Q}_{i}^{(m)}\longrightarrow
0$

By Lemma 5.7 we conclude that $\mathcal{F}_{i}^{(m)}$ is a vector bundle on
$C_{R}$. The discussion above implies that
$\mathcal{W}\vcentcolon=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle on $C_{R}$. We don’t necessarily know that
$\mathcal{W}$ is a parabolic subbundle of $\mathcal{V}$, because we have no
control on the quotients $\mathcal{Q}_{i}^{(m)}$. We do however know that
$\mathcal{Q}_{i}^{(m)}$ $R$-flat. Therefore, the base-change to the special
fiber ${}^{s}\mathcal{W}\hookrightarrow\,^{s}\mathcal{V}$ remains a
monomorphism.

We can now appeal to Lemma 2.11 to obtain a subbundle
${}^{s}\mathcal{W}^{sat}$ of ${}^{s}\mathcal{V}$ containing the subobject
${}^{s}\mathcal{W}$ and satisfying
$\mu(^{s}\mathcal{W}^{sat})\geq\mu(^{s}\mathcal{W})$. By local constancy of
degree (Lemma 2.29) we have $\mu(^{s}\mathcal{W})=\mu(^{\eta}\mathcal{W})$.
Hence $\mu(^{s}\mathcal{W}^{sat})\geq\,\mu(^{\eta}\mathcal{W})$. We conclude
that ${}^{s}\mathcal{W}^{sat}$ is $P$-destabilizing, because
$\text{rank}\,^{s}\mathcal{W}^{sat}=\text{rank}\,^{\eta}\mathcal{W}$ and
${}^{\eta}\mathcal{W}$ is $P$-destabilizing. This implies that
$HN\left({}^{s}\mathcal{V}\right)\nleq P$, as desired. ∎

######  Remark 5.8.

Suppose that $\mu(^{s}\mathcal{W}^{sat})=\mu(^{s}\mathcal{W})$ in the proof
above. Then part (b) of Lemma 2.11 implies that
${}^{s}\mathcal{\mathcal{W}}=\,^{s}\mathcal{W}^{sat}$. In this case
${}^{s}\mathcal{Q}_{i}^{(m)}$ is a vector bundle. Applying Lemma 4.11 to the
$R$-flat sheaf $\mathcal{Q}_{i}^{(m)}$, we see that $\mathcal{Q}_{i}^{(m)}$ is
a vector bundle over $C_{R}$. Therefore $\mathcal{W}\subset\mathcal{V}$ is a
parabolic subbundle over $C_{R}$ in this case.

######  Proposition 5.9.

Let $T$ be a scheme of finite type over $k$. Let $\mathcal{V}$ a parabolic
vector bundle over $C\times S$. Let $P$ be a HN datum. The set $T^{\leq
P}\vcentcolon=\left\\{t\in T\,\mid\,HN(\mathcal{V}|_{C_{t}})\leq P\right\\}$
is Zariski open in $T$.

###### Proof.

By Proposition 5.1, we know that $T^{\leq P}$ is constructible. In order to
prove that it is open we are left to show that it is closed under
generalization. Let $p,t\in T$ be two topological points with
$t\in\overline{\\{p\\}}$. Assume that $t\in T^{\leq P}$. By [Sta, Tag 054F],
there exists a discrete valuation $k$-algebra $R$ and a morphism
$f:\text{Spec}(R)\rightarrow T$ such that $f(\eta)=p$ and $f(s)=t$. (Here we
are using the same notation as in Proposition 5.6, so $s$ is the special point
and $\eta$ is the generic point). By Lemma 2.30, we know that
$HN(\mathcal{V}|_{C_{s}})=HN(\mathcal{V}|_{C_{t}})$. Since $t\in T^{\leq P}$,
we have $HN\left(\mathcal{V}|_{C_{t}}\right)\leq P$. Hence
$HN\left(\mathcal{V}|_{C_{s}}\right)\leq P$. We can apply Proposition 5.6 to
deduce that $HN(\mathcal{V}|_{C_{\eta}})\leq P$. By Lemma 2.30 again, we have
$HN(\mathcal{V}|_{C_{p}})=(\mathcal{V}|_{C_{\eta}})\leq P$. So $p\in T^{\leq
P}$, as desired. ∎

######  Remark 5.10.

In particular, Proposition 5.9 implies that parabolic HN data are upper
semicontinuous.

######  Theorem 5.11.

Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Let $P$ be a
parabolic HN datum. Then $\text{Bun}_{\mathcal{V}}^{\leq P}$ is an open
substack of $\text{Bun}_{\mathcal{V}}$.

###### Proof.

The theorem amounts to showing that for any $k$-scheme $T$ and morphism
$T\rightarrow\text{Bun}_{\mathcal{V}}$, the base-change
$\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T\;\rightarrow T$ is represented by an open
subscheme of $T$. Corollary 3.8 tells us that $\text{Bun}_{\mathcal{V}}$ is
locally of finite type over $k$. So we can assume that $T$ is of finite type
over $k$.

A map $T\rightarrow\text{Bun}_{\mathcal{V}}$ is by definition a parabolic
vector bundle $\mathcal{W}$ of type $\mathcal{V}$ on $C\times T$. We can
describe the base-change $\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T$ as follows. For any scheme $U$ we have

$\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T\,(U)\;=\;\left\\{\begin{matrix}\text{morphisms}\;f:U\rightarrow
T\text{ such that}\\\ \text{ for all $u\in
U$},\;HN\left(\mathcal{W}\,|_{C_{u}}\right)\,\leq P\end{matrix}\right\\}$

For any $u\in U$, Lemma 2.30 implies that
$HN\left(\mathcal{W}\,|_{C_{u}}\right)=HN\left(\mathcal{W}\,|_{C_{f(u)}}\right)$.
We conclude that the functor $\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T$ can be alternatively described by the
set theoretic condition

$\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\mathcal{V}}}T\,(U)\;=\;\left\\{\begin{matrix}\text{morphisms}\;f:U\rightarrow
T\text{ such that}\\\ \text{ for all $u\in U$, $f(u)\in T^{\leq
P}$}\end{matrix}\right\\}$

By Proposition 5.9, the set $T^{\leq P}$ is open. Hence the functor is
represented by $T^{\leq P}$ viewed as an open subscheme of $T$. ∎

The analogue of Theorem 5.11 in the context of vector bundles without
parabolic structure when $P=0$ is classical. This amounts to proving that the
semistable locus of a family of vector bundles is open. See e.g. [HL97][Prop.
2.3.1].

##  6 Quasicompactness of strata

###  6.1 Harder-Narasimhan stratification for the moduli of vector bundles

We recall the Harder-Narasimhan stratification for the classical moduli stack
of vector bundles on a curve.

######  Definition 6.1.

The moduli stack of vector bundles of rank $n$ over $C$ is the pseudofunctor
$\text{Bun}_{\text{GL}_{n}}(C)$ from $k$-schemes into groupoids given as
follows. For each $k$-scheme $T$, we define
$\text{Bun}_{\text{GL}_{n}}(C)\,(T)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid
of rank $n$ vector bundles over $C\times T$}\end{matrix}\right\\}$.

It is well known that $\text{Bun}_{\text{GL}_{n}}(C)$ is a smooth algebraic
stack over $k$. This is just a special case of Proposition 3.5.

Recall that the slope of a rank $n$ vector bundle $\mathcal{E}$ over $C$ is
defined to be $\mu(\mathcal{E})=\frac{1}{n}\,\text{deg}\,\mathcal{E}$. A
vector bundle is called semistable if all nontrivial subbundles
$\mathcal{F}\subset\mathcal{E}$ satisfy
$\mu(\mathcal{F})\leq\mu(\mathcal{E})$. For any given vector bundle
$\mathcal{E}$ over $C$, there exists a filtration
$0=\mathcal{E}_{0}\,\subset\,\mathcal{E}_{1}\,\subset\,\cdots\,\subset\,\mathcal{E}_{l}=\mathcal{E}$
by subbundles such that $\mathcal{E}_{j}\,/\,\mathcal{E}_{j-1}$ is semistable
for all $j$, and such that for all $1\leq j\leq l-1$ we have
$\mu\left(\mathcal{E}_{j}\,/\,\mathcal{E}_{j-1}\right)>\mu\left(\mathcal{E}_{j+1}\,/\,\mathcal{E}_{j}\right)$.
There is a unique filtration satisfying these conditions; it is called the
Harder-Narasimhan filtration.

By a classical HN datum of length $n$ we mean a tuple of $n$ rational numbers
$(\mu_{1},\mu_{2},\cdots,\mu_{n})\in\mathbb{Q}^{n}$ satisfying
$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}$. To any vector bundle $\mathcal{E}$
on $C$ we can associate a HN datum $HN(\mathcal{E})$ given as follows. Let
$0=\mathcal{E}_{0}\,\subset\,\mathcal{E}_{1}\,\subset\,\cdots\,\subset\,\mathcal{E}_{l}=\mathcal{E}$
be the Harder-Narasimhan filtration of $\mathcal{E}$. Then, we set

$\displaystyle
HN(\mathcal{E})=(\;\underbrace{\mu(\mathcal{E}_{1}),\,\cdots,\,\mu(\mathcal{E}_{1})}_{\text{rank}\;\mathcal{E}_{1}/0\,\text{times}},\;\underbrace{\mu(\mathcal{E}_{2}/\,\mathcal{E}_{1}),\,\cdots,\,\mu(\mathcal{E}_{2}/\,\mathcal{E}_{1})}_{\text{rank}\;\mathcal{E}_{2}/\,\mathcal{E}_{1}\,\text{times}},\cdots,\,\underbrace{\mu(\mathcal{E}_{l}/\,\mathcal{E}_{l-1}),\;\cdots,\;\mu(\mathcal{E}_{l}/\,\mathcal{E}_{l-1})}_{\text{rank}\;\mathcal{E}_{l}/\mathcal{E}_{l-1}\,\text{times}}\;)$

If $\mathcal{E}$ has rank $n$, then
$HN(\mathcal{E})\in\left(\frac{1}{n!}\mathbb{Z}\right)^{n}$.

Let $P_{1}=(\mu_{l})$ and $P_{2}=(\nu_{l})$ be two classical HN-data of rank
$n$. We say that $P_{1}\leq P_{2}$ if the following two conditions are
satisfied

  1. (a)

$\sum_{l=1}^{n}\mu_{l}=\sum_{l=1}^{n}\nu_{l}$.

  2. (b)

For all $1\leq m<n$, we have $\sum_{l=1}^{m}\mu_{l}\leq\sum_{l=1}^{m}\nu_{l}$.

######  Definition 6.2.

Let $Q$ be a classical HN datum of rank $n$. We define
$\text{Bun}_{\text{GL}_{n}}^{\leq Q}(C)$ to be the pseudofunctor from
$k$-schemes into groupoids given as follows. For any $k$-scheme $T$,

$\text{Bun}_{\text{GL}_{n}}^{\leq
Q}(C)\,(T)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of rank $n$
vector bundles $\mathcal{E}$ over $C\times T$ such that}\\\ \text{for all
$t\in T$, we have $HN(\mathcal{E}|_{C_{t}})\leq Q$}\end{matrix}\right\\}$

The following is a well known result. See e.g. [Beh91] or [Sch15] for proofs
in the generality of reductive algebraic groups.

######  Theorem 6.3.

Let $Q$ be a classical HN datum. The subfunctor
$\text{Bun}_{\text{GL}_{n}}^{\leq Q}(C)$ is a quasicompact open substack of
$\text{Bun}_{\text{GL}_{n}}(C)$. ∎

###  6.2 Quasicompactness of parabolic Harder-Narasimhan strata

Let us now return to our parabolic setting. We have the following natural map
of stacks.

######  Definition 6.4.

We define
$Forget:\text{Bun}_{\mathcal{V}}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$
to be the map of stacks given as follows. Suppose $T$ is a $k$-scheme and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle in $\text{Bun}_{\mathcal{V}}(T)$. We set
$\phi(T)\,(\mathcal{W})=\mathcal{F}^{(0)}$.

######  Proposition 6.5.

$Forget:\text{Bun}_{\mathcal{V}}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$
is schematic and proper.

In order to prove this proposition, we first recall the definition and
properties of generalized flag varieties.

######  Definition 6.6.

Let $S$ be a scheme and let $\mathcal{G}$ be a locally free sheaf of constant
rank on $S$. Let $(a^{(l)})_{l}$ be a tuple of nonnegative integers
$a^{(1)},a^{(2)},\cdots,a^{(N)}$. The generalized flag variety
$Flag^{(a^{(l)})_{l}}(\mathcal{G})$ is defined to be the functor from
$S$-schemes to sets given as follows. Let $X$ be an $S$-scheme. Then,

$\displaystyle
Flag^{(a^{(l)})_{l}}(\mathcal{G})(X)=\;\left\\{\begin{matrix}\text{filtrations
of $\mathcal{G}|_{X}$ by vector
subbundles}\;\left[\;0\,=\mathcal{P}^{(0)}\,\subset\,\mathcal{P}^{(1)}\subset\cdots\;\subset\mathcal{P}^{(N)}=\mathcal{G}|_{X}\;\right]\\\
\\\ \text{such that}\;\mathcal{P}^{(l)}\,/\,\mathcal{P}^{(l-1)}\;\text{is
locally free of rank $a^{(l)}$ for every $1\leq l\leq
N$}\end{matrix}\right\\}$

The map on morphisms is given by pulling back the filtrations.

######  Proposition 6.7.

Let $S$ be a $k$-scheme and $\mathcal{G}$ a vector bundle of constant rank $n$
on $S$. Let $(a^{(l)})_{l}$ be a tuple of nonnegative integers.
$Flag^{(a^{(l))})_{l}}(\mathcal{G})$ is represented by a proper scheme over
$S$.

###### Proof.

Zariski descent for quasicoherent sheaves implies that
$Flag^{(a^{(l))})_{l}}(\mathcal{G})$ is a sheaf in the Zariski topology. In
particular, it suffices to check representability by a proper scheme after
passing to a Zariski cover of $S$. We can assume that $\mathcal{G}$ is trivial
by passing to an open cover. Then $Flag^{(a^{(l))})_{l}}(\mathcal{G})$ is the
base-change of the classical partial flag variety of flags of type
$(a^{(l))})_{l}$ in the vector space $k^{n}$. This is known to be projective
over $k$. ∎

###### Proof of Proposition 6.5.

Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. Let $T$ be a $k$-scheme and $\mathcal{E}$ a vector bundle of rank $n$ on
$C\times T$. In order to ease notation, let us write
$L\vcentcolon=T\times_{\text{Bun}_{\text{GL}_{n}}(C)}\text{Bun}_{\mathcal{V}}$.
We want to show that $L$ is represented by a proper $T$-scheme. By definition,
$L$ is the functor from $T$-schemes into groupoids given as follows. If $S$ is
a $T$-scheme, then

$\displaystyle L(S)=\;\left\\{\begin{matrix}\text{parabolic vector
bundles}\;\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}\\\ \text{ over $C\times S$ of type $\mathcal{V}$},\text{ with an
isomorphism}\;\phi:\mathcal{E}|_{C\times
S}\xrightarrow{\sim}\mathcal{F}^{(0)}\end{matrix}\right\\}$

The isomorphism $\phi$ is part of the data. Automorphisms in the groupoid are
required to be compatible with the isomorphism $\phi$. So they must be the
identity on $\mathcal{F}^{(0)}$. This in turn implies that they must be the
identity as a morphism of parabolic vector bundles. Therefore, $L$ is
naturally equivalent to the sheaf of sets:

$\displaystyle L(S)=\;\left\\{\begin{matrix}\text{parabolic vector
bundles}\;\mathcal{W}=\left[\;\mathcal{E}|_{C\times
S}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{E}(x_{i})|_{C\times
S}\;\right]_{i\in I}\\\ \text{ over $C\times S$ of type
$\mathcal{V}$}\end{matrix}\right\\}$

We claim that this sheaf is represented by the $T$-scheme $\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)$. By Proposition 6.7, this claim concludes the proof.

In order to prove the claim, we present two natural transformations that are
inverse to each other. We start with $f:L\rightarrow\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)$ given as follows. For a $T$-scheme $S$, and a parabolic bundle
$\mathcal{W}=\left[\;\mathcal{E}|_{C\times
S}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{E}(x_{i})|_{C\times
S}\;\right]_{i\in I}$ in $L(S)$, we define

$\displaystyle
f\left(\mathcal{W}\right)\;\vcentcolon=\;\left(\;0\,\overset{a_{i}^{(1)}}{\subset}\,\left(\mathcal{F}_{i}^{(1)}\,/\,\mathcal{E}|_{C\times
S}\right)|_{x_{i}\times
S}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\left(\mathcal{F}_{i}^{(N_{i})}\,/\,\mathcal{E}|_{C\times
S}\right)|_{x_{i}\times
S}=\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\right)|_{x_{i}\times
S}\;\right)_{i\in I}$

Let $q_{i}:x_{i}\rightarrow C$ denote the inclusion of the closed point
$x_{i}$ into $C$. The inverse $g:\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)\rightarrow L$ is defined as follows. Let $S$ be a $T$-scheme. Let
$(fl_{i})_{i\in I}$ be at tuple of flags in $\prod_{i\in
I}\text{Flag}^{(a_{i}^{(l)})_{l}}\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\,|_{x_{i}\times
T}\right)(S)$. Suppose that $fl_{i}$ is given by a chain of vector bundles
over $S$

$fl_{i}=\;\left[\;0\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{P}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{P}_{i}^{(N_{i})}=\left(\mathcal{E}(x_{i})\,/\,\mathcal{E}\right)|_{x_{i}\times
S}\;\right]$

We define $g(\,(fl_{i})_{i\in I}\,)=\left[\;\mathcal{E}|_{C\times
S}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{E}(x_{i})|_{C\times
S}\;\right]_{i\in I}$, where

$\displaystyle\mathcal{F}^{(l)}_{i}\vcentcolon=\text{Ker}\left(\;\mathcal{E}(x_{i})|_{C\times
S}\xrightarrow{unit}\,(q_{i}\times id_{S})_{*}(q_{i}\times
id_{S})^{*}\mathcal{E}(x_{i})|_{C\times S}\,\twoheadrightarrow\,(q_{i}\times
id_{S})_{*}\left(\mathcal{E}(x_{i})|_{x_{i}\times
S}\,/\,\mathcal{P}^{(l)}_{i}\right)\;\right)$

These functors are inverses of each other by construction. The only nontrivial
thing to check is that $g$ well defined (i.e. $g(\,(fil_{i})_{i\in I}\,)$ is a
parabolic bundle). In order to show this, we need to check that the
$\mathcal{F}_{i}^{(l)}$s are vector bundles on $C\times S$. Each sheaf
$\mathcal{F}^{(l)}_{i}$ is locally finitely presented over $C\times S$ and
flat over $S$, because it fits into the short exact sequence
$0\rightarrow\mathcal{E}|_{C\times
S}\rightarrow\mathcal{F}^{(l)}_{i}\rightarrow(q_{i}\times
id_{S})_{*}\left(\mathcal{P}^{(l)}_{i}\right)\,\rightarrow 0$ and the other
two sheaves in the sequence are finitely presented over $C\times S$ and flat
over $S$. By Lemma 4.11, it suffices to show that for every point $s\in S$ we
have that $\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is flat over $C_{s}$. Since $C_{s}$
is a smooth curve, this is equivalent to proving that
$\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is torsion free. Since
$\mathcal{E}(x_{i})|_{C\times
S}\,/\,\mathcal{F}^{(l)}_{i}\,\cong\,(q_{i}\times
id_{S})_{*}\left(\mathcal{E}|_{x_{i}\times
S}\,/\,\mathcal{P}_{i}^{(l)}\right)$ is $S$-flat, the inclusion
$\mathcal{F}_{i}^{(l)}\subset\mathcal{E}(x_{i})|_{C\times S}$ remains a
monomorphism after passing to the fiber $C_{s}$. Hence
$\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is a subsheaf of the locally free sheaf
$\mathcal{E}(x_{i})|_{C_{s}}$. We conclude that
$\mathcal{F}_{i}^{(l)}|_{C_{s}}$ is torsion free, as desired. ∎

We want to leverage the quasicompactness of the classical Harder-Narasimhan
strata for the moduli of vector bundles in order to prove quasicompactness of
HN strata for parabolic vector bundles. As a first step, we need to understand
how the map $Forget$ interacts with the strata in the source and target. This
is achieved by the following lemma.

######  Lemma 6.8.

Let $P$ be a parabolic HN datum. There exists a finite set
$F(P)\subset\frac{1}{n!}\mathbb{Z}$ of classical HN data of rank $n$ such that
for all fields $K\supset k$ and all parabolic vector bundles
$\mathcal{W}\in\text{Bun}_{\mathcal{V}}^{\leq P}(K)$, the classical HN datum
of the vector bundle $Forget(\mathcal{W})$ belongs to $F(P)$.

###### Proof.

Suppose that $P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$. Let $K\supset k$ be a field
extension. Let
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle of type $\mathcal{V}$ over $C_{K}$. Let
$HN(\mathcal{W})=(\nu_{1},\nu_{2},\cdots,\nu_{n})$. Suppose that
$HN(\mathcal{W})\leq P$. This means in particular that $\mu_{1}\geq\nu_{1}$.
Let
$0=\,_{0}\mathcal{F}^{(0)}\subset\,_{1}\mathcal{F}^{(0)}\,\subset\,_{2}\mathcal{F}^{(0)}\,\subset\cdots\,\subset\,_{l}\mathcal{F}^{(0)}=\mathcal{F}^{(0)}$
be the (classical) Harder-Narasimhan filtration for $\mathcal{F}^{(0)}$.
Suppose that the classical Harder-Narasimhan datum for $\mathcal{F}^{(0)}$ is
given by $HN(\mathcal{F}^{(0)})=(\xi_{1},\xi_{2},\cdots,\xi_{n})$.

Define
${}_{1}\mathcal{F}_{i}^{(m)}\vcentcolon=\,_{1}\mathcal{F}^{(0)}(x_{i})\,\cap\,\mathcal{F}_{i}^{(m)}$.
The diagram of sheaves ${}_{1}\mathcal{W}$ given by

${}_{1}\mathcal{W}=\left[\;{}_{1}\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,_{1}\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;_{1}\overset{a_{i}^{(N_{i})}}{\subset}\,_{1}\mathcal{F}_{i}^{(N_{i})}=_{1}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$

is a parabolic subbundle of $\mathcal{W}$, because any subsheaf of a locally
free sheaf on $C_{K}$ is locally free. By the proof of Proposition 2.18, we
know that $\nu_{1}$ is the maximal slope among subbundles of $\mathcal{W}$. In
particular, $\mu(_{1}\mathcal{W})\leq\nu_{1}\leq\mu_{1}$. By Lemma 2.9, we
know that $\mu(_{1}\mathcal{F}^{(0)})\leq\mu(_{1}\mathcal{W})$. So we have
$\mu_{1}\geq\xi_{1}$.

Next we will bound $\xi_{n}$. By Lemma 2.9, we have
$\text{deg}\,\mathcal{F}^{(0)}\geq\text{deg}\,\mathcal{W}-n|I|$. By additivity
of degree for vector bundles in short exact sequences, we have
$\text{deg}\,\mathcal{F}^{(0)}=\sum_{j=1}^{n}\xi_{j}$. Similarly, additivity
of degree for parabolic bundles gives
$\text{deg}\,\mathcal{W}=\sum_{j=1}^{n}\nu_{j}$. Replacing these in the
inequality above yields $\sum_{j=1}^{n}\xi_{j}\geq\sum_{j=1}^{n}\nu_{j}-n|I|$.
But we know that $\xi_{1}\geq\xi_{2}\geq\cdots\geq\xi_{n}$. Therefore,
$\sum_{j=1}^{n}\xi_{j}\leq(n-1)\xi_{1}+\xi_{n}$. Also, we know that
$HN(\mathcal{W})\leq P$. This means in particular that
$\sum_{j=1}^{n}\nu_{j}=\sum_{j=1}^{n}\mu_{j}$. Hence the inequality above
becomes $(n-1)\xi_{1}+\xi_{n}\geq\sum_{j=1}^{n}\mu_{j}-n|I|$.

We have seen that $\xi_{1}\leq\mu_{1}$. So we can rearrange the last
inequality to obtain $\xi_{n}\geq\sum_{j=1}^{n}\mu_{j}-n|I|-(n-1)\mu_{1}$.
Hence, we get the uniform bounds

$\mu_{1}\geq\xi_{1}\geq\xi_{2}\geq\cdots\geq\xi_{n}\geq\sum_{j=1}^{n}\mu_{j}-n|I|-(n-1)\mu_{1}.$

Since the $\xi_{j}$s must lie in the lattice $\frac{1}{n!}\mathbb{Z}$, there
are finitely many possibilities for
$HN(\mathcal{F}^{(0)})=(\xi_{1},\xi_{2},\cdots,\xi_{n})$. ∎

######  Corollary 6.9.

Let $P$ a parabolic HN datum. There exists finitely many classical HN data
${Q_{j}}$ such that the restriction $Forget:\text{Bun}_{\mathcal{V}}^{\leq
P}\longrightarrow\text{Bun}_{\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
GL}\crcr}}}}_{n}}(C)}$ factors through the open inclusion
$\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq
Q_{j}}(C)\hookrightarrow\text{Bun}_{\text{GL}_{n}}(C)$.

###### Proof.

We can take the finite set $\\{Q_{j}\\}$ to be $F(P)$ as in Lemma 6.8 above.
It suffices to show that for any $k$-scheme $T$ and a vector bundle
$\mathcal{E}$ of rank $n$ on $C\times T$, the fiber product
$Forget:\text{Bun}_{\mathcal{V}}^{\leq
P}\times_{\text{Bun}_{\text{GL}_{n}}(C)}\,T\,\rightarrow T$ factors through
the open subset $\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq
Q_{j}}(C)\times_{\text{Bun}_{\text{GL}_{n}}(C)}\,T$ of $T$.

In order to show that it factors through an open subset, it suffices to show
that it does so set theoretically. So we can check it at the level of points.
We have to show that if $t\in T$ is such that $\mathcal{E}|_{C_{t}}$ does not
satisfy $HN(\mathcal{E}|_{C_{t}})\leq Q_{j}$ for some $Q_{j}\in F(P)$, then
there is no field extesion $K\supset\kappa(t)$ and parabolic vector bundle
$\mathcal{W}\in\text{Bun}_{\mathcal{V}}^{\leq P}(K)$ satisfying
$Forget(\mathcal{W})\cong\mathcal{E}|_{C_{K}}$. This is just a restatement of
Lemma 6.8, because the classical HN type of a vector bundle is preserved under
scalar extension (see [Lan75] for a proof of this last fact). ∎

######  Theorem 6.10.

For any parabolic HN datum $P$, the open substack
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is quasicompact.

###### Proof.

By Corollary 6.9, there are finitely many classical HN data $Q_{j}$ such that
the map $Forget$ factors as $\text{Bun}_{\mathcal{V}}^{\leq
P}\longrightarrow\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq Q_{j}}(C)$. By
Proposition 6.5, this map is the composition of an open immersion and a proper
schematic map. Since everything is locally of finite type over $k$, this map
is quasicompact. Now Theorem 6.3 tells us that
$\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq Q_{j}}(C)$ is a finite union of
quasicompact open substacks of $\text{Bun}_{\text{GL}_{n}}(C)$. Therefore
$\cup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq Q_{j}}(C)$ is itself a quasicompact
algebraic stack. We conclude that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is
quasicompact. ∎

##  7 Completeness of strata

In this section we deal with discrete valuation rings $R$ over $k$. We always
write $\eta$ (resp. $s$) for the generic (resp. special) point of
$\text{Spec}(R)$. We will also work with the base-change $C_{R}$. We denote by
$j:C_{\eta}\hookrightarrow C_{R}$ the open immersion of the generic fiber, and
$\iota:C_{s}\hookrightarrow C_{R}$ the closed immersion of the special fiber.

######  Definition 7.1.

Let $\mathfrak{X}$ be a stack of finite type over $k$. We say that
$\mathcal{X}$ is complete over $k$ if for all complete discrete valuation
$k$-algebras $R$, it satisfies the following lifting criterion. For every map
$f:\eta\rightarrow\mathfrak{X}$ there exists a morphism
$g:\text{Spec}\,R\rightarrow\mathfrak{X}$ making the following diagram commute

$\textstyle{\text{Spec}\,R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\eta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathfrak{X}}$

The goal of this section is to show Theorem 7.6, which states that all the HN-
strata $\text{Bun}_{\mathcal{V}}^{\leq P}$ are complete over $k$. Let us
briefly describe some of the work done in this direction. Mehta and Seshadri
[MS80] prove completeness of the moduli of parabolic vector bundles that are
semistable of parabolic degree $0$. On the other hand, Heinloth [Hei17, Remark
3.20] notes that the valuative criterion for universal closedness [Sta, Tag
0CLK] can be proven for the semistable locus of the moduli stack of torsors
for a parahoric Bruhat-Tits group scheme (under some mild tameness conditions
on the generic fiber). Recall that the semistable locus consits of the union
of the minimal open strata. In our case these are given by
$\text{Bun}_{\mathcal{V}}^{\leq P}$, where $P=(r,r,r,\cdots,r)$ for some real
number $r$. We note that this definition of completeness is slightly stronger
than the valuative criterion for universal closedness, because we do not
require passing to an extension of the fraction field.

We extend the proof in [MS80] so that it applies to unstable strata of
arbitrary degree. First we need a series of lemmas.

######  Lemma 7.2.

Let $Q$ be a parabolic HN datum of rank $n$. Consider the set $B_{Q}$ of
parabolic HN data $P$ satisfying

  1. (a)

$P\leq Q$

  2. (b)

$P=HN(\mathcal{W})$ for some $\mathcal{W}$ of rank $n$ in
$\text{Vect}_{\overline{\lambda}}$

The set $B_{Q}$ is finite.

###### Proof.

Set $Q=(\xi_{1},\xi_{2},\cdots,\xi_{n})$. Choose
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})\in B_{Q}$. Define the finite set of real
numbers

$X\vcentcolon=\left\\{-\sum_{i\in
I}\sum_{m=1}^{N_{i}}b_{i}^{(m)}\lambda_{i}^{(m)}\;\mid\;0\leq b_{i}^{(m)}\leq
n\text{\; for $i\in I$ and $1\leq m\leq N_{i}$}\right\\}$

Let $\mathcal{W}$ in $\text{Vect}_{\overline{\lambda}}$ such that
$HN(\mathcal{W})=P$. Fix $1\leq k\leq n$. By definition, we have that
$\mu_{k}=\mu(\mathcal{P})$ for some parabolic subbundle
$\mathcal{P}\subset\mathcal{W}$. If we have

$\mathcal{P}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$

then by definition
$\mu_{k}=\frac{1}{\text{rank}\,\mathcal{P}}\left(\text{deg}\;\mathcal{F}^{(0)}+\sum_{i\in
I}\left(n-\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,b_{i}^{(j)}\right)\,\right)$.
So $\mu_{k}$ belongs to the discrete subset of real numbers
$\frac{1}{n!}\left(\mathbb{Z}+X\right)$. Since $P\leq Q$, we have
$\mu_{1}\leq\xi_{1}$. By definition, we must also have
$\sum_{j=1}^{n}\xi_{j}=\sum_{j=1}^{n}\mu_{j}$. Since $\mu_{j}\leq\mu_{1}$ for
all $j$, we know that
$\sum_{j=1}^{n-1}\mu_{j}\leq(n-1)\mu_{1}\leq(n-1)\xi_{1}$. So we get the
inequality

$\sum_{j=1}^{n}\xi_{j}=\sum_{j=1}^{n}\mu_{j}\,\leq\,(n-1)\xi_{1}+\mu_{n}$

We conclude that we have the uniform bounds
$\sum_{j=1}^{n}\xi_{j}-(n-1)\xi_{1}\,\leq\,\mu_{k}\,\leq\,\xi_{1}$ for all
$k$. Since $\mu_{k}$ belongs to a discrete set, this shows that there are
finitely many possible values for each $\mu_{k}$. ∎

In order to prove completeness, we need to be able to extend a given parabolic
vector bundle on $C_{\eta}$. This is what the next lemma achieves.

######  Lemma 7.3.

Let $R$ be a discrete valuation $k$-algebra. Let $\mathcal{P}$ be a parabolic
vector bundle over $C_{\eta}$. There exists a parabolic vector bundle
$\mathcal{W}$ over $C_{R}$ such that
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$.

###### Proof.

Suppose that
$\mathcal{P}=\left[\;\mathcal{P}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{P}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{P}_{i}^{(N_{i})}=\mathcal{P}^{(0)}(x_{i})\;\right]_{i\in
I}$. It is well known that exists a vector bundle $\mathcal{F}^{(0)}$ over
$C_{R}$ such that $\mathcal{F}^{(0)}|_{C_{\eta}}\cong\mathcal{P}^{(0)}$ (for
example see Proposition 6 in [Lan75]). Define $\mathcal{F}^{(m)}_{i}$ to be
the kernel

$\mathcal{F}^{(m)}_{i}\vcentcolon=\text{Ker}\,\left(\mathcal{F}^{(0)}(x_{i})\xrightarrow{unit}j_{*}\mathcal{P}^{(0)}(x_{i})\twoheadrightarrow
j_{*}\left(\mathcal{P}^{(0)}(x_{i})\,/\,\mathcal{P}^{(m)}_{i}\right)\,\right)$

We have that
$\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\,\subset\,j_{*}\left(\mathcal{P}^{(0)}(x_{i})\,/\,\mathcal{P}^{(m)}_{i}\right)$
is $R$-torsion free, and hence $R$-flat. By Lemma 5.7, $\mathcal{F}_{i}^{(m)}$
is a vector bundle on $C_{R}$. Also, we have a short exact sequence

$0\longrightarrow\,\mathcal{F}^{(m)}_{i}\,/\,\mathcal{F}^{(m-1)}_{i}\,\longrightarrow\,\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m-1)}_{i}\,\longrightarrow\,\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m)}_{i}\,\longrightarrow
0$

By the $R$-flatness of $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m-1)}_{i}$
and $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}^{(m)}_{i}$ plus the Tor long
exact sequence at each stalk we see that
$\mathcal{F}^{(m)}_{i}\,/\,\mathcal{F}^{(m-1)}_{i}$ is $R$-flat. We conclude
that
$\mathcal{W}\vcentcolon=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle over $C_{R}$. By construction
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$. ∎

######  Lemma 7.4.

Let $\mathcal{W}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. Let $\mathcal{U}$ be the maximal
destabilizing parabolic subbundle of $\mathcal{W}$. Set
$\mathcal{Q}\vcentcolon=\mathcal{W}\,/\,\mathcal{U}$. Suppose that there is a
parabolic vector bundle $\mathcal{W}_{1}$ fitting in a short exact sequence
$0\longrightarrow\mathcal{Q}\longrightarrow\mathcal{W}_{1}\longrightarrow\mathcal{U}\longrightarrow
0$. Then, $HN(\mathcal{W}_{1})\leq HN(\mathcal{W})$.

###### Proof.

Set $P=(\mu_{1},\cdots,\mu_{n})\vcentcolon=HN(\mathcal{W})$. Suppose for the
sake of contradiction that $HN(\mathcal{W}_{1})\nleq P$. The short exact
sequence above show that
$\text{deg}\,\mathcal{W}_{1}=\text{deg}\,\mathcal{W}=\sum_{j=1}^{n}\mu_{j}$.
So we can use Lemma 5.2 to conclude that there is a $P$-destabilizing
parabolic subbundle $\mathcal{M}$ of $\mathcal{W}_{1}$. Let us denote by
$\mathcal{L}$ the image of the composition
$\mathcal{M}\rightarrow\mathcal{W}_{1}\twoheadrightarrow\mathcal{U}$. We have
a short exact sequence
$0\longrightarrow\mathcal{Q}\,\cap\,\mathcal{M}\longrightarrow\mathcal{M}\longrightarrow\mathcal{L}\longrightarrow
0$ of parabolic vector bundles. Since $\mathcal{U}$ is semistable and
$\mathcal{L}\subset\mathcal{U}$ is a parabolic subbundle, we have
$\mu(\mathcal{L})\leq\mu(\mathcal{U})$. We conclude that
$\deg(\mathcal{L})\leq\text{rank}\,\mathcal{L}\cdot\mu_{1}=\sum_{j=1}^{\text{rank}\,\mathcal{L}}\mu_{j}$.

We can also look at the decompostion of $\mathcal{Q}\,\cap\,\mathcal{M}$
obtained by intersecting with the Harder-Narasimahn filtration of
$\mathcal{Q}$. By a similar argument we conclude that
$\text{deg}\,(\mathcal{Q}\,\cap\,\mathcal{M})\leq\sum_{j=\text{rank}\,\mathcal{U}+1}^{\text{rank}\,(\mathcal{Q}\,\cap\,\mathcal{M})+\text{rank}\,\mathcal{U}}\mu_{j}$.
We can put these together to obtain

$\text{deg}\,\mathcal{M}\leq\sum_{j=1}^{\text{rank}\,\mathcal{L}}\mu_{j}\,+\,\sum_{j=\text{rank}\,\mathcal{U}+1}^{\text{rank}\,(\mathcal{Q}\,\cap\,\mathcal{M})+\text{rank}\,\mathcal{U}}\mu_{j}$

Since the $\mu_{j}$ are decreasing, the right hand side is less than or equal
to $\sum_{j=1}^{\text{rank}\,\mathcal{M}}\mu_{j}$. By definition, this
contradicts the fact that $\mathcal{M}$ is $P$-destabilizing. ∎

######  Proposition 7.5.

Let $R$ be a complete discrete valuation ring over $k$. Let $P$ be a HN datum.
Let $\mathcal{P}$ be a parabolic vector bundle of rank $n$ over $C_{\eta}$.
Suppose that $HN(\mathcal{P})=P$. Then there exists a parabolic bundle
$\mathcal{W}$ over $C_{R}$ such that $\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$
and $HN\left(\mathcal{W}|_{C_{s}}\right)=P$.

###### Proof.

Suppose $\mathcal{P}$ has Harder-Narasimhan filtration
$0\subset\,\mathcal{P}_{1}\,\subset\,\cdots\,\subset\,\mathcal{P}_{l}=\mathcal{P}$.
By Lemma 7.3, there exists a parabolic vector bundle $\mathcal{W}$ such that
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$. Let $\overline{\mathcal{U}}$ be the
maximal destabilizing parabolic subbundle of $\mathcal{W}|_{C_{s}}$. Write
$\overline{\mathcal{Q}}\vcentcolon=\mathcal{W}|_{C_{s}}\,/\,\overline{\mathcal{U}}$.
We have a map $g$ of diagrams of sheaves given by the composition
$g:\,\mathcal{W}\mathrel{\ooalign{$\xrightarrow[\mkern 4.0mu]{unit\mkern
4.0mu}$\cr$\rightarrow\mkern
4.0mu$}}\iota_{*}\mathcal{W}|_{C_{s}}\twoheadrightarrow\iota_{*}\overline{\mathcal{Q}}$.

Define $\mathcal{W}_{1}\vcentcolon=\text{Ker}(g)$. We have a short exact
sequence of diagrams of sheaves
$0\longrightarrow\mathcal{W}_{1}\longrightarrow\mathcal{W}\longrightarrow\iota_{*}\overline{\mathcal{Q}}\longrightarrow
0$. The pullback
$\mathcal{W}_{1}|_{C_{\eta}}\hookrightarrow\mathcal{W}|_{C_{\eta}}$ becomes an
isomorphism. In order to pass to the special fiber we use the Tor long exact
sequence for each short exact sequence of sheaves in the diagrams. We get an
exact sequence of diagrams of sheaves over the special fiber
$0\longrightarrow\overline{\mathcal{Q}}\longrightarrow\mathcal{W}_{1}|_{C_{s}}\longrightarrow\mathcal{W}|_{C_{s}}\longrightarrow\overline{\mathcal{Q}}\longrightarrow
0$. To see this recall that
$\text{Tor}^{1}_{\mathcal{O}_{s}}(\mathcal{O}_{s},\,\mathcal{M})$ is given by
the $R$-torsion submodule of $\mathcal{M}$ for any $R$-module $M$. This can be
applied to the stalks of the short exact sequence of sheaves above.

We therefore get short exact sequences

$0\longrightarrow\overline{\mathcal{Q}}\longrightarrow\mathcal{W}_{1}|_{C_{s}}\longrightarrow\overline{\mathcal{U}}\longrightarrow
0$
$0\longrightarrow\overline{\mathcal{U}}\longrightarrow\mathcal{W}|_{C_{s}}\longrightarrow\overline{\mathcal{Q}}\longrightarrow
0$

Using this we see that $\mathcal{W}_{1}|_{C_{s}}$ is a vector bundle over
$C_{s}$. All the sheaves in the diagram
$\mathcal{W}_{1}\hookrightarrow\mathcal{W}$ are $R$-torsion free. Therefore
they are $R$-flat. We can then use Lemma 4.11 and an argument similar to the
one in Lemma 7.3 to conclude that $\mathcal{W}_{1}$ is a parabolic vector
bundle. Lemma 7.4 implies that $HN(\mathcal{W}_{1}|_{C_{s}})\leq
HN(\mathcal{W}|_{C_{s}})$.

Now we can replace $\mathcal{W}$ with $\mathcal{W}_{1}$ and iterate the same
construction. By Lemma 7.2, the Harder-Narasimhan datum
$HN(\mathcal{W}|_{C_{S}})$ can’t decresase indefinitely. So it must eventually
stabilize. Therefore, we can assume without loss of generality that
$HN(\mathcal{W}_{1}|_{C_{s}})=HN(\mathcal{W}|_{C_{s}})$ in the construction
above. Let $\overline{\mathcal{U}}_{1}$ be the maximal destabilizing parabolic
subbundle of $\mathcal{W}_{1}|_{C_{s}}$. Since
$HN(\mathcal{W}_{1}|_{C_{s}})=HN(\mathcal{W}|_{C_{s}})$, we must have that
$\overline{\mathcal{U}}$ and $\overline{\mathcal{U}}_{1}$ have the same rank
and slope. By Lemma 2.17, we see that $\overline{\mathcal{U}}_{1}$ maps
isomorphically to $\overline{\mathcal{U}}$ under the surjection
$\mathcal{W}_{1}\twoheadrightarrow\overline{\mathcal{U}}$. So the short exact
sequence
$0\longrightarrow\overline{\mathcal{Q}}\longrightarrow\mathcal{W}_{1}|_{C_{s}}\longrightarrow\overline{\mathcal{U}}\longrightarrow
0$ splits. Hence we see that
$\mathcal{W}_{1}|_{C_{s}}\cong\overline{\mathcal{U}}\oplus\overline{\mathcal{Q}}$.

Repeating this construction for $\mathcal{W}_{1}$, we can obtain a parabolic
suboject $\mathcal{W}_{2}\hookrightarrow\mathcal{W}_{1}$ such that
$\mathcal{W}_{2}|_{C_{s}}\cong\overline{\mathcal{U}}\oplus\overline{\mathcal{Q}}$.
We can iterate this construction to get an infinite chain of parabolic
subojects

$\cdots\,\mathcal{W}_{n}\,\xhookrightarrow{f_{n}}\,\mathcal{W}_{n-1}\,\xhookrightarrow{f_{n-1}}\,\cdots\,\mathcal{W}_{1}\,\xhookrightarrow{f_{1}}\mathcal{W}$

such that
$\mathcal{W}_{n}|_{C_{s}}\cong\overline{\mathcal{U}}\oplus\overline{\mathcal{Q}}$.
The pullback
$\iota^{*}f_{n}:\mathcal{W}_{n}|_{C_{s}}\rightarrow\mathcal{W}_{n-1}|_{C_{s}}$
induces an isomorphims between the corresponding maximal destabilizing
parabolic subbundles $\overline{\mathcal{U}}$. The proof of Lemma 3.5 in
[MS80] now implies that there exists a parabolic subbundle $\mathcal{U}$ of
$\mathcal{W}$ over $C_{R}$ such that $\mathcal{U}|_{C_{s}}$ is the maximal
destabilizing parabolic subbundle $\overline{\mathcal{U}}$ of
$\mathcal{W}|_{C_{s}}$. By local constancy of slope we know that
$\mu(\mathcal{U}|_{C_{\eta}})=\mu(\overline{\mathcal{U}})$. But
$\mathcal{U}|_{C_{\eta}}$ is a parabolic subbundle of $\mathcal{P}$. Hence the
definition of the maximal destabilizing parabolic subbundle $\mathcal{P}_{1}$
of $\mathcal{P}$ implies that
$\mu(\mathcal{U}|_{C_{\eta}})\leq\mu(\mathcal{P}_{1})$. On the other hand, we
have that $HN(\mathcal{P})\leq HN(\mathcal{W}|_{C_{s}})$ by Proposition 5.6.
So we must actually have $\mu(\overline{U})=\mu(\mathcal{P}_{1})$. The
definition of the maximal destabilizing parabolic subbundle $\mathcal{P}_{1}$
of $\mathcal{P}$ now implies that
$\text{rank}\,\mathcal{U}|_{C_{\eta}}\leq\text{rank}\,\mathcal{P}_{1}$. We can
use again the fact that $HN(\mathcal{P})\leq HN(\mathcal{W}|_{C_{s}})$ to
conclude that we must actually have
$\text{rank}\,\mathcal{U}|_{C_{\eta}}=\text{rank}\,\overline{\mathcal{U}}=\text{rank}\,\mathcal{P}_{1}$.
Hence $\mathcal{U}|_{C_{\eta}}=\mathcal{P}_{1}$, and so the subbundle
$\mathcal{U}$ restricts to the maximal destabilizing parabolic subbundle at
both the special and the generic fiber.

Consider now the parabolic quotient
$\mathcal{M}\vcentcolon=\mathcal{W}\,/\,\mathcal{U}$. We can repeat the same
construction as above to obtain a parabolic subobject
$\mathcal{K}\hookrightarrow\mathcal{M}$ such that

  1. 1.

$\mathcal{K}|_{C_{\eta}}\cong\mathcal{P}\,/\,\mathcal{P}_{1}$.

  2. 2.

There exists a parabolic subbundle $\mathcal{L}$ of $\mathcal{K}$ over $C_{R}$
that restricts to the maximal destabilizing parabolic subbundle at both the
special and the generic fiber.

We can consider the corresponding parabolic suboject
$\mathcal{W}^{1}\hookrightarrow\mathcal{W}$ containing $\mathcal{U}$ such that
$\mathcal{W}^{1}\,/\,\mathcal{U}=\mathcal{M}$. Similarly we can lift
$\mathcal{L}$ to a subbundle $\mathcal{U}^{1}$ of $\mathcal{W}^{1}$ containing
$\mathcal{U}$. By construction the two step filtration
$\mathcal{U}\subset\mathcal{U}^{1}\subset\mathcal{W}^{1}$ restricts to the
first two elements of the Harder-Narasimhan filtration at both the special and
the generic fiber. Now we can repeat the construction with
$\left(\mathcal{W}^{1},\,\mathcal{U}^{1}\right)$ in place of
$\left(\mathcal{W},\,\mathcal{U}\right)$.

Since the length of the Harder-Narasimhan filtration of $\mathcal{P}$ is
finite, this process must eventually stop. We end up with a parabolic vector
bundle $\mathcal{W}$ over $C_{R}$ such that
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$, and $\mathcal{W}$ admits a
filtration by parabolic subbundles
$0\subset\mathcal{U}\subset\mathcal{U}_{1}\subset\cdots\subset\mathcal{W}^{l}$
that restricts to the Harder-Narasimhan filtration at both the special and the
generic fiber. By local constancy of slope, this implies that
$HN(\mathcal{W}^{l}|_{C_{s}})=HN(\mathcal{P})=P$. ∎

######  Theorem 7.6.

Let $P$ be a parabolic HN datum. The moduli stack
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is complete over $k$.

###### Proof.

Let $R$ be a complete discrete valuation ring over $k$. Let
$\mathcal{P}:\eta\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$ be a parabolic
vector bundle over $C_{\eta}$ such that $HN(\mathcal{P})\leq P$. By
Proposition 7.5, there exists a parabolic vector bundle $\mathcal{W}$ of type
$\mathcal{V}$ over $C_{R}$ such that
$HN(\mathcal{W}|_{C_{s}})=HN(\mathcal{P})\leq P$ and
$\mathcal{W}|_{C_{\eta}}\cong\mathcal{P}$. This means that $\mathcal{W}$
represents a morphism
$\mathcal{W}:\text{Spec}\,R\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$
lifting $\mathcal{P}:\eta\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$. ∎

##  8 Harder-Narasimhan filtrations in families

In this section we use the filtration schemes $Fil^{\alpha}_{\mathcal{V}}$
defined in Section 4 in order to give the structure of an algebraic stack to
the locus $\text{Bun}^{=P}_{\mathcal{V}}$ for each HN datum $P$. Since we have
proved that conjecture holds for parabolic vector bundles in arbitrary
characteristic (Proposition 2.31), we have established the result in its
strongest form that
$\text{Bun}_{\mathcal{V}}^{=P}\hookrightarrow\text{Bun}_{\mathcal{V}}^{\leq
P}$ is a closed immersion for every field $k$. See Theorem 8.6 below. This
idea of schematic Harder-Narasimhan stratifications first appeared in [Nit11]
for the moduli of sheaves. Gurjar and Nitsure proved an analogous result for
the moduli stack of principal $G$-bundles on a curve over a field of
characteristic $0$ [GN14], and later for higher dimensional varieties [GN18,
GN16].

We start with a technical definition.

######  Definition 8.1.

Let $T$ be a $k$-scheme. Let $\mathcal{W}$ be a parabolic vector bundle over
$C\times T$. Let $\alpha$ be a filtration datum, and let $P$ be a HN datum. We
write $HN_{\mathcal{W}}(\alpha)=P$ if for all points $x\in
Fil^{\alpha}_{\mathcal{W}}$ we have

  1. (a)

$HN(\mathcal{W}|_{C_{x}})=P$

  2. (b)

$x$ represents the Harder-Narasimhan filtration of $\mathcal{W}|_{C_{x}}$.

######  Lemma 8.2.

Let $T$ be a connected $k$-scheme. Let $\mathcal{W}$ be a parabolic vector
bundle over $C\times T$. Let $P$ be a HN datum. Fix a point $t\in T$. Suppose
that $HN(\mathcal{W}|_{C_{t}})=P$. Let
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{W}|_{C_{t}}$
denote the Harder-Narasimhan filtration of $\mathcal{W}|_{C_{t}}$. Let
$\alpha=\psi(\mathcal{W}_{j})$ denote the associated filtration datum, as in
Definition 4.18. Then, we have $HN_{\mathcal{W}}(\alpha)=P$.

###### Proof.

Suppose that
$\alpha=\left(\left({}_{j}b_{i}^{(m)}\right)_{\begin{subarray}{c}1\leq j\leq
l-1\\\ i\in I\\\ 1\leq m\leq
N_{i}\end{subarray}},\;\left({}_{j}P\right)_{1\leq j\leq l-1}\right)$. Choose
$x\in Fil^{\alpha}_{\mathcal{W}}$. Let
$0=\mathcal{P}_{0}\subset\mathcal{P}_{1}\subset\cdots\,\subset\mathcal{P}_{l-1}\subset\mathcal{P}_{l}=\mathcal{W}|_{C_{x}}$
be the filtration of $\mathcal{W}|_{C_{x}}$ corresponding to $x$. We know that
$\theta(\mathcal{P}_{j})=\theta(\mathcal{W}_{j})=(_{j}b_{i}^{(m)},\,_{j}P)$
for all $1\leq j\leq l-1$. By connectedness of $T$, we know that
$\theta(\mathcal{P}_{l})=\theta(\mathcal{W}|_{C_{x}})=\theta(\mathcal{W}|_{C_{t}})=\theta(\mathcal{W}_{l})$
(see Remark 4.14). Since the rank and slope of a parabolic vector bundle can
be determined from its associated parabolic Quot datum, we conclude that
$\mu(\mathcal{P}_{j})=\mu(\mathcal{W}_{j})$ and
$\text{rank}\,\mathcal{P}_{j}=\text{rank}\,\mathcal{W}_{j}$ for all $1\leq
j\leq l$. Since $\\{\mathcal{W}_{j}\\}$ is a Harder-Narasimhan filtration, we
must have that $\\{\mathcal{P}_{j}\\}$ is a Harder-Narasimhan filtration. We
have $HN(\mathcal{W}|_{C_{x}})=HN(\mathcal{W}|_{C_{t}})=P$, because both
filtrations have the same ranks and slopes. ∎

######  Proposition 8.3.

Let $\mathcal{V}$ be a parabolic vector bundle of rank $n$ over $C$. Let
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum. Let $\alpha$ be
filtration datum. Fix a scheme $S$ of finite type over $k$. Let
$\mathcal{W}:\,S\longrightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$ be a
parabolic vector bundle. Suppose that $HN_{\mathcal{W}}(\alpha)=P$. Then, the
structure map $\pi:\,Fil^{\alpha}_{\mathcal{W}}\longrightarrow S$ is a closed
immersion.

###### Proof.

We will use [Sta, Tag 04XV], which says that a morphism that is universally
injective, unramified and proper is a closed immersion. By Proposition 4.17,
we know that $\pi:\,Fil^{\alpha}_{\mathcal{W}}\longrightarrow S$ is of finite
type. Let $t\in S$ such that the fiber
$\pi^{*}(t)=\text{Fil}^{\alpha}_{\mathcal{W}}\times_{S}t$ is nonempty. By
definition, the $\kappa(t)$-scheme
$\pi^{*}(t)=\text{Fil}^{\alpha}_{\mathcal{W}|_{C_{t}}}$ classifies filtrations
of $\mathcal{W}|_{C_{t}}$ with filtration datum $\alpha$. A point
$x\in\pi^{*}(t)$ would give us a filtration of $\mathcal{W}|_{C_{\kappa(x)}}$
with associated HN datum $P$. By Lemma 2.30, we know that
$HN(\mathcal{W}|_{C_{\kappa(x)}})=HN(\mathcal{W}_{C_{t}})\leq P$. Hence, we
must have $HN(\mathcal{W}|_{C_{\kappa(x)}})=P$. So the filtration must be the
Harder-Narasimhan filtration of $\mathcal{W}|_{C_{\kappa(x)}}$. By the
uniqueness of the Harder-Narasimhan filtration for any field extension (Lemma
2.30 again), $\pi^{*}(t)$ has a single point and this point is defined over
$\kappa(t)$. So $\pi^{*}(t)$ must be of the form $\text{Spec}(A)$ for some
local artinian $\kappa(t)$-algebra $A$ with residue field $\kappa(t)$. This
shows that $\pi$ radicial. Hence $\pi$ is universally injective.

We claim that $A=\kappa(t)$. In order to see this, it suffices to show that
$\Omega_{A\,/\,\kappa(t)}^{1}=0$. We have that $\Omega_{A\,/\,\kappa(t)}^{1}$
is dual to the tangent space
$\text{Hom}_{\kappa(t)}\left(\text{Spec}(\kappa(t)[\epsilon]),\,\pi^{*}(t)\right)$.
By Proposition 2.31, the Harder-Narasimhan filtration has no notrivial first
order deformations. This means that
$\text{Hom}_{\kappa(t)}\left(\text{Spec}(\kappa(t)[\epsilon]),\,\pi^{*}(t)\right)=0$.
Therefore $A=\kappa(t)$. We conclude that $\pi$ is unramified.

We are left to show that $\pi$ is proper. We will use the valuative criterion
for properness. The argument is similar to the proof of 5.6. We use the same
notation as in that proof (look there for any piece of unexplained notation).
Let $R$ be a discrete valuation $k$-algebra with generic point $\eta$ and
special point $s$. Let $f:\text{Spec}\;R\rightarrow S$. The fiber product
$\text{Fil}^{\alpha}_{\mathcal{W}}\times_{S}\text{Spec}\;R$ is by definition
the $R$-scheme
$\pi_{R}:\text{Fil}^{\alpha}_{\mathcal{W}|_{C_{R}}}\rightarrow\text{Spec}\;R$.
To simplify notation, let us set
$\mathcal{U}\vcentcolon=\mathcal{W}|_{C_{R}}$. Suppose that we have a section
$p:\eta\rightarrow\text{Fil}^{\alpha}_{\mathcal{U}}$ of $\pi_{R}$ defined over
$\eta$. We want to show that $p$ extends to a unique section over
$\text{Spec}\;R$. Uniqueness follows from the fact that
$\text{Fil}^{\alpha}_{\mathcal{U}}$ is a separated $R$-scheme (Proposition
4.17). We are left to show existence.

By hypothesis, we know that $HN(^{\eta}\mathcal{U}),HN(^{s}\mathcal{U})\leq
P$. The point $p$ is a filtration of ${}^{\eta}\mathcal{U}$ by parabolic
subbundles
$0=\,^{\eta}\mathcal{W}_{0}\,\subset\,^{\eta}\mathcal{W}_{1}\,\subset\,\cdots\,\subset\,^{\eta}\mathcal{W}_{l}=\,^{\eta}\mathcal{U}$
with filtration datum $\alpha$. Set
$k_{j}\vcentcolon=\text{rank}^{\eta}\mathcal{W}_{j}$. Arguing as in the proof
of Proposition 5.6, we get set of inclusions
$0=\,\mathcal{W}_{0}\,\hookrightarrow\,\mathcal{W}_{1}\,\hookrightarrow\,\cdots\,\hookrightarrow\,\mathcal{W}_{l}=\,\mathcal{U}$
of parabolic subobjects over $C_{R}$. These are not parabolic subbundles a
priori. As in the proof of Proposition 5.6, passing to the special fiber
preserves the subobject inclusions
$0=\,^{s}\mathcal{W}_{0}\,\hookrightarrow\,^{s}\mathcal{W}_{1}\,\hookrightarrow\,\cdots\,\hookrightarrow\,^{s}\mathcal{W}_{l}=\,^{s}\mathcal{U}$.

Using Lemma 2.11, we get a filtration of ${}^{s}\mathcal{U}$ by parabolic
subbundles

$0=\,^{s}\mathcal{W}^{sat}_{0}\,\subset\,^{s}\mathcal{W}_{1}^{sat}\,\subset\,\cdots\,\subset\,\mathcal{W}_{l}^{sat}=\,^{s}\mathcal{U}$

with $\mu(^{s}\mathcal{W}_{j})\leq\mu(^{s}\mathcal{W}^{sat}_{j})$ for all
$1\leq j\leq l$. By local constancy of degree we have
$\mu(^{s}\mathcal{W}_{j})=\mu(^{\eta}\mathcal{W}_{j})$. Since $HN(\alpha)=P$,
we have $\mu(^{s}\mathcal{W}_{j})=\mu_{k_{j}}$. So
$\mu_{k_{j}}\leq\mu(^{s}\mathcal{W}^{sat}_{j})$. We know that
$HN(^{s}\mathcal{U})\leq P$, hence we must have
$\mu_{k_{j}}=\mu(^{s}\mathcal{W}^{sat}_{j})$. By Remark 5.8, this implies that
$\mathcal{W}_{j}$ is actually a parabolic subbundle of $\mathcal{U}$ over
$C_{R}$. The filtration
$0=\,\mathcal{W}_{0}\,\subset\,\mathcal{W}_{1}\,\subset\,\cdots\,\subset\,\mathcal{W}_{l}=\,\mathcal{U}$
gives us a section of $\pi_{R}$ over $\text{Spec}\;R$, as desired. ∎

######  Definition 8.4.

Let $S$ be a $k$-scheme and $\mathcal{V}$ be a parabolic vector bundle on
$C\times S$. A relative Harder-Narasimhan filtration (over $S$) is a
filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
by parabolic subbundles $\mathcal{W}_{j}$ over $C\times S$ such that for all
$s\in S$, the restriction
$0=\mathcal{W}_{0}|_{C_{s}}\subset\mathcal{W}_{1}|_{C_{s}}\subset\cdots\,\subset\mathcal{W}_{l-1}|_{C_{s}}\subset\mathcal{W}_{l}|_{C_{s}}=\mathcal{V}|_{C_{s}}$
to the fiber $C_{s}$ is the Harder-Narasimhan filtration of
$\mathcal{V}|_{C_{s}}$.

######  Definition 8.5.

Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Let $P$ be a HN
datum. $\text{Bun}_{\mathcal{V}}^{=P}$ denotes the functor from $k$-schemes
into groupoids given as follows. For a $k$-scheme $S$,
$\text{Bun}_{\mathcal{V}}^{=P}(S)$ is the groupoid of parabolic bundles over
$C\times S$ of type $\mathcal{V}$ along with a relative Harder-Narasimhan
filtration (over $C\times S$) such that for all $s\in S$, we have
$HN(\mathcal{V}|_{C_{s}})=P$.

We will denote by
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}$ the
morphism given by forgetting the relative Harder-Narasimhan filtration.

######  Theorem 8.6.

Let $P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum. The morphism
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}$
exhibits $\text{Bun}_{\mathcal{V}}^{=P}$ as a locally closed substack of
$\text{Bun}_{\mathcal{V}}$.

###### Proof.

By definition
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}$
factors through the open substack $\text{Bun}_{\mathcal{V}}^{\leq P}$. We
shall show that
$\iota:\text{Bun}_{\mathcal{V}}^{=P}\rightarrow\text{Bun}_{\mathcal{V}}^{\leq
P}$ is a closed immersion.

Let $T$ be a connected scheme of finite type over $k$. Let
$\mathcal{W}:T\rightarrow\text{Bun}_{\mathcal{V}}^{\leq P}$ be a parabolic
vector bundle over $C\times T$. Define the set
$T^{=P}\vcentcolon=\left\\{\;t\in
T\;\mid\;HN(\mathcal{W}|_{C_{t}})=P\right\\}$. Recall that for any $t\in T$
and any filtration $\\{\mathcal{W}_{j}^{t}\\}$ of $\mathcal{W}|_{C_{t}}$, we
have an associated filtration datum $\psi(\mathcal{W}^{t}_{j})$ as in
Definition 4.18. In particular, we can do this with the Harder-Narasimhan
filtration $HN\text{Fil}\,\mathcal{W}|_{C_{t}}$ of $\mathcal{W}|_{C_{t}}$.

Let’s define a set
$\mathfrak{S}=\left\\{\;\psi\left(\,HN\text{Fil}\,\mathcal{W}|_{C_{t}}\,\right)\;\mid\;t\in
T^{=P}\right\\}$. We claim that $\mathfrak{S}$ is a finite set. Let $t\in
T^{=P}$ and let
$0=\mathcal{W}_{0}^{t}\subset\mathcal{W}_{1}^{t}\subset\cdots\,\subset\mathcal{W}_{l-1}^{t}\subset\mathcal{W}_{l}^{t}=\mathcal{W}|_{C_{t}}$
be the Harder-Narasimhan filtration of $\mathcal{W}|_{C_{t}}$. Suppose that
${}_{j}\mathcal{W}^{t}=\left[\;{}_{j}\mathcal{F}^{(0)}\,\overset{{}_{j}b_{i}^{(1)}}{\subset}\,_{j}\mathcal{F}_{i}^{(1)}\overset{{}_{j}b_{i}^{(2)}}{\subset}\cdots\;\overset{{}_{j}b_{i}^{(N_{i})}}{\subset}\,_{j}\mathcal{F}_{i}^{(N_{i})}=\,_{j}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. We know that $HN(\mathcal{W}|_{C_{t}})=P$. Therefore the length $l$ of
the filtration and the rank $k_{j}$ of each $\mathcal{W}_{j}^{t}$ do not
depend on $t$. Since there is a finite number of possibilities for the numbers
${}_{j}b_{i}^{(m)}$, it suffices to show that for each $j$ there is a finite
number of possibilities for $\text{deg}\,_{j}\mathcal{F}^{(0)}$. We know that
$\text{deg}\,_{j}\mathcal{W}=k_{j}\,\mu_{k_{j}}$ is fixed. By Lemma 2.9, we
have

$k_{j}\,\mu_{k_{j}}-\,k_{j}\,|I|\leq\text{deg}\,_{j}\mathcal{F}^{(0)}\leq\,k_{j}\,\mu_{k_{j}}$

Hence there are finitely many possible choices for the integer
$\text{deg}\,_{j}\mathcal{F}^{(0)}$. We conclude that the set $\mathfrak{S}$
is finite.

Consider the $T$-scheme
$X\vcentcolon=\sqcup_{\alpha\in\mathfrak{S}}\,\text{Fil}_{\mathcal{V}}^{\alpha}$.
By construction, $X$ represents the fiber product
$\text{Bun}_{\mathcal{V}}^{=P}\times_{\text{Bun}_{\mathcal{V}}^{\leq P}}T$. It
therefore suffices to show that $X\rightarrow T$ is a closed immersion. For
every $\alpha\in\mathfrak{S}$, we have $HN_{\mathcal{W}}(\alpha)=P$ by Lemma
8.2. By Proposition 8.3, $X$ is a finite disjoint union of closed subschemes
of $T$. It therefore suffices to show that if $\alpha\neq\beta$ are two
filtration data in $\mathfrak{S}$, then the closed subschemes
$\text{Fil}_{\mathcal{V}}^{\alpha}$ and $\text{Fil}_{\mathcal{V}}^{\beta}$ are
set theoretically disjoint inside $T$. But we can’t have a point $t\in T$ in
the intersection, since that would violate the uniqueness of the Harder-
Narasimhan filtration for $\mathcal{W}|_{C_{t}}$. ∎

The result that the relative Harder-Narasimhan stacks are locally closed
substacks does not hold for $\text{Bun}_{\mathcal{G}}$ for exceptional
parahoric Bruhat-Tits group schemes $\mathcal{G}$ over $C$ when the
characteristic of $k$ is small. This is because of the failure of Behrend’s
conjecture, even in the case when the group scheme is reductive (see [Hei08]).

We end this section by stating an explicit corollary of Theorem 8.6. A weaker
version of this corollary in the context of coherent sheaves without parabolic
structure can be found in [HL97][Thm. 2.3.2].

######  Corollary 8.7.

Fix a parabolic vector bundle $\mathcal{V}$ over $C$. Let $S$ be a $k$-scheme
and let $\mathcal{W}$ be a parabolic vector bundle of type $\mathcal{V}$ over
$C\times S$. There is a surjective morphism $\bigsqcup_{P}S^{P}\rightarrow S$
where

  1. (i)

The $P$ run over the set of all HN data of rank $n$.

  2. (ii)

Each $S^{P}$ is a locally closed subscheme of $S$.

  3. (iii)

$\mathcal{W}|_{S^{P}}$ admits a relative Harder-Narasimhan filtration.

  4. (iv)

For all $s\in S^{P}$, we have $HN(\mathcal{W}|_{C_{s}})=P$.

###### Proof.

The parabolic vector bundle $\mathcal{W}$ is the same a as morphism
$S\rightarrow\text{Bun}_{\mathcal{V}}$. Theorem 8.6 implies that there is a
surjective map of stacks
$\bigsqcup_{P}\text{Bun}_{\mathcal{V}}^{=P}\,\rightarrow\text{Bun}_{\mathcal{V}}$,
where the $P$ run over the set of HN data of rank $\text{rank}\,\mathcal{V}$.
Moreover, each $\text{Bun}_{\mathcal{V}}^{=P}$ is a locally closed substack of
$\text{Bun}_{\mathcal{V}}$.

Define $S^{P}$ to be the locally closed subscheme of $S$ given by
$S^{P}\vcentcolon=S\times_{\text{Bun}_{\mathcal{V}}}\text{Bun}_{\mathcal{V}}^{=P}$.
We have a fiber product diagram

$\textstyle{\bigsqcup_{P}S^{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup_{P}\text{Bun}_{\mathcal{V}}^{=P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Bun}_{\mathcal{V}}}$

Since the map
$\bigsqcup_{P}\text{Bun}_{\mathcal{V}}^{=P}\,\rightarrow\text{Bun}_{\mathcal{V}}$
is surjective, we conclude that the base-change $\bigsqcup_{P}S^{P}\rightarrow
S$ is surjective as well. Properties $(iii)$ and $(iv)$ follow directly from
the definition of $\text{Bun}_{\mathcal{V}}^{=P}$ above. ∎

##  Appendix A Descent for finite covers

For the convenience of the reader, we give an explicit description of descent
data for certain finite flat covers as done in [Gro95][B.3]. All sheaves
considered are quasicoherent.

Let $\pi:Y\rightarrow X$ be a faithfully-flat finite morphims of schemes such
that $\pi_{*}(\mathcal{O}_{Y})$ is a locally free $\mathcal{O}_{X}$-module. We
want to explicitly describe descent data for quasicoherent sheaves on $Y$
relative to $X$. Since $\pi$ is affine, the functor $\pi_{*}$ sets up an
equivalence between $\text{QCoh}(Y)$ and the category of quasicoherent sheaves
on $X$ with a $\pi_{*}(\mathcal{O}_{Y})$-module structure. Consider the fiber
product:

$\textstyle{Y\times_{X}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;\;\;\;pr_{1}}$$\scriptstyle{pr_{2}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X}$

By the same reasoning, a quasicoherent sheaf on $Y\times_{X}Y$ is the same as
a quasicoherent
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
-module.

Given a $\pi_{*}(\mathcal{O}_{Y})$-module $\mathcal{F}$, we can obtain two
different
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
-modules via pullback:

$pr_{1}^{*}(\mathcal{F})=\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
$pr^{*}_{2}(\mathcal{F})=\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}$

A descent datum for $\mathcal{F}$ would correspond to an homorphism of
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
-modules $\phi:pr_{2}^{*}(\mathcal{F})\rightarrow pr_{1}^{*}(\mathcal{F})$
satisfying a cocycle condition.

By assumption $\pi_{*}(\mathcal{O}_{Y})$ is locally free as a
$\mathcal{O}_{X}$-module, so it is $\mathcal{O}_{X}$-dualizable. Using this
fact plus the tensor-Hom adjunction, we get a chain of isomorphisms:

$\begin{split}\text{Hom}_{\mathcal{O}_{X}}\left(pr_{2}^{*}(\mathcal{F}),\,pr_{1}^{*}(\mathcal{F})\right)&=\,\text{Hom}_{\mathcal{O}_{X}}\left(\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F},\;\;\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\right)\\\
&\cong\,\text{Hom}_{\mathcal{O}_{X}}\left(\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})^{\vee}\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F},\;\;\;\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\right)\\\
&\cong\,\text{Hom}_{\mathcal{O}_{X}}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}\,\right)\end{split}$

What is the image of the set of
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$\-
homomorphisms? Tracing back the chain isomorphisms above, one can check that
this is
$\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}\,\right)$.
Here
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
acts on each End group by $(s\otimes t)\cdot f(x)\vcentcolon=tf(sx)$ for any
sections $s,t$ and $x$ of the corresponding sheaves over a given open set.

We are left to describe the cocycle condition for a given
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\,$-homomorphims
$\phi:pr_{2}^{*}(\mathcal{F})\rightarrow pr_{1}^{*}(\mathcal{F})$. Recall that
the cocycle condition amounts to the commutativity of the following diagram:

$\textstyle{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\otimes\phi}$$\scriptstyle{id\otimes\text{swap}}$$\textstyle{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes
id}$$\textstyle{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes
id}$$\textstyle{\mathcal{F}\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}$

Tracing back the construction of the isomorphim

$\psi:\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(pr_{2}^{*}(\mathcal{F}),\,pr_{1}^{*}(\mathcal{F})\right)\,\xrightarrow{\sim}\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}\,\right)$

we see that the cocycle condition corresponds to $\psi(\phi)$ being an
homomorphism of unital algebras.

Let us give two examples when there is a nice explicit presentation of the
unital algebra $\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$.

######  Example A.1 (Galois descent).

Suppose that $\pi:Y\rightarrow X$ is a Galois cover with Galois group $G$. By
definition, $G$ is a group of $\mathcal{O}_{X}$-linear automorphisms of
$\pi_{*}(\mathcal{O}_{Y})$. Then it can be checked that we have

$\text{End}\left(\,\pi_{*}(\mathcal{O}_{Y})\,\right)=\bigoplus_{g\in
G}\left(1\otimes\pi_{*}(\mathcal{O}_{Y})\right)\,g$

Since everything is $\mathcal{O}_{X}$ coherent, it suffices to check this
equality on fibers, This reduces the claim to Galois theory for separable
field extensions.

By definition of the
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$-module
structure on $\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$, we have
$(s\otimes 1)\cdot g=(1\otimes g(s))\cdot g$ for any $g\in G$ and any section
$s$ of $\pi_{*}(\mathcal{O}_{Y})$ on an open. So in this case descent data
correspond to homomorphisms of groups
$\psi:G\rightarrow\text{Aut}_{\mathcal{O}_{X}}\,\mathcal{F}$ such that
$\psi(g)$ is $g$-semilinear for all $g\in G$.

######  Example A.2 (Inseparable descent/ Frobenius descent).

Let us suppose that $X$ is a scheme over a finite field $\kappa$ of
characteristic $p$ (in other words, $p\,\mathcal{O}_{X}=0$). Let
$\pi:Y\rightarrow X$ as above. Suppose that
$\pi_{*}(\mathcal{O}_{Y})^{p}\subset\mathcal{O}_{X}$, so the cover is of
”exponent 1”. Furthermore, suppose that $\pi_{*}(\mathcal{O}_{Y})$ locally
admits a $p$-basis over $\mathcal{O}_{X}$ (for Noetherian schemes this is
known to be equivalent to $\Omega^{1}_{Y\,/\,X}$ being locally free, see
[Tyc88]).

Let $\mathcal{D}$ be the $\pi_{*}(\mathcal{O}_{Y})$-module of
$\mathcal{O}_{X}$-linear derivations of $\pi_{*}(\mathcal{O}_{Y})$. Here the
$\pi_{*}(\mathcal{O}_{Y})$-module structure comes from the action of the
second coordinate in
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$
when we view $\mathcal{D}$ as a subset of
$\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$. By assumption, we
know that $\mathcal{D}=(\Omega^{1}_{Y\,/\,X})^{\vee}$ is locally free. Observe
that $\mathcal{D}$ has the structure of a restricted p-Lie algebra over
$\mathcal{O}_{X}$. We will denote by $[-,-]$ and $(-)^{(p)}$ the p-Lie algebra
operations in $\mathcal{D}$.

We claim that $\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})$ is
generated as a
$\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})$-algebra
by $\mathcal{D}$ subject to the relations

$(1\otimes s)\cdot\delta\,=\,(1\otimes\delta(s))\cdot\,id\,+\,(s\otimes
1)\cdot\delta$ $\delta\gamma-\gamma\delta\,=\,[\delta,\,\gamma]$
$\delta^{p}\,=\,\delta^{(p)}$

Here $\delta$ and $\gamma$ are sections of $\mathcal{D}$ and $s$ is a section
of $\pi_{*}(\mathcal{O}_{Y})$. The claim is a local statement. Since
everything is $\mathcal{O}_{X}$ coherent, we can check it on fibers. We are
reduced to the case of a finite algebra of exponent 1 over a field. This can
be done by hand by using the result for field extensions ([Jac89] Theorem
8.45) and a computation for the ring of dual numbers.

Let $\mathcal{F}$ be a quasicoherent sheaf on $Y$. The discussion above shows
that a descent datum
$\phi:\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})\,\rightarrow\,\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}$
is equivalent to a map
$\nabla:\,\mathcal{D}\,\rightarrow\,\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}$
of restricted p-Lie algebras over $\mathcal{O}_{X}$ satisfying the Leibniz
condition

$(s\otimes 1)\cdot\nabla(\delta)=(1\otimes\delta(s))\cdot id\,+\,(1\otimes
s)\cdot\nabla(\delta)$

for all sections $\delta$ of $\mathcal{D}$ and sections $s$ of
$\pi_{*}(\mathcal{O}_{Y})$. This is by definition a $X$-relative connection on
$\mathcal{F}$ with vanishing $p$-curvature.

We will use this version of descent in the special case of inseparable field
extensions of exponent 1. This goes back to the work of Jacobson (see [Jac89]
Theorem 8.45). Another example is the case when $\pi$ is a Frobenius morphism.
This is worked out in [Kat70] Theorem 5.1.

Finally, let us describe descent for morphims of quasicoherent sheaves in this
language. Let quasicoherent $\pi_{*}(\mathcal{O}_{Y})$-modules
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, along with descent data

$\phi_{i}\in\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}_{i}\,\right)$

for $i=1,2$. A homomorphims of $\pi_{*}(\mathcal{O}_{Y})$-modules
$f:\mathcal{F}_{2}\rightarrow\mathcal{F}_{2}$ descends if and only if we have
$f^{*}(\phi_{2})=f_{*}(\phi_{1})$, where

$\displaystyle
f^{*}:\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}_{2}\,\right)\,\longrightarrow\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{Hom}_{\mathcal{O}_{X}}(\mathcal{F}_{1},\,\mathcal{F}_{2})\,\right)$
$\displaystyle
f_{*}:\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{F}_{1}\,\right)\,\longrightarrow\,\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{Hom}_{\mathcal{O}_{X}}(\mathcal{F_{1}},\,\mathcal{F}_{2})\,\right)$

are given by precomposing (resp. postcomposing) with $f$. Let us give a few
examples/consequences of this.

######  Example A.3 (Subsheaves).

Let $\mathcal{E}$ be a $\pi_{*}(\mathcal{O}_{Y})$-module along with a descent
datum

$\phi\in\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}\,\mathcal{E}\,\right)$

A subsheaf $\mathcal{F}\subset\mathcal{E}$ descends if and only if for all
sections $s\in\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y})(U)$ over
an open $U\subset X$, we have that $\mathcal{F}|_{U}$ is stable under the
corresponding homomorphism $\phi(s)$.

If $\text{End}_{\mathcal{O}_{X}}$ is generated by global sections, it suffices
to check global sections. This is the case for descent along field extensions
(or base-changes of field extensions).

######  Example A.4 (Diagrams of sheaves).

Let $I$ be a small category. A diagram of sheaves on $Y$ of shape $I$ is a
covariant functor $F:I\rightarrow\text{QCoh}(Y)$. A morphism of diagrams of
sheaves of shape $I$ is a natural transformation. Descent data for a diagram
of sheaves $F$ correspond to homomorphisms of unital algebras

$\phi\in\text{Hom}_{\pi_{*}(\mathcal{O}_{Y})\underset{\mathcal{O}_{X}}{\otimes}\pi_{*}(\mathcal{O}_{Y})}\left(\,\text{End}_{\mathcal{O}_{X}}\,\pi_{*}(\mathcal{O}_{Y}),\,\;\text{End}_{\mathcal{O}_{X}}F\,\right)$

Here $\text{End}_{\mathcal{O}_{X}}F$ is the algebra of endomorphisms of $F$
viewed as a diagram in $\text{QCoh}(X)$ via $\pi_{*}$.

######  Example A.5 (Parabolic vector bundles).

Parabolic vector bundles are a particular case of diagrams of sheaves. Given a
parabolic vector bundle $\mathcal{W}$ over $C\times Y$, descent data
correspond to homomorphisms of unital algebras from
$\text{End}_{\mathcal{O}_{C\times X}}\,(id\times\pi)_{*}(\mathcal{O}_{C\times
Y})$ into the algebra of $\mathcal{O}_{C\times X}$-linear endomorphisms of
$\mathcal{W}$.

##  Appendix B Comparison with previous work

In this appendix we compare the stratification described in this article with
the $\Theta$-stratifications defined in [Hei17] and [AHLH18][§8]. For this
appendix we will freely use the notation from these papers.

For simplicity of notation, we will stick with the case when there is a single
point of degeneration $x\in C(k)$. Our arguments can be modified easily to
deal with the case when there are several points of degeneration. Let’s denote
by $q:x\hookrightarrow C$ the closed immerion of $x$ into $C$. Let
$\mathcal{V}$ be a parabolic vector bundle on $C$ with parabolic structure at
$x$. Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a^{(1)}}{\subset}\,\mathcal{E}^{(1)}\overset{a^{(2)}}{\subset}\cdots\;\overset{a^{(N)}}{\subset}\,\mathcal{E}^{(N)}=\mathcal{E}^{(0)}(x)\;\right]$.
Fix a collection of parabolic weights
$0<\lambda^{(1)}<\lambda^{(2)}<\cdots<\lambda^{(N)}<1$. This collection
defines a Harder-Narasimhan stratification on $\text{Bun}_{\mathcal{V}}$, as
in Definition 3.13. Our goal in this appendix is to explain how, in the case
when $\overline{\lambda}$ is “admissible”, one can view this stratification as
the $\Theta$-stratification coming from a $\mathbb{R}$-line bundle on the
stack $\text{Bun}_{\mathcal{V}}$, as in [AHLH18][§8].

Set $n=\text{rank}\,\mathcal{V}$. By Definition 3.6,
$\text{Bun}_{\mathcal{V}}$ only depends on the type of the parabolic bundle
$\mathcal{V}$. Hence we can, without loss of generality, assume that
$\mathcal{E}^{(0)}=\mathcal{O}_{C}^{\oplus n}$ and

$\mathcal{E}^{(i)}=\mathcal{O}_{C}(x)^{\oplus\sum_{j=1}^{i}a^{(j)}}\,\oplus\,\mathcal{O}_{C}^{\oplus\left(n-\sum_{j=1}^{i}a^{(j)}\right)}$

By Proposition 3.7, we have
$\text{Bun}_{\mathcal{V}}\simeq\text{Bun}_{\text{GL}(\mathcal{V})}$. Set
$\mathcal{G}\vcentcolon=\text{GL}(\mathcal{V})$. Let $\text{Lie}(\mathcal{G})$
denote the Lie algebra of $\mathcal{G}$. Since $\mathcal{G}$ is smooth over
$C$, this Lie algebra is a vector bundle on $C$. Following [Hei17], we define
$\mathcal{L}_{det}$ to be the line bundle on the stack
$\text{Bun}_{\mathcal{G}}$ defined as follows. For any $k$-scheme $T$ and
every map $f:T\rightarrow\text{Bun}_{\mathcal{G}}$ represented by a
$\mathcal{G}_{T}$-torsor $\mathcal{P}$ on $C\times T$, we let

$f^{*}(\mathcal{L}_{det})=\text{det}\left(R(\pi_{T})_{*}\,\mathcal{P}\times^{\mathcal{G}}\text{Lie}(\mathcal{G})\right)^{\vee}$

Here det denotes the determinant in the $K$-theoretic sense. This makes sense
because the derived pushforward is a perfect complex [Sta, Tag 0A1H].

As in [HL14], we let
$\Theta=\left[\,\mathbb{A}^{1}_{k}/\,\mathbb{G}_{m}\,\right]$. We write $0$
for the origin in $\mathbb{A}^{1}_{k}$. We can view $0$ as a point of
$\Theta$. Let $\varphi:\Theta\rightarrow\text{Bun}_{\mathcal{G}}$ be a
filtration of a parabolic vector bundle
$\mathcal{W}\in\text{Bun}_{\mathcal{V}}$. We would like to compute the weight
of the $0$-fiber of $\mathcal{L}_{det}$. Say

$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a^{(1)}}{\subset}\,\mathcal{F}^{(1)}\overset{a^{(2)}}{\subset}\cdots\;\overset{a^{(N)}}{\subset}\,\mathcal{F}^{(N)}=\,\mathcal{F}^{(0)}(x)\;\right]$

Via the Rees construction, a filtration of $\mathcal{W}$ corresponds to a
decreasing $\mathbb{Z}$-filtration of $\mathcal{F}^{(0)}$ by subbundles
$\left(\mathcal{F}^{(0)}_{m}\right)_{m\in\mathbb{Z}}$. More precisely, for all
$m\in\mathbb{Z}$ we have that $\mathcal{F}^{(0)}_{m}$ is a subbundle of
$\mathcal{F}^{(0)}$ and $\mathcal{F}_{m}^{(0)}\supset\mathcal{F}^{(0)}_{m+1}$.
Moreover we require that $\mathcal{F}^{(0)}_{m}=0$ for $m>>0$ and
$\mathcal{F}^{(0)}_{m}=\mathcal{F}^{(0)}$ for all $m<<0$. Alternatively, we
can view this as a filtration of $\mathcal{W}$ by parabolic subbundles
$(\mathcal{W}_{m})_{m\in\mathbb{Z}}$ by setting
$\mathcal{F}^{(i)}_{m}\vcentcolon=\mathcal{F}_{m}^{(0)}\cap\mathcal{F}^{(i)}$
and defining

$\mathcal{W}_{m}\vcentcolon=\left[\;\mathcal{F}_{m}^{(0)}\,\overset{a_{m}^{(1)}}{\subset}\,\mathcal{F}_{m}^{(1)}\overset{a_{m}^{(2)}}{\subset}\cdots\;\overset{a_{m}^{(N)}}{\subset}\,\mathcal{F}_{m}^{(N)}=\,\mathcal{F}_{m}^{(0)}(x)\;\right]$

In order to compute $wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)$, we
will set up a short exact sequence for $\text{Lie}(\mathcal{G})$. There is a
monomorphism of group schemes
$\mathcal{G}\hookrightarrow\text{Aut}(\mathcal{E}^{(0)}(x))$, given by
forgetting the automorphisms of $\mathcal{E}^{(i)}$ for $i<N$. By
[MRR20][Lemma 3.7], we have a short exact sequence of pointed ètale sheaves

$1\,\rightarrow\,\mathcal{G}\,\rightarrow\,\text{Aut}(\mathcal{E}^{(0)}(x)\,\rightarrow\text{Aut}(\mathcal{E}^{(0)}(x)|_{x})\,/\,P\,\rightarrow\,1$

where $P$ is the parabolic subgroup of $\text{Aut}(\mathcal{E}^{(0)}(x)|_{x})$
that preserves the flag

$0\subset\mathcal{E}^{(1)}/\,\mathcal{E}^{(0)}\,\subset\,\mathcal{E}^{(2)}/\,\mathcal{E}^{(0)}\,\subset\cdots\,\subset\mathcal{E}^{(N)}/\,\mathcal{E}^{(0)}\cong\mathcal{E}^{(0)}(x)|_{x}$

We remark that in this case the short exact sequence can be proven by hand.

Under a choice of isomorphism
$\mathcal{E}^{(0)}(x)|_{x}\cong\mathcal{O}^{\oplus n}_{C}(x)|_{x}\cong k^{n}$,
the group $P$ is the standard parabolic subgroup of $\text{GL}_{n}$
corresponding to the partition $(a^{(1)},a^{(2)},\cdots,a^{(N)})$. In other
words, $P$ consists of block upper-triangular matrices in $\text{GL}_{n}$ with
blocks of sizes given by the tuple $(a^{(1)},a^{(2)},\cdots,a^{(N)})$.

It is known that $\text{GL}_{n}/P$ admits sections Zariski locally. Let
$\mathfrak{gl}_{n}$ (resp. $\mathfrak{p}$) denote the Lie algebra of
$\text{GL}_{n}$ (resp. $P$). By looking at points over the dual numbers, one
can see that the short exact sequence of étale sheaves above induces a short
exact sequence of coherent sheaves on $C$

$0\,\rightarrow\,\text{Lie}(\mathcal{G})\,\rightarrow\,\text{Lie}(\text{Aut}(\mathcal{E}^{(0)}(x))\,\rightarrow\,q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})\,\rightarrow\,0$

Note that
$\text{Lie}(\text{Aut}(\mathcal{E}^{(0)}(x))=\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))$.
The group scheme $\mathcal{G}$ acts on both
$\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))$ and
$q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})$ via the adjoint representation. The
action on $q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})$ factors through the
composition $\mathcal{G}\rightarrow q_{*}(\mathcal{G}|_{x})\rightarrow
q_{*}(P)$.

Let’s compute the weight of $\varphi^{*}(\mathcal{L}_{det})|_{0}$. The
composition $0\rightarrow\Theta\xrightarrow{\varphi}\text{Bun}_{\mathcal{G}}$
is represented by the $\mathbb{Z}$-graded parabolic vector bundle
$\bigoplus_{m\in\mathbb{Z}}\mathcal{W}_{m}/\mathcal{W}_{m+1}$. Here
$\mathbb{G}_{m}$ acts with weight $m$ on $\mathcal{W}_{m}/\mathcal{W}_{m+1}$.
Let $\mathcal{P}_{univ}$ denote the universal $\mathcal{G}$-torsor on
$C\times\text{Bun}_{\mathcal{G}}$. We can use flat base-change to write

$wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)=-wt\left(\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}\text{Lie}(\mathcal{G})\right)$

After using the short exact sequence of vector bundles above and the
additivity of the (K-theoretic) determinant, we get

$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)=-wt\left(\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))\right)+wt\left(\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})\right)$

We are left to compute each term separately. For the first term, one can
unravel the definitions to obtain

$\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}\mathcal{E}\mathit{nd}(\mathcal{E}^{(0)}(x))=\mathcal{E}\mathit{nd}\left(\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{(0)}_{m}(x)/\,\mathcal{F}^{(0)}_{m+1}(x)\right)$

The computation in [Hei17][1.E.c] shows that

$\displaystyle-
wt\left(\text{det}\,R\pi_{*}\left(\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{(0)}_{m}(x)/\,\mathcal{F}^{(0)}_{m+1}(x)\right)\right)=2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)$

This takes care of the first term above. On the other hand, we can write

$\displaystyle\text{det}\,R\pi_{*}\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
C}\times^{\mathcal{G}}q_{*}(\mathfrak{gl}_{n}/\mathfrak{p})\,\cong\,\text{det}\,R\pi_{*}q_{*}\left(\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P\times^{P}\mathfrak{gl}_{n}/\mathfrak{p})\right)$

where we use the natural map $\mathcal{G}|_{x}\rightarrow P$ to form the
associated $P$-torsor $\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P$. We can further rewrite this as

$\text{det}\,R(\pi q)_{*}\left(\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P\times^{P}\mathfrak{gl}_{n}/\mathfrak{p})\right)\,\cong\,\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}P\times^{P}\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})$

The character $\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})$ of $P$ factors
through the Levi quotient $M=P/\,U=\prod_{i=1}^{N}\text{GL}_{a^{(i)}}$.
Therefore we can rewrite the expression above as

$\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}M\times^{M}\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})$

As a $M$-representation, we have

$\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})\,\cong\,\text{det}\left(\bigoplus_{i<j}\left(St_{\text{GL}_{a^{(i)}}}\right)^{\vee}\otimes\left(St_{\text{GL}_{a^{(j)}}}\right)\right)$

Here $St_{\text{GL}_{a^{(i)}}}$ denotes the standard $a^{(i)}$-dimensional
representation of the $\text{GL}_{a^{(i)}}$-component of $M$. Using this, we
get

$\displaystyle\varphi_{C}^{*}(\mathcal{P}_{univ})|_{0\times
x}\times^{\mathcal{G}|_{x}}M\times^{M}\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})\,\cong\,\bigotimes_{i<j}\,\text{det}\left(\left(\bigoplus_{m\in\mathbb{Z}}\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\bigoplus_{l\in\mathbb{Z}}\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)$

Here $\mathbb{G}_{m}$ acts on
$\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)$
with weight $l-m$. We are reduced to computing for each $i,j$ the weight

$wt\left(\text{det}\left(\left(\bigoplus_{m\in\mathbb{Z}}\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\bigoplus_{l\in\mathbb{Z}}\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)\right)$

This is the weight of the $0$-fiber of a line bundle
$\mathcal{L}_{\chi_{i,j}}$ in $\text{Bun}_{\mathcal{G}}$ corresponding to a
character $\chi_{i,j}$ of $\mathcal{G}|_{x}$, as in [Hei17][3.B]. Indeed we
can take
$\chi_{i,j}=\text{det}\left(\,\left(St_{\text{GL}_{a^{(i)}}}\right)^{\vee}\otimes\left(St_{\text{GL}_{a^{(j)}}}\right)\,\right)$,
which we can view as a representation of $\mathcal{G}|_{x}$ via the natural
map $\mathcal{G}|_{x}\rightarrow M$. Let’s compute the corresponding weight.
We have

$\displaystyle
wt\left(\text{det}\left(\left(\bigoplus_{m\in\mathbb{Z}}\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)^{\vee}\otimes\left(\bigoplus_{l\in\mathbb{Z}}\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)\right)\,=\,\sum_{m,l\in\mathbb{Z}}(l-m)\cdot\text{rank}\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)\cdot\text{rank}\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)$

Let’s rewrite this as

$\displaystyle\sum_{m<l\in\mathbb{Z}}(l-m)\cdot\text{rank}\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)\cdot\text{rank}\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\,-\,\sum_{m<l\in\mathbb{Z}}(l-m)\cdot\text{rank}\left(\frac{\mathcal{F}^{(i)}_{l}/\mathcal{F}^{(i)}_{l+1}}{\mathcal{F}^{(i-1)}_{l}/\mathcal{F}^{(i-1)}_{l+1}}\right)\cdot\text{rank}\left(\frac{\mathcal{F}^{(j)}_{m}/\mathcal{F}^{(j)}_{m+1}}{\mathcal{F}^{(j-1)}_{m}/\mathcal{F}^{(j-1)}_{m+1}}\right)$

We can use the same trick as in [Hei17][1.E.c] to express this as a sum

$\displaystyle\sum_{h\in\mathbb{Z}}\left(\sum_{m\leq
h}\text{rank}\left(\frac{\mathcal{F}^{(i)}_{m}/\mathcal{F}^{(i)}_{m+1}}{\mathcal{F}^{(i-1)}_{m}/\mathcal{F}^{(i-1)}_{m+1}}\right)\right)\cdot\left(\sum_{l>h}\text{rank}\left(\frac{\mathcal{F}^{(j)}_{l}/\mathcal{F}^{(j)}_{l+1}}{\mathcal{F}^{(j-1)}_{l}/\mathcal{F}^{(j-1)}_{l+1}}\right)\right)\,-\,\sum_{h\in\mathbb{Z}}\left(\sum_{l>h}\text{rank}\left(\frac{\mathcal{F}^{(i)}_{l}/\mathcal{F}^{(i)}_{l+1}}{\mathcal{F}^{(i-1)}_{l}/\mathcal{F}^{(i-1)}_{l+1}}\right)\right)\cdot\left(\sum_{m\leq
h}\text{rank}\left(\frac{\mathcal{F}^{(j)}_{m}/\mathcal{F}^{(j)}_{m+1}}{\mathcal{F}^{(j-1)}_{m}/\mathcal{F}^{(j-1)}_{m+1}}\right)\right)$

This can be rewritten as

$\displaystyle\sum_{h\in\mathbb{Z}}a^{(i)}_{h}\cdot\left(a^{(j)}-a^{(j)}_{h}\right)\,-\,\sum_{h\in\mathbb{Z}}a^{(j)}_{h}\cdot\left(a^{(i)}-a^{(i)}_{h}\right)$

After some cancellation, we end up with
$wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)=\sum_{h\in\mathbb{Z}}\left(a^{(i)}_{h}\cdot
a^{(j)}-a^{(j)}_{h}\cdot a^{(i)}\right)$. So after all of our computations, we
conclude that

$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)\,+\,\sum_{i<j}wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)$

Set $\chi$ to be the element in
$\text{Hom}\left(\mathcal{G}|_{x},\mathbb{G}_{m}\right)\otimes_{\mathbb{Z}}\mathbb{R}$
given by
$\chi=\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})^{\vee}\,\otimes\,\bigotimes_{i,j}\chi_{i,j}^{\otimes-2\,\lambda^{(i)}}$.
By linearizing over $\mathbb{R}$ the definition in [Hei17][3.B], we can define
an element $\mathcal{L}_{\chi}$ in the $\mathbb{R}$-Picard group of the stack
$\text{Bun}_{\mathcal{G}}$. Using our computations from above, we can get

$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)\,-\,2\sum_{i,j}\lambda^{(i)}\cdot
wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)$

We can further plug in our formula for
$wt\left(\varphi^{*}(\mathcal{L}_{\chi_{i,j}})|_{0}\right)$ to get

$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))\cdot\text{rank}(\mathcal{F}^{(0)})-\text{deg}(\mathcal{F}^{(0)}(x))\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)\,-\,2\sum_{i,j}\lambda^{(i)}\sum_{h\in\mathbb{Z}}\left(a^{(i)}_{h}\cdot
a^{(j)}-a^{(j)}_{h}\cdot a^{(i)}\right)$

The second term can be rewritten as

$\displaystyle
2\sum_{i,j}\lambda^{(i)}\sum_{h\in\mathbb{Z}}\left(a^{(i)}_{h}\cdot
a^{(j)}-a^{(j)}_{h}\cdot
a^{(i)}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\left(\sum_{i}a^{(i)}_{h}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}^{(0)})-\left(\sum_{i}a^{(i)}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)$

Plugging this back in, we end up with

$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\left(\text{deg}(\mathcal{F}^{(0)}_{h}(x))-\sum_{i}a^{(i)}_{h}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}^{(0)})-\left(\text{deg}(\mathcal{F}^{(0)}(x))-\sum_{i}a^{(i)}\lambda^{(i)}\right)\cdot\text{rank}(\mathcal{F}_{h}^{(0)})\right)$

By our definition of parabolic degree, this is the same as

$\displaystyle
wt\left(\varphi^{*}(\mathcal{L}_{det}\otimes\mathcal{L}_{\chi})|_{0}\right)\,=\,2\sum_{h\in\mathbb{Z}}\left(\,\text{deg}(\mathcal{W}_{h})\cdot\text{rank}(\mathcal{W})-\text{deg}(\mathcal{W})\cdot\text{rank}(\mathcal{W}_{h})\,\right)$

Using this formula, one can see that a parabolic vector bundle is
$\mathcal{L}_{det}\otimes\mathcal{L}_{\chi}$-(semi)stable in the sense of
[Hei17] if and only if it is slope (semi)stable as in our definition. In order
to see that the HN filtrations match up, one can use the deletion lemma in
[HL14][Lemma 5.32], which can be verified to work the same way in our context
of parabolic vector bundles.

It should be noted that the results in [Hei17] [AHLH18][§8] are valid whenever
the character $\chi$ used to define $\mathcal{L}_{\chi}$ is “admissible” (see
[Hei17][3.F] for a definition). Let’s study whether the character we defined
above is admissible. The diagonal torus $T$ of $\text{GL}_{n}$ over $k(C)$ can
be viewed as a maximal split torus of the generic fiber of $\mathcal{G}$ under
the identification provided by the proof of Proposition 3.2. This torus
extends to a split torus $\mathcal{T}$ of $\mathcal{G}$ defined over all of
$C$ (take the scheme theoretic closure). The fiber
$\mathcal{T}|_{x}\subset\mathcal{G}|_{x}$ maps isomorphically onto the
diagonal torus $T_{x}$ of the Levi quotient
$M=\prod_{i=1}^{N}\text{GL}_{a^{(i)}}$. We will use the standard basis of the
character lattice of $T_{x}$, namely
$X^{*}(T_{x})=\bigoplus_{j=1}^{n}\mathbb{Z}e_{j}$, where $e_{j}$ is the
character that projects to the $j$th component of $T_{x}$ (under the canonical
ordered basis of $k^{n}$). We can view each $\chi_{i,j}$ as a
$T_{x}$-representation via restriction. Then, we can write

$\chi_{i,j}=\text{det}\left(\left(St_{\text{GL}_{a^{(i)}}}\right)^{\vee}\otimes\left(St_{\text{GL}_{a^{(j)}}}\right)\right)=-\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}a^{(j)}e_{l}\;\;+\sum_{l=(\sum_{m=1}^{j-1}a^{(m)})+1}^{\sum_{m=1}^{j}a^{(m)}}a^{(i)}e_{l}$

We conclude that

$\bigotimes_{i,j}\chi_{i,j}^{\otimes-2\,\lambda^{(i)}}=-2\sum_{i=1}^{N}\left(\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}e_{l}\right)\cdot\left(\left(\sum_{m=1}^{N}a^{(m)}\lambda^{(m)}\right)-n\lambda^{(i)}\right)$

and

$\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})\,=\,\bigotimes_{i<j}\chi_{i,j}\,=\,\sum_{i=1}^{N}\left(\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}e_{l}\right)\cdot\left(2\sum_{m=1}^{i-1}a^{(m)}+a^{(i)}-n\right)$

Therefore

$\displaystyle\chi=\text{det}(\mathfrak{gl}_{n}/\mathfrak{p})^{\vee}\,\otimes\,\bigotimes_{i,j}\chi_{i,j}^{\otimes-2\,\lambda^{(i)}}=\sum_{i=1}^{N}\left(\sum_{l=(\sum_{m=1}^{i-1}a^{(m)})+1}^{\sum_{m=1}^{i}a^{(m)}}e_{l}\right)\cdot\left(n-a^{(i)}-2\sum_{m=1}^{i-1}a^{(m)}-2(\sum_{m=1}^{N}a^{(n)}\lambda^{(m)})+2n\lambda^{(i)}\right)$

To determine admissibility, now we need to compute the inner product with each
coroot of $\text{GL}_{n}$. We know that the coroots are of the form
$\alpha_{ij}=e_{i}-e_{j}$ for some $i\neq j$ with $1\leq i,j\leq n$. In this
case the admissibility condition means that
$\langle\chi,\alpha_{i,j}\rangle\leq 1$ for all $i\neq j$. Note that if the
indices $i,j$ lie in the same “block” determined by the partition
$(a^{(1)},a^{(2)},\cdots,a^{(N)})$, then $\langle\chi,\alpha_{i,j}\rangle=0$.
Otherwise, if $i$ is in the $l$-th block and $j$ is in the $k$-th block we
have

$\langle\chi,\alpha_{i,j}\rangle=2n\lambda^{(l)}-2\sum_{m=1}^{l-1}a^{(m)}-a^{(l)}-2n\lambda^{(m)}+2\sum_{m=1}^{k-1}a^{(m)}+a^{(k)}$

Hence the admissibility condition reads

$2n\lambda^{(l)}-2\sum_{m=1}^{l-1}a^{(m)}-a^{(l)}-2n\lambda^{(m)}+2\sum_{m=1}^{k-1}a^{(m)}+a^{(k)}\leq
1$

This condition can be expressed as follows. For all $1\leq k\leq l\leq N$, we
must have the pair of inequalities

$-\frac{1}{2n}+\frac{1}{n}\sum_{m=k}^{l-1}a^{(m)}+\frac{a^{(l)}-a^{(k)}}{2n}\,\leq\,\lambda^{(l)}-\lambda^{(m)}\,\leq\,\frac{1}{2n}+\frac{1}{n}\sum_{m=k}^{l-1}a^{(m)}+\frac{a^{(l)}-a^{(k)}}{2n}$

In particular this implies that for all $l$ we must have

$\frac{a^{(l)}+a^{(l-1)}}{2n}-\frac{1}{2n}\,\leq\,\lambda^{(l)}-\lambda^{(l-1)}\,\leq\,\frac{a^{(l)}+a^{(l-1)}}{2n}+\frac{1}{2n}$

From these inequalities we conclude that $\lambda^{(l)}>\lambda^{(l-1)}$ and
that the maximum difference $\lambda^{(N)}-\lambda^{(1)}$ is strictly smaller
than $1$. The notion of
$\mathcal{L}_{det}\otimes\mathcal{L}_{\chi}$-(semi)stability and the
admissibility of $\chi$ do not change if we translate all stability parameters
$\lambda^{(l)}$ by the same constant
$\lambda^{(l)}\,\mapsto\,\lambda^{(l)}+c$. In order to normalize, we can
translate so that $0<\lambda^{(1)}<\lambda^{(2)}<\cdots<\lambda^{(N)}<1$. We
therefore see that, up to translation, all of the admissible tuples of
parameters $\lambda^{(l)}$ in the sense of [Hei17] are parabolic weights in
our sense. However the admissibility conditions described above are not
satisfied by all parabolic weights. In other words, the admissibility of the
stability parameters from [Hei17] is more restrictive that the conditions in
the definition of parabolic weights (Definition 2.5).

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Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, New
York 14853, USA

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