spacing=nonfrench

# The moduli stack of principal $\rho$-sheaves and Gieseker-Harder-Narasimhan
filtrations

Andres Fernandez Herrero  Department of Mathematics, Cornell University, 310
Malott Hall, Ithaca, New York 14853, USA<EMAIL_ADDRESS>,  Tomás L.
Gómez  Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera
15, Campus Cantoblanco UAM, 28049 Madrid, Spain<EMAIL_ADDRESS>and
Alfonso Zamora  Departamento de Matemática Aplicada a las TIC, ETSI
Informáticos, Universidad Politécnica de Madrid, Campus de Montegancedo, 28660
Madrid, Spain<EMAIL_ADDRESS>
###### Abstract.

Given a smooth projective variety $X$ and a connected reductive group $G$
defined over a field of characteristic $0$, we define a moduli stack of
principal $\rho$-sheaves that compactifies the stack of $G$-bundles on $X$. We
apply the theory developed in [HL14, AHLH19] to construct a moduli space of
Gieseker semistable principal $\rho$-sheaves. This provides an intrinsic
stack-theoretic construction of the moduli space of semistable singular
principal bundles [Sch02, GLSS08].

Our second main result is the definition of a schematic Gieseker-Harder-
Narasimhan filtration for $\rho$-sheaves, which induces a stratification of
the stack by locally closed substacks. This filtration for a general reductive
group $G$ is a refinement of the canonical slope parabolic reduction
previously considered at the level of points in [AAB02] and as a
stratification of the stack in [GN18, GN16]. In an appendix, we apply the same
techniques to define Gieseker-Harder-Narasimhan filtrations in arbitrary
characteristic and show that they induce a stratification of the stack by
radicial morphisms.

###### Key words and phrases:

Moduli spaces, principal bundles, stacks, $\Theta$-stratifications, Harder-
Narasimhan filtration, Gieseker stability, principal $\rho$-sheaves

###### 2020 Mathematics Subject Classification:

14D20, 14D23, 14F06, 14J60, 14L24

###### Contents

  1. 1 Introduction
    1. 1.1 Stack-theoretic approach vs. GIT
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Big open sets and torsion-free sheaves
    3. 2.3 The scheme of $G$-reductions
  3. 3 Principal $\rho$-sheaves
    1. 3.1 Principal $\rho$-sheaves and the moduli stack $\text{Bun}_{\rho}(X)$
    2. 3.2 Filtrations of $\rho$-sheaves
    3. 3.3 Filtrations of $\rho$-sheaves in the split case
  4. 4 $\Theta$-stratification on $\operatorname{Bun}_{\rho}(X)$
    1. 4.1 The numerical invariant
    2. 4.2 Affine Grassmannians for $\rho$-sheaves
    3. 4.3 Monotonicity of the numerical invariant
    4. 4.4 HN boundedness
    5. 4.5 $\Theta$-stratification, semistability and leading term filtrations
  5. 5 Completeness of the stack $\operatorname{Bun}_{\rho}(X)$
  6. 6 Moduli space for the semistable locus
    1. 6.1 The degree of a $\rho$-sheaf
    2. 6.2 Boundedness of the semistable locus
    3. 6.3 Moduli space for the semistable locus
  7. 7 Generalization: $\rho$-sheaves for products of general linear groups
  8. 8 Gieseker-Harder-Narasimhan filtrations for $\rho$-sheaves
    1. 8.1 Lexicographic filtrations
    2. 8.2 Multi-weighted parabolic reductions
    3. 8.3 The GHN filtration for $\rho$-sheaves
    4. 8.4 The GHN filtration in families
  9. 9 Examples
    1. 9.1 Torsion-free sheaves
    2. 9.2 Example of Gieseker-Harder-Narasimhan filtration
  10. 10 Comparison with previous work
    1. 10.1 Comparison with previous notions of semistability
    2. 10.2 Comparison of the leading term filtration with previous notions in the literature
  11. A Gieseker-Harder-Narasimhan filtrations in positive characteristic

##  1\. Introduction

Fix a smooth projective variety $X$ and a connected reductive group $G$
defined over a field of characteristic $0$ (see Appendix A and Theorem 3 for
positive characteristic). When $\dim X=1$, Ramanathan [Ram75] constructed a
projective moduli space of semistable principal $G$-bundles on $X$ using
Geometric Invariant Theory (GIT). The stability condition in this case is
analogous to the notion of slope stability for a vector bundle (introduced by
Mumford), in the sense that it involves the degree and the rank of certain
vector bundles.

If $\dim X>1$, it is necessary to consider also degenerations of principal
$G$-bundles in order to construct a proper moduli space. These degenerations
are called principal $\rho$-sheaves, where $\rho$ is a choice of
representation of $G$. A principal $\rho$-sheaf is a triple
$(P,\mathcal{E},\psi)$ consisting of a torsion-free sheaf $\mathcal{E}$ on
$X$, a principal $G$-bundle $P$ defined on the open set $U$ where
$\mathcal{E}$ is locally free, and an isomorphism $\psi$ between the
restriction $\mathcal{E}|_{U}$ and the vector bundle associated to $P$ using
the representation $\rho$ (see Definition 3.1 and Proposition 2.18). More
generally, we allow $\rho$ to be a faithful homomorphism from $G$ into a
product of general linear groups (see Definition 7.1, Definition 10.1, and
Section 7)111If we want our definition of stability to coincide with
Ramanathan’s in the case of curves, then we have to require that the
representation $\rho$ is central (Definition 10.5).. These $\rho$-sheaves have
also been called in the literature “singular principal bundles” by Schmitt
[Sch02, GLSS08] (in the case when $\rho$ is valued in a special linear group),
and “principal $G$-sheaves” by Gómez-Sols [GS05] (when $\rho$ is the adjoint
representation, not injective in general). We show that there is an algebraic
stack $\text{Bun}_{\rho}(X)$ that parametrizes principal $\rho$-sheaves on
$X$.

When going from $\dim X=1$ to $\dim X>1$, an additional consideration needs to
be introduced in order to construct the moduli space. Instead of using slope
semistability, we have to use a notion where the degree is substituted by the
Hilbert polynomial. This is Gieseker semistability for torsion-free sheaves.
An analogous notion of semistability was defined in [GS05] and [GLSS08] for
principal $\rho$-sheaves. Using the methods of GIT, Gómez-Sols [GS05], Schmitt
[Sch02] and Gómez-Langer-Schmitt-Sols [GLSS08] constructed moduli spaces of
semistable $\rho$-sheaves. We remark that the moduli space obtained will in
general depend on the choice of representation $\rho$.

Alper, Halpern-Leistner and Heinloth [AHLH19] have recently introduced an
alternative approach to GIT for the construction of moduli spaces. Their
approach is intrinsic to the algebraic stack associated to the moduli problem
of interest. In this article we employ their techniques instead of the GIT
methods that were previously used in the literature.

In this direction, our first main result in this article is an intrinsic-stack
theoretic construction of the moduli space of semistable $\rho$-sheaves, where
$\rho$ is any faithful representation of $G$ into a product of general linear
groups. In order to obtain a quasicompact moduli space, it is helpful to fix a
tuple $P^{\bullet}=(P^{1},P^{2},\ldots,P^{b})$ of rational polynomials
$P^{i}\in\mathbb{Q}[t]$ and restricts to the locus of $\rho$-sheaves where the
$i^{th}$ underlying sheaf has Hilbert polynomial $P^{i}$. We state the result
here in the case when the representation $\rho$ is central (Definition
10.5)222A more general modified statement is proven in the body of the
document, see Theorem 7.2.:

######  Theorem 1.

Fix a central faithful representation
$\rho:G\hookrightarrow\prod_{i}\operatorname{GL}_{n_{i}}$. Then

  1. (a)

The moduli stack $\text{Bun}_{\rho}(X)$ of $\rho$-sheaves admits a
$\Theta$-stratification induced by the polynomial numerical invariant $\nu$
defined in Subsection 4.1.

  2. (b)

(Semistable reduction) The semistable locus $\text{Bun}_{\rho}(X)^{ss}$
satisfies the existence part of the valuative criterion for properness.

  3. (c)

For any choice of tuple of rational polynomials $P^{\bullet}$, the open
substack $\text{Bun}_{\rho}(X)^{ss,P^{\bullet}}$ of $\rho$-sheaves with tuple
$P^{\bullet}$ admits a proper good moduli space (in the sense of [Alp13]).

If the representation $\rho$ is valued in a single special linear group (resp.
$\rho$ is the adjoint representation), then we recover the moduli space
constructed in [Sch02, GLSS08] (resp. [GS05]).

Another important aspect of the theory of semistable vector bundles on a curve
is the existence of a canonical filtration for any unstable vector bundle.
This is called the Harder-Narasimhan filtration, and it is such that the
associated graded object is a direct sum of semistable sheaves. This is
helpful because unstable vector bundles are then constructed as extensions of
semistable bundles, thus allowing us to do recursive arguments to study
properties of the moduli space, as done by Harder-Narasimhan [HN75] and
Atiyah-Bott [AB83]. More generally, for principal $G$-bundles on a curve
Behrend [Beh95] and Biswas-Holla [BH04] constructed the Harder-Narasimhan
reduction to a parabolic subgroup $P\subset G$. An important property is that
the associated principal $L$-bundle is semistable, where $L=G/P$ is the Levi
factor of $P$.

For torsion free sheaves over $X$ with $\dim X>1$, we can construct two
different Harder-Narasimhan filtrations, depending on the stability condition
we use, slope or Gieseker stability. They are respectively called slope-
Harder-Narasimhan and Gieseker-Harder-Narasimhan.

For principal $\rho$-sheaves over $X$ with $\dim X>1$ we only find in the
literature the slope version of the Harder-Narasimhan filtration, constructed
at the level of points by Anchouche-Azad-Biswas [AAB02] and in families by
Gurjar-Nitsure [GN18, GN16]. Biswas and Zamora [BZ16] showed that the notion
of Gieseker filtration is not well-behaved with respect to change of group.
The strategy of considering the canonical filtration of the adjoint vector
bundle does not yield in general a well-defined notion of Gieseker-Harder-
Narasimhan reduction in higher dimensions. Therefore a new approach needs to
be introduced.

Our second main result is the definition of a Gieseker-Harder-Narasimhan
parabolic reduction in the context of $\rho$-sheaves.

######  Theorem 2.

Every principal $\rho$-sheaf admits a uniquely defined multi-weighted
filtration called the Gieseker-Harder-Narasimhan filtration (abbreviated GHN
filtration), see Definition 8.10.

This filtration is schematic, in the sense that the universal stacks for
relative GHN filtrations induce a locally closed stratification of
$\text{Bun}_{\rho}(X)$.

If the group $G$ is split, then the GHN filtration comes from a multi-weighted
parabolic reduction defined over a big open subset (cf. Definition 2.1). The
associated Levi $\rho_{\vec{\lambda}}$-sheaf is semistable.

If the representation $\rho$ is central, then our GHN filtration refines the
canonical slope reduction constructed in [AAB02]. In this case, the resulting
stratification of the stack $\text{Bun}_{\rho}(X)$ refines the canonical slope
stratification on the stack of $G$-bundles that was recently constructed in
[GN18].

Our main tools originate in some recent work on the theory of stacks. Alper,
Halpern-Leistner and Heinloth [AHLH19] have recently developed a theory which
produces moduli spaces for Artin stacks, generalizing results of Keel-Mori on
Deligne-Mumford stacks. This can be combined with the theory of
$\Theta$-stability and $\Theta$-stratifications by Halpern-Leistner [HL14] to
produce a powerful stack-theoretic approach to the construction of moduli
spaces and canonical filtrations (see [HL14, §5] and [HLHJ21, §2.5] for more
details). Halpern-Leistner defines in [HL14] a notion of filtration of an
object in an Artin stack. Given certain cohomology classes on the stack, one
can define a numerical invariant on filtrations. An object in the stack is
called semistable if the numerical function is non-positive for all
filtrations. If the numerical invariant satisfies certain properties, then the
open substack of semistable objects admits a _good moduli space_ (as defined
in [Alp13]). This is an intrinsic way of constructing the moduli space, in the
sense that we do not need to choose a parameter space nor the action of a
group.

Here are the main steps involved in this general approach:

  1. (Step 1)

Define a (polynomial) numerical invariant $\nu$ on the stack (Subsection 4.1).

  2. (Step 2)

Prove that the numerical invariant $\nu$ satisfies some monotonicity
properties (Subsection 4.3).

  3. (Step 3)

Show that $\nu$ satisfies the HN boundedness condition (Subsection 4.4).

  4. (Step 4)

Prove boundedness of the semistable locus (Subsection 6.2).

In this article we carry out this program for the moduli stack of principal
$\rho$-sheaves, thus obtaining the intrinsic stack-theoretic construction of
the moduli space described in Theorem 1. Furthermore, this theory provides a
$\Theta$-stratification of $\text{Bun}_{\rho}(X)$, which yields a canonical
filtration (called the leading term HN filtration) for each unstable object.
We apply this construction iteratively to the centers of the
$\Theta$-stratification to define the GHN reduction from Theorem 2.

In Appendix A we explain how to apply the same techniques to define the GHN
filtration for $\rho$-sheaves in arbitrary characteristic.

######  Theorem 3.

Suppose that $k$ is a field of arbitrary characteristic. Every principal
$\rho$-sheaf admits a uniquely defined multi-weighted filtration called the
GHN filtration, after perhaps passing to a finite purely inseparable extension
of the field of definition.

Relative GHN filtrations are represented by a schematic morphism
$H\to\text{Bun}_{\rho}(X)$ that is surjective, radicial and of finite type.
Each connected component of $H$ is finite over its image.

If the group $G$ is split, then the GHN filtration comes from a multi-weighted
parabolic reduction defined over a big open subset. The associated Levi
$\rho_{\vec{\lambda}}$-sheaf is semistable.

Here is an outline of the paper:

In Section 2 we discuss some of the necessary background on big open
subschemes and torsion-free sheaves. We define the scheme of $G$-reductions
(Definition 2.14), which is a version of the fiber bundle of $G$-reductions in
the context of torsion-free sheaves.

In Section 3 we define the notion of principal $\rho$-sheaf (Definition 3.1)
and define the stack $\text{Bun}_{\rho}(X)$ (Definition 3.5) in the case of a
representation $\rho:G\to\operatorname{GL}(V)$ into a single general linear
group. Furthermore, we use the work of Heinloth [Hei17] to produce an explicit
description of $\Theta$-filtrations on $\text{Bun}_{\rho}(X)$ (Proposition
3.16).

Section 4 is the technical heart of the paper. We define a polynomial
numerical invariant $\nu$ on the stack $\text{Bun}_{\rho}(X)$. Following
[HLHJ21], we define a higher dimensional analogue of the affine Grassmannian
for $\rho$-sheaves and use it to prove the relevant monotonicity properties of
$\nu$. We also prove the HN boundedness condition via an optimization argument
on cones inside the vector space of cocharacters of $G$ (see Proposition
4.26). This way we obtain a $\Theta$-stratification. In Subsection 4.5 we give
descriptions of the semistable locus and the $\Theta$-stratification. We call
the resulting canonical filtration of $\rho$-sheaves the “leading term HN
filtration”.

In Section 5 we prove that the stack $\text{Bun}_{\rho}(X)$ satisfies the
existence part of the valuative criterion for properness. The key lemma is a
result about extension of $G$-bundles over a DVR after a purely ramified
extension (Proposition 5.2). The proof boils down to a description of Galois
cohomology classes in [BT87] and the behavior of the Bruhat-Tits building
under purely ramified extensions.

Section 6 includes a proof of boundedness of the semistable locus, and
finalizes the intrinsic construction of the moduli space (Theorem 6.8) in the
case of a representation $\rho:G\to\operatorname{GL}(V)$ into a single general
linear group.

In Section 7 we generalize the results of previous subsections to the case
when $\rho$ is a faithful homomorphism into a product of general linear
groups.

Section 8 introduces the notion of lexicographic filtration which provides a
convenient framework for the construction of the Gieseker-Harder-Narasimhan
filtration in Subsection 8.3. We end the section by defining a relative
version of the Gieseker-Harder-Narasimhan filtration and proving that it
induces a locally closed stratification of the stack $\text{Bun}_{\rho}(X)$.

In Section 9 we present several examples to illustrate the theory developed.
In Subsection 9.1 we consider the case when $G$ is a general linear group and
$\rho$ is the identity. In this case a $\rho$-sheaf is the same thing as a
torsion-free sheaf. We describe explicitly the construction of our Gieseker-
Harder-Narasimhan filtration in this case, and show that it coincides with the
classical notion for torsion-free sheaves (Proposition 9.3). In Subsection 9.2
we work out explicitly the case of a $\rho$-sheaf for $\operatorname{SO}(7)$
and compare its leading term HN filtration with its Gieseker-Harder-Narasimhan
filtration.

In Section 10 we compare our notions with previous ones in the literature. In
Subsection 10.1 we introduce central representations (Definition 10.5). If
$\rho$ is central, then we prove that our notion of Gieseker semistability is
compatible with previous notions in the literature, and our Gieseker-Harder-
Narasimhan stratification of the stack refines the slope stratification
constructed in [GN18, GN16].

In Appendix A we show how to apply our techniques to define the Gieseker-
Harder-Narasimhan stratification in arbitrary characteristic. In this case the
GHN filtrations induce a “stratification” of the stack by radicial morphisms
of finite type.

###  1.1. Stack-theoretic approach vs. GIT

We end our introduction with a discussion comparing the stack theoretic
approach in [HL14, AHLH19] with the usual construction of moduli spaces using
Geometric Invariant Theory. We hope that this can provide some intuition for
those readers that are more familiar with the GIT approach.

The construction of the moduli space of torsion free sheaves (and principal
$\rho$-sheaves) follows the usual application of GIT. We first construct a
scheme $R$ parametrizing all objects together with an extra data (in this case
an isomorphism between a fixed vector space $W$ and the space of sections
$H^{0}(\mathcal{E}(n))$ for a chosen $n\gg 0$ and the sheaf $\mathcal{E}$
associated to each object). The group $\operatorname{SL}(W)$ acts on the
scheme $R$. We choose an ample line bundle on $R$ together with a
linearization of the action of $\operatorname{SL}(W)$. Mumford’s GIT shows
that there is an open subset $R^{ss}\subset R$ which admits a _good quotient_
[Ses72], and this is the moduli space of semistable torsion-free sheaves we
are looking for.

To identify the subset $R^{ss}$ of semistable points we use the Hilbert-
Mumford criterion. For each point $p\in R$ and for each one-parameter subgroup
$\lambda:\mathbb{G}_{m}\to\operatorname{SL}(W)$ we consider the weight
$\mu(\lambda,p)$ of the action of the one-parameter subgroup $\lambda$ on the
fiber of the limiting point $\lim_{t\to 0}\lambda(t)\cdot p$. The point $p$ is
semistable if and only if $\mu(\lambda,p)\leq 0$ for all one-parameter
subgroups $\lambda$. We remark that, in the usual applications of GIT to the
construction of moduli spaces involving sheaves, the one-parameter subgroup
$\lambda$ produces a filtration of the sheaf, and the Hilbert-Mumford
criterion becomes equivalent to checking certain numerical condition over all
filtrations.

In [HL14], the one-parameter subgroups appearing in the Hilbert-Mumford
criterion of GIT are replaced by _filtered objects_. For any algebraic stack
$\mathcal{M}$, a filtration of a point $p:\operatorname{Spec}k\to\mathcal{M}$
is a morphism

(1) $f:\Theta:=[\mathbb{A}^{1}/\mathbb{G}_{m}]\to\mathcal{M}$

and an identification $f(1)\simeq p$. Given $f$, there is a group homomorphism
from $\mathbb{G}_{m}$ to the automorphism group of the object $f(0)$ in
$\mathcal{M}$, which can be thought of as the graded object associated to the
filtration $f$.

For understanding these notions it is useful to let $\mathcal{M}$ be the
moduli stack of torsion free sheaves on a projective scheme $X$. By the Rees
construction (cf. Proposition 3.8), a decreasing $\mathbb{Z}$-indexed
filtration

$0\subset\cdots\subset\mathcal{E}_{i+1}\subset\mathcal{E}_{i}\subset\cdots\subset\mathcal{E}$

of a torsion free sheaf $\mathcal{E}$ on $X$ determines an
$\mathbb{A}^{1}$-family $\widetilde{\mathcal{E}}$ of torsion free sheaves on
$X$. We have $\widetilde{\mathcal{E}}|_{X_{t}}\cong\mathcal{E}$ for all $t\neq
0$, and
$\widetilde{\mathcal{E}}|_{X_{0}}\cong\oplus\mathcal{E}_{i}/\mathcal{E}_{i+1}$.
The group $\mathbb{G}_{m}$ has a natural action on the graded object
$\widetilde{\mathcal{E}}|_{X_{0}}$, with weight $t^{i}$ on each piece
$\mathcal{E}_{i}/\mathcal{E}_{i+1}$. The family $\widetilde{\mathcal{E}}$
gives a morphism $\mathbb{A}^{1}\to\mathcal{M}$ and the standard action of
$\mathbb{G}_{m}$ on $\mathbb{A}^{1}$ lifts to $\widetilde{\mathcal{E}}$. Hence
this morphism descends to a map from the quotient stack
$[\mathbb{A}^{1}/\mathbb{G}_{m}]$, providing a filtration $f$ as in (1). One
can define a function $\nu$ which assigns a real number for each filtration
$f$ as in (1). This is done in a similar way to GIT: we choose a line bundle
$L$ on $\mathcal{M}$, and define $\nu(f)$ to be the weight of the action of
$\mathbb{G}_{m}$ on the fiber of $L|_{f(0)}$.

In practice this approach has an important advantage: in the GIT case we are
looking at the action of $\mathbb{G}_{m}$ on $\operatorname{SL}(W)$. This
produces a filtration on the vector space $W\cong H^{0}(E(n))$, which in turn
produces a filtration on $\mathcal{E}$. To show that the numerical condition
of the Hilbert-Mumford criterion given by $\mu(\lambda,p)$ matches the
stability condition on Hilbert polynomials is an involved calculation. On the
other hand, if we follow the intrinsic approach of [HL14], we are directly
looking at a filtration on $\mathcal{E}$ and the action of $\mathbb{G}_{m}$ on
the associated graded sheaf. This makes it easier to obtain the stability
condition.

In the case of principal $G$-bundles the advantage of the new approach is even
more clear. In the GIT setup we are looking at all filtrations of $W\cong
H^{0}(E(n))$, and it is not easy to show that it is enough to look at
filtrations coming from 1-parameter subgroups of $G$. In contrast, in the
theory of [HL14, Hei17] we only look at these subgroups from the very
beginning.

Both GIT and [HL14] can be used to produce filtrations of unstable objects. We
recall how this works in GIT. Let $p\in R$ be an unstable point. Among all the
1-parameter subgroups $\lambda$ of $\operatorname{SL}(W)$ with
$\mu(\lambda,p)>0$, we can find one for which this weight is maximal (after
some normalization). This defines the Kempf filtration (cf. [Kem78]) of $p$.
This filtration depends on the choices that were made in the GIT construction
(the number $n$, the parameter space $R$ and the linearization). For torsion
free sheaves it can be shown that it is independent of the choice of $n$ for
$n\gg 0$, and it coincides with the Gieseker-Harder-Narasimhan filtration
[GSZ15, Zam13]. Unfortunately, the computations become too involved in the
case of principal $G$-bundles, so we do not obtain a Gieseker-Harder-
Narasimhan filtration following this approach [Zam16].

In the theory in [HL14], given an unstable object $p\in\mathcal{M}(k)$ we can
also find a filtration $f$ with $f(1)\simeq p$ such that $\nu(f)$ is maximal.
This program was carried out for $\Lambda$-modules and pairs in [HLHJ21]. It
should be pointed out that the filtration $f$ which maximizes $\nu$ does not
have the property that the graded pieces of the filtration are semistable. But
if we apply this procedure inductively we obtain the Gieseker-Harder-
Narasimhan filtration.

In this article we show that a similar story applies in the case of principal
$\rho$-sheaves. We use this to define what we call the Gieseker-Harder-
Narasimhan reduction of an unstable $\rho$-sheaf. We refer the reader to
Section 9 for examples.

Acknowledgments: This project was started during the workshop “Moduli problems
beyond geometric invariant theory” at the American Institute of Mathematics.
We would like to thank the American Institute of Mathematics and the
organizers that made the workshop possible. We would like to thank V. Balaji,
Federico Fuentes, Daniel Halpern-Leistner and Jochen Heinloth for helpful
remarks.

This work is supported by grants CEX2019-000904-S and PID2019-108936GB-C21
(funded by MCIN/AEI/ 10.13039/501100011033), and NSF grants DMS-1454893 and
DMS-2001071.

##  2\. Preliminaries

###  2.1. Notation

We will work over a field $k$ of characteristic $0$. We denote by
$\operatorname{Aff}_{k}$ the category of affine schemes over $k$. More
generally, for any $k$-scheme $T$ we let $\operatorname{Aff}_{T}$ be the
category of (absolutely) affine schemes equipped with a morphism to $T$. We
write $\operatorname{Sch}_{k}$ and $\operatorname{Sch}_{T}$ to denote the
category of all schemes over $k$ and $T$ respectively. Unless otherwise
stated, all schemes will be understood to belong to $\operatorname{Sch}_{k}$.
An undecorated product of schemes (e.g. $X\times S$) is understood to be a
fiber product over $k$. We might sometimes write $X_{S}$ instead of $X\times
S$.

All of the sheaves that we consider in this paper are quasicoherent. Whenever
we write “$\mathcal{O}_{Y}$-module” we mean a quasicoherent
$\mathcal{O}_{Y}$-module. In order to ease notation for pullbacks, we use the
following convention. Whenever we have a map of schemes $f:Y\rightarrow T$ and
a sheaf $\mathcal{G}$ on $T$, we will write $\mathcal{G}|_{Y}$ to denote the
pullback $f^{*}\mathcal{G}$, if the morphism $f$ is clear from context.

We fix a smooth projective geometrically connected variety $X$ over $k$. We
denote by $d$ the dimension of $X$. We also choose once and for all an ample
line bundle $\mathcal{O}(1)$ on $X$. For any coherent sheaf $\mathcal{F}$ on
$X$, we denote by $P_{\mathcal{F}}$ the Hilbert polynomial of $\mathcal{F}$
with respect to the line bundle $\mathcal{O}(1)$. If $\mathcal{F}$ is a
torsion-free sheaf on $X$, then this is a polynomial of degree $d$ in the
variable $n$ which can be written in the form

$P_{\mathcal{F}}(n)=\sum_{i=0}^{d}\frac{a_{i}}{i!}n^{i}$

for some sequence of rational numbers $a_{i}$ (see [HL97, Lemma 1.2.1]). We
define the reduced Hilbert polynomial to be
$\overline{p}_{\mathcal{F}}\vcentcolon=\frac{1}{a_{d}}P_{\mathcal{F}}$. For
every $0\leq i\leq d-1$, we set
$\widehat{\mu}_{i}(\mathcal{F})\vcentcolon=\frac{a_{i}}{a_{d}}$. We call this
the $i^{th}$ slope of $\mathcal{F}$. The number
$\widehat{\mu}_{d-1}(\mathcal{F})=\frac{a_{d-1}}{a_{d}}$ is simply called the
slope of $\mathcal{F}$.

We define an ordering for polynomials where, for any two
$p_{1},p_{2}\in\mathbb{R}[n]$, we write $p_{1}\geq p_{2}$ if $p_{1}(n)\geq
p_{2}(n)$ for $n\gg 0$.

Fix a reductive linear algebraic group $G$ over $k$. We write $k[G]$ to denote
the coordinate ring of $G$, which has the structure of a commutative Hopf
algebra. We will denote by $\operatorname{Bun}_{G}(X)$ the moduli stack of
principal $G$-bundles on $X$. This is an algebraic stack that has affine
diagonal and is locally of finite type over $k$ (by [HR19, Thm. 1.2] taking
$Z=X$ and $X=BG$).

Fix a $k$-vector space $V$ of dimension $r$. Let us denote by
$\operatorname{GL}(V)$ the group of linear automorphisms of $V$. We choose
once and for all a faithful representation
$\rho:G\rightarrow\operatorname{GL}(V)$.

We denote by $\operatorname{SL}(V)$ the closed algebraic subgroup of
$\operatorname{GL}(V)$ consisting of linear automorphisms $g$ of $V$ such that
the $r^{th}$ exterior power $\wedge^{r}g$ induces the identity on
$\bigwedge^{r}V$.

Set $\Theta$ to be the quotient stack $[\mathbb{A}^{1}_{k}/\,\mathbb{G}_{m}]$.
Here by convention we always equip
$\mathbb{A}^{1}_{k}=\operatorname{Spec}(k[t])$ with the
$\mathbb{G}_{m}$-action that gives the coordinate function $t$ weight $-1$.

###  2.2. Big open sets and torsion-free sheaves

######  Definition 2.1.

Let $S$ be a $k$-scheme. An open subscheme $U\subset X_{S}$ is said to be big
(relative to $S$) if for all points $s\in S$ the complement $X_{s}\setminus
U_{s}$ of the fiber $U_{s}$ is a closed subset of codimension at least 2 in
$X_{s}$.

######  Lemma 2.2.

Let $S$ be a $k$-scheme. Let $j:U\hookrightarrow X_{S}$ be a big open subset
in $X_{S}$. Then the following are satisfied

  1. (a)

For all points $x\in X_{S}\setminus U$, the local ring $\mathcal{O}_{X_{S},x}$
has depth at least 2.

  2. (b)

$U$ is scheme theoretically dense in $X_{S}$.

  3. (c)

Let $p:Y\rightarrow X_{S}$ be an affine morphism of finite type. Any section
of $p$ defined over $U$ extends uniquely to a section defined over the whole
$X_{S}$.

###### Proof.

  1. (a)

Let $x\in X_{S}\setminus U$ be a point with image $s$ in $S$. By assumption
the point $x$ has codimension at least 2 in $X_{s}$. Since $X_{s}$ is smooth,
this means that $\mathcal{O}_{X_{s},x}$ has depth at least $2$. So there
exists a regular sequence $\overline{f}_{1},\overline{f}_{2}$ of length $2$ in
$\mathcal{O}_{X_{s},x}$. Lift this sequence to a pair of elements
$f_{1},f_{2}\in\mathcal{O}_{X_{S},x}$. Since $\mathcal{O}_{X_{S},x}$ is flat
over $\mathcal{O}_{S,s}$, we can apply [Sta21, Tag 00ME] twice to conclude
that $f_{1},f_{2}$ is a regular sequence. First we apply [Sta21, Tag 00ME]
with $N=M=\mathcal{O}_{X_{S},x}$ and $u=f_{1}$. Then we apply this result a
second time with $N=M=\mathcal{O}_{X_{S},x}/(f_{1})$ and $u=f_{2}$.

  2. (b)

By definition, the scheme theoretic closure of $U$ is cut out by the ideal
sheaf of $\mathcal{O}_{X_{S}}$ that locally consists of sections whose
restriction to $U$ vanishes. In other words, it is cut out by the kernel of
the unit $\mathcal{O}_{X_{S}}\rightarrow j_{*}j^{*}\mathcal{O}_{X_{S}}$. By
part $(a)$ every point outside $U$ has depth at least $2$. Therefore the unit
map $\mathcal{O}_{X_{S}}\rightarrow j_{*}j^{*}\mathcal{O}_{X_{S}}$ is an
isomorphism by [Sta21, Tag 0E9I]. We conclude that the scheme theoretic
closure is cut out by the $0$ ideal, i.e. it is the whole $X_{S}$.

  3. (c)

Uniqueness of the extension follows because $U$ is scheme theoretically dense
in $X_{S}$ and the morphism $Y\rightarrow X_{S}$ is separated. It remains to
show existence. Since extensions are unique, any set of local extensions will
glue. Therefore we can work Zariski locally. Let $V$ be an affine open subset
of $X_{S}$. Since $Y$ is of finite type over $X_{S}$, the scheme $Y_{V}$ can
be viewed as a closed subscheme of $\mathbb{A}^{n}_{V}$ for some $n$. Since
$U\cap V$ is scheme theoretically dense inside $V$, any extension of the
section to $\mathbb{A}^{n}_{V}$ will automatically factor through the closed
subscheme $Y_{V}$. It therefore suffices to extend the composition $p:U\cap
V\rightarrow Y_{U\cap V}\hookrightarrow\mathbb{A}^{n}_{U\cap V}$ to a map
$\widetilde{p}:V\rightarrow\mathbb{A}^{n}_{V}$. This amounts to taking a
section of $\mathcal{O}_{U\cap V}^{\oplus n}$ and extending to a section of
$\mathcal{O}_{V}^{\oplus n}$. This in turn follows from [Sta21, Tag 0E9I]
applied to the sheaf $\mathcal{O}_{V}^{\oplus n}$.

∎

For the following definition, recall that the rank of a torsion-free sheaf on
an integral scheme is the dimension of the generic fiber.

######  Definition 2.3.

Let $S$ be a $k$-scheme. Let $\mathcal{F}$ be a $\mathcal{O}_{X_{S}}$-module.
We say that $\mathcal{F}$ is a relative torsion-free sheaf on $X_{S}$ if it is
$S$-flat, finitely presented, and for all points $s\in S$ the fiber
$\mathcal{F}|_{X_{s}}$ is torsion-free. We say that $\mathcal{F}$ has rank $h$
if all of the fibers $\mathcal{F}|_{X_{s}}$ have rank $h$.

######  Remark 2.4.

From now on we will drop the adjective “relative” whenever it is clear from
context.

######  Lemma 2.5.

Let $\mathcal{F}$ be a torsion-free sheaf on $X_{S}$. Then, there exists a big
open subset $U\subset X_{S}$ such that $\mathcal{F}|_{U}$ is locally-free.

###### Proof.

Let $U$ denote the (maximal) open subset of $X_{S}$ where $\mathcal{F}$ is
locally-free. Choose $s\in S$. Since $\mathcal{F}$ is $S$-flat and finitely
presented, we can apply [Sta21, Tag 0CZR] to conclude that $U\cap X_{s}$
coincides with the set of points of $X_{s}$ where the fiber
$\mathcal{F}|_{X_{s}}$ is locally-free. Since $X_{s}$ is smooth, all points of
codimension $1$ are discrete valuation rings. Therefore the torsion-free sheaf
$\mathcal{F}|_{X_{s}}$ is automatically free at all codimension $\leq 1$
points. We conclude that the closed complement of $U\cap X_{s}$ has
codimension at least $2$, as desired. ∎

Next we recall the notion of determinant for torsion-free sheaves, as in
[Kob87, 5.6]. Let $Y$ be a $k$-scheme and $\mathcal{G}$ be a finitely
presented $\mathcal{O}_{Y}$-module. Let $h$ be a nonnegative integer. There is
a natural action of the symmetric group $S_{h}$ of $h$-letters on the $h$-fold
tensor $\mathcal{G}^{\otimes h}$, given by permuting the indexes. Let
$\text{sign}:S_{h}\rightarrow\\{1,-1\\}$ denote the sign character.

######  Definition 2.6.

Let $Y$ and $\mathcal{G}$ be as in the paragraph above. We denote by
$\mathcal{G}_{Alt,h}$ the $\mathcal{O}_{Y}$-submodule of $\mathcal{G}^{\otimes
h}$ that is locally generated over each affine open by sections of the form
$\sigma\cdot(x_{1}\otimes x_{2}\otimes\cdots\otimes
x_{h})-\text{sign}(\sigma)\,x_{1}\otimes x_{2}\otimes\cdots\otimes x_{h}$ with
$\sigma\in S_{h}$. We denote by $\bigwedge^{h}\mathcal{G}$ the quotient sheaf
$\mathcal{G}^{\otimes h}/\,\mathcal{G}_{\text{Alt, h}}$.

Note that $\bigwedge^{h}\mathcal{G}$ is a finitely presented
$\mathcal{O}_{Y}$-sheaf by definition. The formation of
$\bigwedge^{h}\mathcal{G}$ is compatible with arbitrary base-change, because
taking pullbacks is right exact and commutes with tensor products.

######  Definition 2.7.

Let $S$ be a $k$-scheme and let $\mathcal{F}$ be a torsion-free sheaf of rank
$h$ on $X_{S}$. We define
$\det(\mathcal{F})\vcentcolon=\left(\bigwedge^{h}\mathcal{F}\right)^{\vee\vee}$.

For all morphisms of $k$-schemes $T\rightarrow S$ there is a canonical
morphism
$\det(\mathcal{F})|_{X_{T}}\rightarrow\det\left(\mathcal{F}|_{X_{T}}\right)$.
It turns out that this is an isomorphism, as shown in the following lemma.

######  Lemma 2.8.

Let $S$ be a $k$-scheme and let $\mathcal{F}$ be a torsion-free sheaf on
$X_{S}$ such that the rank of the fibers is constant. Then:

  1. (a)

Let $j:U\hookrightarrow X_{S}$ be a big open subscheme such that $\mathcal{F}$
is locally-free. Then
$\det(\mathcal{F})=j_{*}\left(\det(\mathcal{F}|_{U})\right)$, where
$\det(\mathcal{F}|_{U})$ on the right hand side denotes the usual determinant
of a locally-free sheaf.

  2. (b)

$\det(\mathcal{F})$ is a line bundle on $X_{S}$.

  3. (c)

For all morphisms of $k$-schemes $T\rightarrow S$, the canonical morphism
$\det(\mathcal{F})|_{X_{T}}\rightarrow\det\left(\mathcal{F}|_{X_{T}}\right)$
is an isomorphism.

###### Proof.

  1. (a)

By construction, $\det(\mathcal{F})|_{U}$ coincides with the usual determinant
of $\mathcal{F}|_{U}$ as a locally-free sheaf. Therefore it suffices to show
that the unit $\det(\mathcal{F})\rightarrow
j_{*}\left(\det(\mathcal{F})|_{U}\right)$ is an isomorphism. By [Sta21, Tag
0E9I] it suffices to check that the sheaf
$\det(\mathcal{F})=\operatorname{Hom}\left(\left(\bigwedge^{h}\mathcal{F}\right)^{\vee},\mathcal{O}_{X_{S}}\right)$
has depth at least $2$ at all points in the complement of $U$. Notice that
this follows from $(2)$ in [Sta21, Tag 0AV5], since $\mathcal{O}_{X_{S}}$ has
depth at least $2$ at all points in the complement of $U$ by Lemma 2.2.

  2. (b)

One can define a notion of determinant $\text{d}(\mathcal{F})$ using a finite
resolution by locally-free sheaves, as in [HL97, pg. 36,37]. By definition,
$\text{d}(\mathcal{F})$ is a line bundle that agrees with $\det(\mathcal{F})$
on the open subset $U$. By the same proof as in part $(a)$ applied to the line
bundle $\text{d}(\mathcal{F})$, it follows that $\text{d}(\mathcal{F})\cong
j_{*}(\det(\mathcal{F}|_{U})$. We conclude that
$\det(\mathcal{F})=j_{*}\left(\det(\mathcal{F}|_{U})\right)=\text{d}(\mathcal{F})$
is a line bundle.

  3. (c)

This follows because the notion of determinant $\text{d}(\mathcal{F})$ defined
in [HL97, pg. 36,37] using a fixed choice of a resolution can be seen to be
compatible with base-change, by pulling back the resolution.

∎

Recall that $r$ is the dimension of the vector space $V$ that we use to define
the faithful representation of the group $G$. The following is an open
substack of the algebraic stack of coherent sheaves on $X$ as in [LMB00,
4.6.2].

######  Definition 2.9.

We define $\operatorname{Coh}_{r}^{tf}(X)$ to be the pseudofunctor from
$\left(\operatorname{Aff}_{k}\right)^{op}$ into groupoids defined as follows.
For any affine scheme $T\in\operatorname{Aff}_{k}$, we set

$\displaystyle\operatorname{Coh}_{r}^{tf}(X)\,(T)\;=\;\left[\begin{matrix}\;\;\text{groupoid
of $T$-flat relative torsion-free sheaves $\mathcal{F}$ of rank $r$ on
$X_{T}$}\;\;\end{matrix}\right]$

######  Proposition 2.10.

$\operatorname{Coh}_{r}^{tf}(X)$ is an algebraic stack with affine diagonal
and locally of finite type over $k$.

###### Proof.

Since the rank of a family of torsion-free sheaves is a locally constant
function on the base, $\operatorname{Coh}^{tf}_{r}(X)$ is an open substack of
the stack $\operatorname{Coh}^{tf}(X)$ that parametrizes all torsion-free
sheaves. The proposition follows, because $\operatorname{Coh}^{tf}(X)$ is an
algebraic stack with affine diagonal and locally of finite type over $k$
[HLHJ21, Prop. 2.5]. ∎

######  Remark 2.11.

The stack $\operatorname{Coh}_{r}^{tf}(X)$ contains the stack
$\operatorname{Bun}_{r}(X)$ of rank $r$ vector bundles on $X$ as an open
substack.

###  2.3. The scheme of $G$-reductions

Let $S$ be a $k$-scheme and let
$\mathcal{F}\in\operatorname{Coh}_{r}^{tf}(X)(S)$ be a rank $r$ torsion-free
sheaf on $X_{S}$. In this subsection we modify the construction in [Sch02] in
order to define a scheme of $G$-reductions of $\mathcal{F}$. In our convention
the representation $\rho$ is the dual of the representation in [Sch02].

Let $Y$ be a $k$-scheme. Let $\mathcal{G}$ be a finitely presented
$\mathcal{O}_{Y}$-module. We can form the universal graded associative algebra
$T(\mathcal{G}):=\bigoplus_{j=0}^{\infty}\mathcal{G}^{\otimes j}$. There is a
graded ideal that on each affine open is generated by sections of the form
$x\otimes y-y\otimes x$. The quotient of $T(\mathcal{G})$ by this ideal is the
symmetric algebra $\operatorname{Sym}^{\bullet}(\mathcal{G})$. By definition
$\operatorname{Sym}^{\bullet}(\mathcal{G})$ is a graded commutative
$\mathcal{O}_{Y}$-algebra that is locally finitely presented as an algebra.

Suppose that the sheaf $\mathcal{G}$ admits a $k[G]$-comodule structure. This
induces a comodule structure on $T(\mathcal{G})$, where we equip each graded
component $\mathcal{G}^{\otimes j}$ with the tensor comodule structure. The
ideal used to form $\operatorname{Sym}^{\bullet}(\mathcal{G})$ is a
$k[G]$-subcomodule. It follows that the quotient
$\operatorname{Sym}^{\bullet}(\mathcal{G})$ acquires a $k[G]$-comodule
structure that is compatible with the multiplication and the grading.

######  Definition 2.12.

Let $S$ be a $k$-scheme and let $\mathcal{F}$ be a torsion-free sheaf of rank
$r$ on $X_{S}$. We set

$E_{V}(\mathcal{F})\vcentcolon=\operatorname{Sym}^{\bullet}(\mathcal{F}\otimes_{k}V^{\vee})\otimes_{\mathcal{O}_{X_{S}}}\operatorname{Sym}^{\bullet}(\det(\mathcal{F})^{\vee}\otimes_{k}\det(V))$

We also define
$H(\mathcal{F},V)\vcentcolon=\underline{\operatorname{Spec}}_{X_{S}}(E_{V}(\mathcal{F}))$.
Note that the structure morphism $H(\mathcal{F},V)\rightarrow X_{S}$ is affine
and of finite presentation.

The vector space $V$ acquires a left $k[G]$-comodule structure $V\rightarrow
V\otimes k[G]$ via the representation $\rho:G\rightarrow\operatorname{GL}(V)$.
We equip $V^{\vee}$ with the dual right comodule structure
$c:V^{\vee}\rightarrow V^{\vee}\otimes k[G]$. This induces a comodule
structure
$\mathcal{F}\otimes_{k}V^{\vee}\rightarrow(\mathcal{F}\otimes_{k}V^{\vee})\otimes
k[G]$ given by $\text{id}_{\mathcal{F}}\otimes c$. By the discussion above,
the graded algebra
$\operatorname{Sym}^{\bullet}(\mathcal{F}\otimes_{k}V^{\vee})$ inherits a
right $k[G]$-comodule structure.

On the other hand, $\det(V)$ acquires a left $k[G]$-comodule structure via the
determinant representation of $V$. By post-composing with the antipode of the
Hopf algebra $k[G]$ (i.e. pre-composing the representation with inversion on
$G$), we get a right $k[G]$-comodule structure on $\det(V)$. In a similar way
as above, this induces a right $k[G]$-comodule structure on the algebra
$\operatorname{Sym}^{\bullet}(\det(\mathcal{F})^{\vee}\otimes_{k}\det(V))$.

By taking the tensor comodule structure for the $k[G]$-comodules defined
above, we can equip $E_{V}(\mathcal{F})$ with a right $k[G]$-comodule
structure

$c_{E_{V}(\mathcal{F})}:E_{V}(\mathcal{F})\rightarrow
E_{V}(\mathcal{F})\otimes_{k}k[G]$

Since this comodule structure is compatible with the algebra multiplication,
it translates into a left action of the algebraic group $G$ on the scheme
$H(\mathcal{F},V)$.

Consider the subalgebra $E_{V}(\mathcal{F})^{G}$ of $G$-invariants of
$E_{V}(\mathcal{F})$. Recall that this is defined over each affine open as the
$\mathcal{O}_{X_{S}}$-subalgebra of $E_{V}(\mathcal{F})$ consisting of
sections $s$ such that $c_{E_{V}(\mathcal{F})}(s)=s\otimes 1_{k[G]}$. Since
the characteristic of $k$ is $0$ and $G$ is reductive, the algebra of
invariants $E_{V}(\mathcal{F})^{G}$ is a locally finitely presented
$\mathcal{O}_{X_{S}}$-algebra and its formation is compatible with arbitrary
base-change. This can be checked affine locally, and over each affine it
follows from the existence of the Reynolds operator [MFK94, Chpt. 1, §1,2]. By
definition, the scheme
$\underline{\operatorname{Spec}}_{X_{S}}\left((E_{V}(\mathcal{F}))^{G}\right)$
is the affine GIT quotient $H(\mathcal{F},V)/\\!\\!/\,G$ relative to $X_{S}$.

Let $U$ be a big open subscheme of $X_{S}$ where $\mathcal{F}$ is locally-
free. The restriction $H(\mathcal{F},V)|_{U}$ is, by definition, the scheme

$H\vcentcolon=\operatorname{Hom}\left(\mathcal{F}|_{U},V\otimes_{k}\mathcal{O}_{U}\right)\times\operatorname{Hom}\left(\det(\mathcal{F}|_{U})^{\vee},\,\det(V^{\vee})\otimes_{k}\mathcal{O}_{U}\right)$

parametrizing pairs $(f_{1},f_{2})$ of homomorphisms of vector bundles

$f_{1}:\mathcal{F}|_{U}\rightarrow
V\otimes_{k}\mathcal{O}_{U}\;\;\;\;\text{and}\;\;\;\;f_{2}:\det(\mathcal{F}|_{U})^{\vee}\rightarrow\det(V^{\vee})\otimes_{k}\mathcal{O}_{U}$

This means that there is a universal pair $(u_{1},u_{2})$ of morphisms

$u_{1}:\mathcal{F}|_{H}\rightarrow
V\otimes_{k}\mathcal{O}_{H}\;\;\;\;\text{and}\;\;\;\;u_{2}:\det(\mathcal{F}|_{H})^{\vee}\rightarrow\det(V^{\vee})\otimes\mathcal{O}_{H}$

We take the $r^{th}$ wedge of $u_{1}$ to get a universal determinant morphism.

$\wedge^{r}u_{1}:\det(\mathcal{F}|_{H})\rightarrow\det(V)\otimes\mathcal{O}_{H}$

We set $\delta_{U}\vcentcolon=(\wedge^{r}u_{1})\otimes u_{2}$. Using the
canonical trivializations $\det(V)\otimes\det(V^{\vee})\xrightarrow{\sim}k$
and
$\det(\mathcal{F}|_{H})\otimes\det(\mathcal{F}|_{H})^{\vee}\xrightarrow{\sim}\mathcal{O}_{H}$,
we can view this as a map
$\delta_{U}:\mathcal{O}_{H}\rightarrow\mathcal{O}_{H}$. We can then form the
map $\delta_{U}-1:\mathcal{O}_{H}\rightarrow\mathcal{O}_{H}$ obtained by
subtracting the identity of $\mathcal{O}_{H}$. Notice that $\delta_{U}-1$ cuts
out the divisor on $H$ given by the locus where $u_{1}$ is an isomorphism and
$u_{2}$ is the determinant of the inverse dual of $u_{1}$. In particular the
vanishing locus of $\delta_{U}-1$ is canonically isomorphic to the scheme
parametrizing linear isomorphisms of vector bundles
$\operatorname{Iso}(\mathcal{F}|_{U},V\otimes_{k}\mathcal{O}_{U})$. This
identification is given by forgetting $u_{2}$.

By construction, the composition
$\mathcal{O}_{H/\\!\\!/\,G}\hookrightarrow\mathcal{O}_{H}\xrightarrow{\delta_{U}-1}\mathcal{O}_{H}$
lands in the $\mathcal{O}_{U}$-subalgebra of invariants
$(E_{V}(\mathcal{F})|_{U})^{G}$. Therefore this composition induces a well-
defined map
$\alpha_{U}:\mathcal{O}_{H/\\!\\!/\,G}\rightarrow\mathcal{O}_{H/\\!\\!/\,G}$
that cuts out an effective divisor on $H/\\!\\!/\,G$. We denote this closed
subscheme by $(H/\\!\\!/\,G)_{\alpha_{U}=0}\hookrightarrow H/\\!\\!/\,G$.

######  Remark 2.13.

Alternatively, we can think of the construction above as follows. The divisor
$D$ cut out by the equation $\delta_{U}-1$ on $H$ is $G$-stable. Then
$(H/\\!\\!/\,G)_{\alpha_{U}=0}$ is the unique closed subscheme in the GIT
quotient $H/\\!\\!/\,G$ such that the preimage recovers $D$. It follows that
$(H/\\!\\!/\,G)_{\alpha_{U}=0}=(H_{\delta_{U}=1})/\\!/\\!\,G$.

######  Definition 2.14.

$\operatorname{Red}_{G}(\mathcal{F})$ is defined to be the scheme theoretic
closure of $(H/\\!\\!/\,G)_{\alpha_{U}=0}$ inside
$H(\mathcal{F},V)/\\!\\!/\,G$. In other words,
$\operatorname{Red}_{G}(\mathcal{F})$ is the scheme theoretic image of the
composition

$(H/\\!\\!/\,G)_{\alpha_{U}=0}\hookrightarrow H/\\!\\!/\,G\hookrightarrow
H(\mathcal{F},V)/\\!\\!/\,G$

where the second map $H/\\!\\!/\,G\hookrightarrow H(\mathcal{F},V)/\\!\\!/\,G$
is the open immersion that comes from identifying
$H/\\!\\!/\,G\xrightarrow{\sim}\left(H(\mathcal{F},V)/\\!\\!/\,G\right)|_{U}$
via the compatibility of taking $G$-invariants with flat base-change.

######  Remark 2.15.

It should be noted that the construction of
$\operatorname{Red}_{G}(\mathcal{F})$ does not depend on the choice of $U$.
This can be seen as follows. For any two big open subsets $U_{1}$ and $U_{2}$
of $X_{S}$, the intersection $U_{1}\cap U_{1}$ is also big. Since the element
$\alpha_{U}$ is canonically defined above, the restrictions of
$\alpha_{U_{1}}$ and $\alpha_{U_{2}}$ to the subset $U_{1}\cap U_{2}$ agree
with $\alpha_{U_{1}\cap U_{2}}$. Notice that $H$ is always flat over $U$,
because it is locally given by a polynomial algebra. It follows that
$H/\\!/\\!\,G$ is also flat over $U$, because taking $G$ invariants is exact
by the existence of the Reynolds operator. Since $U$ is scheme theoretically
dense in $X_{S}$ by Lemma 2.2, it follows that the open
$(H/\\!/\\!\,G)|_{U_{1}\cap U_{2}}$ is scheme theoretically dense in
$(H/\\!/\\!\,G)|_{U_{1}}$ and $(H/\\!/\\!\,G)|_{U_{2}}$. In particular the
closures of $(H/\\!/\\!\,G)_{\alpha_{U_{1}}=0}$ and
$(H/\\!/\\!\,G)_{\alpha_{U_{2}}=0}$ both agree with the closure of
$(H/\\!/\\!\,G)_{\alpha_{U_{1}\cap U_{2}}=0}$.

The following lemma is an immediate consequence of the constructions above.

######  Lemma 2.16.

The scheme $\operatorname{Red}_{G}(\mathcal{F})$ is affine and of finite
presentation over $X_{S}$. The formation of
$\operatorname{Red}_{G}(\mathcal{F})$ is compatible with base-change. In other
words, for any morphism of $k$-schemes $T\rightarrow S$ there is a canonical
isomorphism
$\operatorname{Red}_{G}(\mathcal{F})\times_{X_{S}}X_{T}\xrightarrow{\sim}\operatorname{Red}_{G}\left(\mathcal{F}|_{X_{T}}\right)$.
∎

Let $U$ be a big open subset of $X_{S}$ where $\mathcal{F}$ is locally-free.
Since $\mathcal{F}|_{U}$ is a vector bundle of rank $r$ on $U$, we can
naturally associate to it a $\operatorname{GL}(V)$-bundle $\mathcal{P}$ on
$U$.

######  Proposition 2.17.

Let $\mathcal{F}$ and $U$ be as above. Let $\mathcal{P}$ denote the
corresponding $\operatorname{GL}(V)$-bundle on $U$. Then there is a canonical
identification of $U$-schemes
$\operatorname{Red}_{G}(\mathcal{F})|_{U}\xrightarrow{\sim}\mathcal{P}/G$.
Here the quotient on the right is formed via the inclusion
$\rho:G\hookrightarrow\operatorname{GL}(V)$.

###### Proof.

Let $H_{\delta_{U}=1}$ denote the closed subscheme of $H$ cut out by the
equation $\delta_{U}-1$, as described in the construction of
$\operatorname{Red}_{G}(\mathcal{F})$ above. By definition,
$\operatorname{Red}_{G}(\mathcal{F})|_{U}$ is the affine GIT quotient
$(H_{\delta_{U}=1})/\\!/\\!\,G$. We have seen that $H_{\delta_{U}=1}$ is
canonically isomorphic to the $U$-scheme
$\operatorname{Iso}(\mathcal{F}|_{U},V\otimes_{k}\mathcal{O}_{U})$
parametrizing isomorphisms of vector bundles. This identification respects the
natural left $G$-actions defined on both schemes via the inclusions
$\rho:G\rightarrow\operatorname{GL}(V)$.

The map that takes an isomorphism
$\varphi:\mathcal{F}|_{U}\oplus\det(\mathcal{F})^{\vee}|_{U}\xrightarrow{\sim}(V\oplus\det(V^{\vee}))\otimes_{k}\mathcal{O}_{U}$
to its inverse in
$\operatorname{Hom}\left((V\oplus\det(V^{\vee}))\otimes_{k}\mathcal{O}_{U},\,\mathcal{F}|_{U}\oplus\det(\mathcal{F})^{\vee}|_{U}\right)$
induces an isomorphism of schemes

$(-)^{-1}:\;\operatorname{Iso}(\mathcal{F}|_{U},V\otimes_{k}\mathcal{O}_{U})\xrightarrow{\sim}\operatorname{Iso}\left(V\otimes_{k}\mathcal{O}_{U},\,\mathcal{F}|_{U}\right)$

The right-hand side admits a right action of $\operatorname{GL}(V)$ given by
precomposition. By definition
$\operatorname{Iso}(V\otimes_{k}\mathcal{O}_{U},\,\mathcal{F}|_{U})$ is the
total space of the $\operatorname{GL}(V)$-bundle $\mathcal{P}$ equipped with
its right $\operatorname{GL}(V)$-action. The inclusion
$\rho:G\hookrightarrow\operatorname{GL}(V)$ induces a right $G$-action on
$\mathcal{P}$. The inversion map $(-)^{-1}$ above intertwines the left
$G$-action on the left hand side and the right $G$-action on the right hand
side.

Since the representation $\rho$ is faithful, it follows that $G$ acts properly
and freely on $\mathcal{P}$. Therefore the GIT quotient
$\mathcal{P}/\\!/\\!\,G$ and the quotient as étale sheaves $\mathcal{P}/G$
coincide. This way we get the sought after canonical identification of
$\operatorname{Red}_{G}(\mathcal{F})|_{U}\xrightarrow{\sim}\mathcal{P}/G$. ∎

The following proposition explains the connection between the scheme
$\operatorname{Red}_{G}(\mathcal{F})$ and $G$-reductions of structure group.

######  Proposition 2.18.

For any big open subset $U\subset X_{S}$ where $\mathcal{F}$ is locally-free,
there is a bijection between the following two sets $(A),(B)$:

  1. (A)

The set of sections of the structure morphism
$\operatorname{Red}_{G}(\mathcal{F})\rightarrow X_{S}$.

  2. (B)

The set of $G$-reductions of structure group for the
$\operatorname{GL}(V)$-bundle $\mathcal{P}$ corresponding to
$\mathcal{F}|_{U}$.

###### Proof.

$\operatorname{Red}_{G}(\mathcal{F})\rightarrow X_{S}$ is an affine morphism
of finite type. Lemma 2.2 shows that the restriction of maps induces a
bijection between sections
$\sigma:X_{S}\rightarrow\operatorname{Red}_{G}(\mathcal{F})$ defined over the
whole $X_{S}$ and sections
$\sigma_{U}:U\rightarrow\operatorname{Red}_{G}(\mathcal{F})$ defined over $U$.
By Proposition 2.17, sections of
$\operatorname{Red}_{G}(\mathcal{F})|_{U}\rightarrow U$ are in canonical
correspondence with sections of $\mathcal{P}/G\rightarrow U$ via the inversion
map. A section of $\mathcal{P}/G\rightarrow U$ is the same as a $G$-reduction
of structure group for $\mathcal{P}$. This establishes the canonical bijection
between $(A)$ and $(B)$. ∎

##  3\. Principal $\rho$-sheaves

###  3.1. Principal $\rho$-sheaves and the moduli stack $\text{Bun}_{\rho}(X)$

######  Definition 3.1.

Let $S$ be a $k$-scheme. A principal $\rho$-sheaf on $X_{S}$ consists of a
pair $(\mathcal{F},\sigma)$, where

  1. (1)

$\mathcal{F}$ is a family of torsion-free sheaves of rank $r$ on $X_{S}$.

  2. (2)

$\sigma$ is a section of the structure morphism
$\operatorname{Red}_{G}(\mathcal{F})\rightarrow X_{S}$.

######  Remark 3.2.

We will often omit the word “principal” and refer to these objects simply as
$\rho$-sheaves.

By Proposition 2.18, one can think of a $\rho$-sheaf over $X_{S}$ as the data
of

  1. (1)

A torsion-free sheaf $\mathcal{F}$ of rank $r$ on $X_{S}$.

  2. (2)

A reduction of structure group to $G$ for the $\operatorname{GL}(V)$ bundle
$\mathcal{F}|_{U}$, where $U$ is (any) big open subscheme of $X_{S}$ such that
$\mathcal{F}|_{U}$ is locally-free.

######  Remark 3.3.

Assume that the image of the representation $\rho$ is contained in the
subgroup $\operatorname{SL}(V)\subset\operatorname{GL}(V)$. Then our
$\rho$-sheaves can be viewed as singular principal $G$-bundles [GLSS08] for
the dual representation $\rho^{\vee}$. This readily follows from the
description given above.

######  Definition 3.4.

Let $(\mathcal{F}_{1},\sigma_{1})$ and $(\mathcal{F}_{2},\sigma_{2})$ be two
$\rho$-sheaves on $X_{S}$. By construction, an isomorphism
$\varphi:\mathcal{F}_{1}\xrightarrow{\sim}\mathcal{F}_{2}$ induces an
isomorphism of $G$-reduction schemes
$\operatorname{Red}_{G}(\mathcal{F}_{1})\xrightarrow{\sim}\operatorname{Red}_{G}(\mathcal{F}_{2})$.
We say that $\varphi$ is an isomorphism of the $\rho$-sheaves if the sections
$\sigma_{1}$ and $\sigma_{2}$ are identified under this induced isomorphism.

Using this notion of isomorphism, we endow the class of $\rho$-sheaves on
$X_{S}$ with the structure of a groupoid. By Lemma 2.16, the formation of
$\operatorname{Red}_{G}(\mathcal{F})$ is compatible with base-change. In
particular, it makes sense to pull back principal $\rho$-sheaves under a
morphism of $k$-schemes $T\rightarrow S$.

######  Definition 3.5.

We define $\operatorname{Bun}_{\rho}(X)$ to be the pseudofunctor from
$\left(\operatorname{Aff}_{k}\right)^{op}$ into groupoids defined as follows.
For any affine scheme $T\in\operatorname{Aff}_{k}$, we set

$\operatorname{Bun}_{\rho}(X)\,(T)\;=\;\left[\;\;\text{groupoid of
$\rho$-sheaves $(\mathcal{F},\sigma)$ on $X_{T}$}\;\;\;\right]$

There is a natural forgetful morphism
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ given
by $(\mathcal{F},\sigma)\mapsto\mathcal{F}$.

######  Proposition 3.6.

The forgetful morphism
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ is
schematic, affine and of finite type. In particular
$\operatorname{Bun}_{\rho}(X)$ is an algebraic stack with affine diagonal and
locally of finite type over $k$.

###### Proof.

Let $T\in\operatorname{Aff}_{k}$. Choose a morphism
$T\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ represented by a relative
torsion-free sheaf $\mathcal{F}$ on $X_{T}$. The fiber product
$\operatorname{Bun}_{\rho}(X)\times_{\operatorname{Coh}_{r}^{tf}(X)}T$ is the
functor that parametrizes sections of the structure morphism
$\operatorname{Red}_{G}(\mathcal{F})\rightarrow X_{T}$. Since we know that
$\operatorname{Red}_{G}(\mathcal{F})\rightarrow X_{T}$ is affine and of finite
presentation, we can apply [Beh91, Prop. 4.4.1] (or [HR15, Lemma 2.3 (i)] \+
[HR19, Thm. 1.3 (ii)]) to conclude that
$\operatorname{Bun}_{\rho}(X)\times_{\operatorname{Coh}_{r}^{tf}(X)}T$ is
represented by a scheme that is relatively affine and of finite type over $T$.
∎

Recall that we have an open immersion
$\operatorname{Bun}_{r}(X)\hookrightarrow\operatorname{Coh}_{r}^{tf}(X)$ from
the stack $\operatorname{Bun}_{r}(X)$ of rank $r$ vector bundles on $X$. The
following shows that the same holds for $\operatorname{Bun}_{\rho}(X)$ and the
stack of principal $G$-bundles.

######  Proposition 3.7.

There is a Cartesian square

${\operatorname{Bun}_{G}(X)}$${\operatorname{Bun}_{\rho}(X)}$${\operatorname{Bun}_{r}(X)}$${\operatorname{Coh}_{r}^{tf}(X)}$$\xhookrightarrow{\;\;}$$\scriptstyle{\rho_{*}}$$\xhookrightarrow{\;\;}$

where $\operatorname{Bun}_{G}(X)$ is the stack of principal $G$-bundles on $X$
and $\rho_{*}$ is the map on stacks induced by extension of the structure
group via the representation $\rho$.

###### Proof.

Let $S\in\operatorname{Aff}_{k}$. The groupoid
$\operatorname{Bun}_{r}(X)\,(S)$ consists of points
$\mathcal{F}\in\operatorname{Coh}_{r}^{tf}(X)\,(S)$ where $\mathcal{F}$ is
locally-free on the entirety of $X_{S}$. Applying Proposition 2.18 with
$U=X_{S}$, we see that the fiber product
$\operatorname{Bun}_{r}(X)\times_{\operatorname{Coh}_{r}^{tf}(X)}\operatorname{Bun}_{\rho}(X)\,(S)$
parametrizes pairs $(\mathcal{P},\theta)$, where $\mathcal{P}$ is a
$\operatorname{GL}(V)$-bundle on $X_{S}$ and $\theta$ is a reduction of
structure group to $G$. This groupoid is naturally isomorphic to the groupoid
of $G$-bundles on $X_{S}$. ∎

###  3.2. Filtrations of $\rho$-sheaves

We will “destackify” the notion of $\Theta$-filtration [HL14, Hei17, AHLH19]
and explain what it means in the context of $\rho$-sheaves. For this
subsection we will work over a field extension $K\supset k$.

Let us recall the definition of a $\Theta$-filtration. We denote by
$\Theta_{K}$ the base-change
$\left[\,\mathbb{A}^{1}_{K}/\mathbb{G}_{m}\right]$. The stack $\Theta_{K}$ has
a unique closed point $0$ corresponding to the origin of $\mathbb{A}^{1}_{K}$.
It also has an open point $1$, which is the image of the $K$-point $1$ in
$\mathbb{A}^{1}_{K}$. Let $\mathcal{M}$ be a stack over $k$ and let
$p\in\mathcal{M}(K)$ be a $K$-point. A $\Theta$-filtration of the point $p$ is
a morphism of stacks $f:\Theta_{K}\rightarrow\mathcal{M}$ along with an
isomorphism $f(1)\xrightarrow{\sim}p$. The graded point
$f|_{[0/\mathbb{G}_{m}]}$ is called the associated graded of the filtration.
We say that the filtration is nondegenerate if the action of $\mathbb{G}_{m}$
on the object $f(0)$ is nontrivial.

We start by giving a concrete description of $\Theta$-filtrations in the stack
$\operatorname{Coh}_{r}^{tf}(X)$.

######  Proposition 3.8.

Let $\mathcal{F}$ be a $K$-point of $\operatorname{Coh}^{tf}_{r}(X)$. A
$\Theta$-filtration $f:\Theta_{K}\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ of
$\mathcal{F}$ corresponds to a $\mathbb{Z}$-filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ by $\mathcal{O}_{X_{K}}$-subsheaves of
$\mathcal{F}$ satisfying the following properties

  1. (1)

$\mathcal{F}_{m+1}\subset\mathcal{F}_{m}$.

  2. (2)

$\mathcal{F}_{m}=0$ for $m\gg 0$ and $\mathcal{F}_{m}=\mathcal{F}$ for $m\ll
0$.

  3. (3)

$\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is torsion-free.

An isomorphism between two such filtrations is an automorphism of
$\mathcal{F}$ that identifies the filtrations. The graded object at the origin
$f|_{[0/\mathbb{G}_{m}]}:[0/\mathbb{G}_{m}]\rightarrow\operatorname{Coh}_{r}^{tf}(X)$
is the associated graded sheaf
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$.

###### Proof.

By definition, a $\Theta$-filtration of $\mathcal{F}$ consists of the
following data

  1. (a)

A $\mathbb{G}_{m}$-equivariant family $\widetilde{\mathcal{F}}$ of torsion-
free sheaves on $X\times\mathbb{A}^{1}_{K}$.

  2. (b)

An identification $\widetilde{\mathcal{F}}|_{X\times
1}\xrightarrow{\sim}\mathcal{F}$.

For any $\mathbb{Z}$-filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$
satisfying $(1)$ and $(2)$ above, we can associate a
$\mathbb{G}_{m}$-equivariant coherent sheaf on $X\times\mathbb{A}^{1}_{K}$ via
the Rees construction [HL14, Prop. 1.0.1]. Namely, we can set
$\widetilde{\mathcal{F}}\vcentcolon=\bigoplus_{m\in\mathbb{Z}}t^{-m}\mathcal{F}_{m}$.
Here $\mathbb{G}_{m}$ acts on $t^{-m}\mathcal{F}_{m}$ with weight $m$. By
construction we have that
$\widetilde{\mathcal{F}}|_{X\times(\mathbb{A}^{1}_{K}\setminus 0)}$ is the
constant family $\mathcal{F}|_{X\times(\mathbb{A}^{1}_{K}\setminus 0)}$. All
$\mathbb{G}_{m}$-equivariant coherent sheaves $\widetilde{\mathcal{F}}$ on
$X\times\mathbb{A}^{1}_{K}$ along with an isomorphism
$\widetilde{\mathcal{F}}|_{X\times 1}\xrightarrow{\sim}\mathcal{F}$ arise by
applying the Rees construction to a $\mathbb{Z}$-filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ satisfying conditions $(1)$ and $(2)$
above [HL14, Prop. 1.0.1].

The restriction $\widetilde{\mathcal{F}}|_{X\times 0}$ is the associated
graded $\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$. So we
see that $\widetilde{\mathcal{F}}$ is a family of torsion-free sheaves if and
only if $\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is
torsion-free on $X_{K}$. This, in turn, is equivalent to condition $(3)$
above. ∎

For any filtration $\Theta_{K}\rightarrow\operatorname{Bun}_{\rho}(X)$ we can
post-compose with the forgetful morphism
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ to get
a filtration $\Theta_{K}\rightarrow\operatorname{Coh}_{r}^{tf}(X)$. This
defines a functor $F_{K}$ between the groupoid of $\Theta$-filtrations of
$K$-points of $\operatorname{Bun}_{\rho}(X)$ and the groupoid of
$\Theta$-filtrations of $K$-points of $\operatorname{Coh}_{r}^{tf}(X)$.

######  Proposition 3.9.

The functor $F_{K}$ defined above is faithful.

###### Proof.

This follows from [AHLH19, Lem. 3.7 (3)], which applies here because the
forgetful morphism is affine by Proposition 3.6. ∎

This tells us that we can view filtrations of $\operatorname{Bun}_{\rho}(X)$
as a subcategory of filtrations in $\operatorname{Coh}_{r}^{tf}(X)$. In other
words, if $(\mathcal{F},\sigma)$ is a $\rho$-sheaf, then a filtration of
$(\mathcal{F},\sigma)$ is the same thing as a $\mathbb{Z}$-filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$ (as in Proposition 3.8)
satisfying some “compatibility conditions”. We will often refer to
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ as the underlying associated filtration.

Our goal for the remainder of this section is to explain these “compatibility”
conditions. First, we need to explain what is a filtration of
$\operatorname{GL}(V)$-bundles on any scheme $U$.

######  Proposition 3.10.

Let $U$ be a scheme over $K$. Let $\mathcal{F}$ be a
$\operatorname{GL}(V)$-bundle on $U$. Then the following two groupoids are
equivalent.

  1. (1)

The groupoid of $\operatorname{GL}(V)$-bundles $\mathcal{P}$ on
$U\times_{K}\Theta_{K}$ along with an isomorphism of the restriction
$\mathcal{P}|_{U\times_{K}1}$ with $\mathcal{F}$.

  2. (2)

The groupoid of decreasing $\mathbb{Z}$-weighted filtrations
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$ by vector subbundles
$\mathcal{F}_{m}$. An isomorphism between two filtrations is just an
automorphism of $\mathcal{F}$ that identifies the filtrations.

###### Proof.

The equivalence is given by the Rees construction. The argument is the same as
in Proposition 3.8, we omit the details. ∎

Now we are ready to give a first description of $\Theta$-filtrations for a
$\rho$-sheaf. Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. A
filtration of $(\mathcal{F},\sigma)$ will consist of a pair
$((\mathcal{F}_{m})_{m\in\mathbb{Z}},\,\mathcal{P})$. Here
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ is a filtration of $\mathcal{F}$, as
described in Proposition 3.8. Let $U\hookrightarrow X_{K}$ be the maximal big
open subset of $X_{K}$ where the associated graded
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is locally-free.
The second element of the pair $\mathcal{P}$ is a $G$-bundle on
$U\times_{K}\Theta_{K}$ such that the associated $\operatorname{GL}(V)$-bundle
$\rho_{*}(\mathcal{P})$ on $U\times_{K}\Theta_{K}$ corresponds to the
restricted filtration $(\mathcal{F}_{m}|_{U})_{m\in\mathbb{Z}}$ under the
equivalence described in Proposition 3.10.

The filtration $(\mathcal{F}_{m}|_{U})_{m\in\mathbb{Z}}$ on $U$ completely
determines the underlying associated filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ on $X_{K}$. This is because each
$\mathcal{F}_{m}$ is saturated on $\mathcal{F}$, so it is determined in
codimension 1 [Lan75, pg. 90-92]. We can describe $\mathcal{F}_{m}$ more
explicitly in terms of reflexive hulls. Let $\widetilde{\mathcal{F}}_{m}$
denote the pushforward of the vector bundle $\mathcal{F}_{m}|_{U}$ under the
open immersion $U\hookrightarrow X_{K}$. We have that
$\widetilde{\mathcal{F}}_{m}$ is a subsheaf of the reflexive hull
$\mathcal{F}^{\vee\vee}$ of $\mathcal{F}$. Then we can recover
$\mathcal{F}_{m}$ as the intersection
$\widetilde{\mathcal{F}}_{m}\cap\mathcal{F}$ taken inside of
$\mathcal{F}^{\vee\vee}$. In particular the filtration is completely
determined by the $G$-bundle $\mathcal{P}$ on $U\times_{K}\Theta_{K}$. We
will, however, keep the underlying associated filtration as part of the
notation.

A morphism between two such filtrations
$((\mathcal{F}_{m})_{m\in\mathbb{Z}},\mathcal{P})$ and
$((\mathcal{F}^{\prime}_{m})_{m\in\mathbb{Z}},\mathcal{P}^{\prime})$ consists
of an isomorphism of $G$-bundles
$\varphi:\mathcal{P}\xrightarrow{\sim}\mathcal{P}^{\prime}$ defined over some
big open subset $U\subset X_{K}$ where both $\mathcal{P}$ and
$\mathcal{P}^{\prime}$ are defined. We require that the associated isomorphism
of $\operatorname{GL}(V)$-bundles on $U\times_{K}\Theta_{K}$ given by
$\rho_{*}(\varphi):(\mathcal{F}_{m}|_{U})_{m\in\mathbb{Z}}\xrightarrow{\sim}(\mathcal{F}^{\prime}_{m}|_{U})_{m\in\mathbb{Z}}$
admits a (unique) extension to an isomorphism of filtrations
$\widetilde{\varphi}:(\mathcal{F}_{m})_{m\in\mathbb{Z}}\xrightarrow{\sim}(\mathcal{F}^{\prime}_{m})_{m\in\mathbb{Z}}$
defined over all of $X_{K}$. By the description of the filtrations as
intersections of reflexive hulls, it follows that we only need to check that
the isomorphism $\rho_{*}(\varphi)|_{U\times_{K}1}$ extends to an automorphism
$\mathcal{F}\xrightarrow{\sim}\mathcal{F}$. This is automatically satisfied if
$\mathcal{F}$ is reflexive, but not otherwise.

######  Definition 3.11.

Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. We will denote by
$\operatorname{Filt}(\mathcal{F},\sigma)$ the groupoid of pairs
$((\mathcal{F}_{m})_{m\in\mathbb{Z}},\mathcal{P})$, with isomorphisms as
described above.

######  Proposition 3.12.

Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. There is a natural
equivalence between the following two groupoids.

  1. (1)

The groupoid of $\Theta$-filtrations
$f:\Theta_{K}\rightarrow\operatorname{Bun}_{\rho}(X)$ of the $\rho$-sheaf
$(\mathcal{F},\sigma)$.

  2. (2)

The groupoid $\operatorname{Filt}(\mathcal{F},\sigma)$.

###### Proof.

The groupoid of $\Theta$-filtrations classifies $\mathbb{G}_{m}$-equivariant
$\rho$-sheaves $(\widetilde{\mathcal{F}},\widetilde{\sigma})$ on
$X\times\mathbb{A}^{1}_{K}$ along with an isomorphism of the restriction
$(\widetilde{\mathcal{F}},\widetilde{\sigma})|_{X\times 1}$ with the original
$(\mathcal{F},\sigma)$. So $\widetilde{\mathcal{F}}$ is a $\Theta$-filtration
of $\mathcal{F}$ and $\widetilde{\sigma}$ is a $\mathbb{G}_{m}$-equivariant
section
$\widetilde{\sigma}:X\times\mathbb{A}^{1}_{K}\rightarrow\operatorname{Red}_{G}(\widetilde{\mathcal{F}})$
extending $\sigma:X\times 1\rightarrow\operatorname{Red}_{G}(\mathcal{F})$.

By Proposition 3.8, the bundle $\widetilde{\mathcal{F}}$ is obtained by
applying the Rees construction to some $\mathbb{Z}$-filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$. Let $U$ be the maximal open in $X_{K}$
where $\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is
locally-free. Then $U\times_{K}\mathbb{A}^{1}_{K}$ is a big open in
$X\times\mathbb{A}^{1}_{K}$ and
$\widetilde{\mathcal{F}}|_{U\times_{K}\mathbb{A}^{1}_{K}}$ is locally-free. By
Proposition 2.18, the $\mathbb{G}_{m}$-equivariant section
$\widetilde{\sigma}$ is equivalent to the data of a
$\mathbb{G}_{m}$-equivariant $G$-reduction of the
$\operatorname{GL}(V)$-bundle
$\widetilde{\mathcal{F}}|_{U\times_{K}\mathbb{A}^{1}_{K}}$. This is by
definition a $\mathbb{G}_{m}$-equivariant $G$-bundle $\mathcal{P}$ on
$U\times_{K}\mathbb{A}^{1}_{K}$ such that the associated
$\operatorname{GL}(V)$-bundle $\rho_{*}(\mathcal{P})$ on
$U\times_{K}\Theta_{K}$ recovers the restriction of the filtration
$(\mathcal{F}_{m}|_{U})_{m\in\mathbb{Z}}$ under the equivalence in Proposition
3.10. This shows how to define the functor on objects, and it also shows that
it is essentially surjective. The definition of the functor on morphisms and
the fact that it is fully faithful follows from the definition of the
isomorphisms in the groupoid $\operatorname{Filt}(\mathcal{F},\sigma)$. ∎

###  3.3. Filtrations of $\rho$-sheaves in the split case

We will describe the elements of $\operatorname{Filt}(\mathcal{F},\sigma)$
more explicitly in the case when $G_{K}$ is split. For that purpose, we need a
better description of $G$-bundles on $U\times_{K}\Theta_{K}$. This is done in
[Hei17, 1.F], which gives a group theoretic description in terms of parabolic
reductions and a generalized Rees construction333We note that some of the
discussion in [Hei17, 1.F] applies without the hypothesis that $G_{K}$ is
split. However, the results are more concrete under this assumption..

We start by recalling some facts about parabolic subgroups of reductive
groups. Let $H$ be a reductive group over $K$. Let $\lambda$ be a cocharacter
$\lambda:\mathbb{G}_{m}\rightarrow H$. We can associate a corresponding
parabolic subgroup $P_{\lambda}$. For any $K$-scheme $T$, its $T$-points are

$P_{\lambda}(T)=\left\\{\,p\in H(T)\,\mid\,\lim\limits_{x\to
0}\,\lambda(x)\,p\,\lambda(x)^{-1}\;\text{exists}\,\right\\}$

Furthermore, the cocharater $\lambda$ induces a Levi decomposition
$P_{\lambda}=L_{\lambda}\ltimes U_{\lambda}$, where

$U_{\lambda}(T)=\left\\{\,u\in H(T)\,\mid\,\lim\limits_{x\to
0}\,\lambda(x)\,u\,\lambda(x)^{-1}=1_{H}\,\right\\}$
$L_{\lambda}(T)=\left\\{\,l\in H(T)\,\mid\,\lim\limits_{x\to
0}\,\lambda(x)\,l\,\lambda(x)^{-1}=l\,\right\\}$

######  Definition 3.13.

Let $H$ be a reductive linear algebraic group over $K$. Let $Y$ be a
$K$-scheme and let $\mathcal{H}$ be a $H$-bundle on $Y$. A weighted parabolic
reduction is a pair $(\lambda,\mathcal{H}_{\lambda})$, where
$\lambda:\mathbb{G}_{m}\rightarrow H$ is a cocharacter and
$\mathcal{H}_{\lambda}$ is a reduction of structure group of $\mathcal{H}$ to
the parabolic subgroup $P_{\lambda}$.

Let $\theta:H\rightarrow F$ be a homomorphism of reductive groups over $K$.
For every cocharacter $\lambda:\mathbb{G}_{m}\rightarrow H$, we can compose
with $\theta$ to obtain a cocharacter $\theta\circ\lambda$ of $F$. Notice that
$\theta$ restricts to a well defined homomorphism of parabolic subgroups
$\theta_{P}:P_{\lambda}\rightarrow P_{\theta\circ\lambda}$. Suppose that we
have an $H$-bundle $\mathcal{H}$ defined on a $K$-scheme $Y$. Then we can
define an $F$-bundle $\theta_{*}\mathcal{H}$ via extension of structure group.
If we have a parabolic reduction $\mathcal{H}_{\lambda}$ of $\mathcal{H}$ to
the subgroup $P_{\lambda}$, then by extension of structure group using
$\theta_{P}$ we get a parabolic reduction of $\theta_{*}\mathcal{H}$ to the
subgroup $P_{\theta\circ\lambda}$. This way we can define a map $\theta_{*}$
between weighted parabolic reductions of $\mathcal{H}$ and weighted parabolic
reductions of $\theta_{*}\mathcal{H}$.

Having set up the notation for weighted parabolic reductions, we proceed to
explain how this notion relates to $G$-bundles on $U\times_{K}\Theta_{K}$
following [Hei17]. Let $U$ be a $K$-scheme. Suppose that $\mathcal{G}$ is a
$G$-bundle on $U$. We can equivalently view this as a $G_{K}$-bundle on $U$.
Let $(\lambda,\mathcal{G}_{\lambda})$ be a weighted parabolic reduction of
$\mathcal{G}$. The morphism $\lambda:\mathbb{G}_{m}\rightarrow G_{K}$ can be
used to define a homomorphism
$\text{conj}_{\lambda}:P_{\lambda}\times\mathbb{G}_{m}\rightarrow
P_{\lambda}\times\mathbb{G}_{m}$ of group schemes over $\mathbb{G}_{m}$
defined by $(p,x)\mapsto(\lambda(x)p\lambda(x)^{-1},\,x)$. This extends to a
homomorphism
$\text{gr}_{\lambda}:P_{\lambda}\times_{K}\mathbb{A}^{1}_{K}\rightarrow
P_{\lambda}\times_{K}\mathbb{A}^{1}_{K}$. Let $t\in\mathbb{A}^{1}_{K}$ be a
closed point. The restriction of $\text{gr}_{\lambda}$ to a slice
$P_{\lambda}\times\\{t\\}$ is the identity when $t=1$, it is an isomorphism
when $t\neq 0$, and it is the projection of the parabolic subgroup
$P_{\lambda}$ to its Levi $L_{\lambda}$ when $t=0$. Now we can define a
$P_{\lambda}$-bundle on $U\times_{K}\Theta_{K}$ as follows. Note that
$\mathcal{G}_{\lambda}\times_{K}\mathbb{A}^{1}_{K}$ is a
$P_{\lambda}\times_{K}\mathbb{A}^{1}_{K}$-torsor on
$X\times\mathbb{A}^{1}_{K}$. Using the homomorphism $\text{gr}_{\lambda}$ we
obtain, by extension of structure group, a new
$P_{\lambda}\times_{K}\mathbb{A}^{1}_{K}$-torsor which is denoted
$(\mathcal{G}_{\lambda}\times_{K}\mathbb{A}^{1}_{K})\times^{\text{gr}_{\lambda}}_{\mathbb{A}^{1}_{K}}(P_{\lambda}\times_{K}\mathbb{A}^{1}_{K})$.
This bundle is $\mathbb{G}_{m}$-equivariant because the homomorphism
$\text{gr}_{\lambda}$ is $\mathbb{G}_{m}$-equivariant (with the adjoint action
on $P_{\lambda}$ and multiplication on $\mathbb{A}^{1}_{K}$), and therefore we
can take the quotient by $\mathbb{G}_{m}$

$\operatorname{Rees}(\mathcal{G}_{\lambda},\lambda)\vcentcolon=\left[\,\left((\mathcal{G}_{\lambda}\times_{K}\mathbb{A}^{1}_{K})\times^{\text{gr}_{\lambda}}_{\mathbb{A}^{1}_{K}}(P_{\lambda}\times_{K}\mathbb{A}^{1}_{K})\right)\,/\,\mathbb{G}_{m}\,\right]$

By construction, the restriction
$\operatorname{Rees}(\mathcal{G}_{\lambda},\lambda)|_{U\times 1}$ is
canonically isomorphic to the $P_{\lambda}$-bundle $\mathcal{G}_{\lambda}$,
and the restriction to $U\times 0$ is a $P_{\lambda}$-bundle which admits a
reduction of structure group to $L_{\lambda}$.

We can extend the structure group from $P_{\lambda}$ to $G_{K}$ in order to
obtain a $G_{K}$-bundle
$\operatorname{Rees}(\mathcal{G}_{\lambda},\,\lambda)\times^{P_{\lambda}}G_{K}$.
By construction, the restriction
$\operatorname{Rees}(\mathcal{G}_{\lambda},\,\lambda)\times^{P_{\lambda}}G_{K}|_{U\times_{K}1}$
is canonically isomorphic to the original $G$-bundle $\mathcal{G}$.

If $G_{K}$ is split, then it turns out that all $\Theta$-filtrations of
$G$-bundles on $U$ arise this way.

######  Proposition 3.14.

Suppose that $G_{K}$ is split. Let $U$ be a connected scheme of finite type
over $K$. Let $\mathcal{G}$ be a $G$-bundle on $U$. Then every $G$-bundle
$\mathcal{E}$ on $U\times_{K}\Theta_{K}$ such that
$\mathcal{E}|_{U\times_{K}1}\cong\mathcal{P}$ is of the form
$\operatorname{Rees}(\mathcal{G}_{\lambda},\,\lambda)\times^{P_{\lambda}}G_{K}$,
where

  1. (a)

$\lambda:\mathbb{G}_{m}\rightarrow G_{K}$ is a cocharacter that is uniquely
determined up to conjugation.

  2. (b)

$\mathcal{G}_{\lambda}$ is a $P_{\lambda}$-reduction of $\mathcal{G}$.

In other words, all $\Theta$-filtrations of $\mathcal{G}$ arise from weighted
parabolic reductions of $\mathcal{G}$.

###### Proof.

This is [Hei17, Lem. 1.13]. The lemma found there is stated for a connected
smooth projective curve over $K$, but the same argument applies to any
connected scheme of finite type over $K$. ∎

Suppose that $G_{K}$ is split. Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf
over $X_{K}$. Let $U$ be a big open subset of $X_{K}$ where $\mathcal{F}$ is
locally-free. Let $\mathcal{G}$ denote the $G_{K}$-bundle on $U$ corresponding
to the $\rho$-sheaf $(\mathcal{F},\sigma)$. By Proposition 3.14, all
$\Theta$-filtrations of $\mathcal{G}$ arise from the Rees construction applied
to a weighted parabolic reduction $(\lambda,\mathcal{G}_{\lambda})$ of
$\mathcal{G}$. We have an associated weighted parabolic reduction
$(\rho_{*}(\lambda),\rho_{*}(\mathcal{G}_{\lambda}))$ of the
$\operatorname{GL}(V)$-bundle $\mathcal{F}|_{U}$. The Rees construction
applied to this weighted parabolic reduction yields a filtration
$(\mathcal{F}^{\prime}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}|_{U}$. This
induces a filtration $(\mathcal{F})_{m\in\mathbb{Z}}$ of $\mathcal{F}$ as
follows. Let $\widetilde{\mathcal{F}^{\prime}}_{m}$ denote the pushforward of
$\mathcal{F}^{\prime}_{m}$ under the open inclusion $U\hookrightarrow X_{K}$.
Note that $\widetilde{\mathcal{F}^{\prime}}_{m}$ is a subsheaf of the
reflexive hull $\mathcal{F}^{\vee\vee}$. Then, we set
$\mathcal{F}_{m}\vcentcolon=\widetilde{\mathcal{F}^{\prime}}_{m}\cap\mathcal{F}^{\vee\vee}$.
It follows by construction that $(\mathcal{F})_{m\in\mathbb{Z}}$ satisfies the
properties in Proposition 3.8, so it is indeed a filtration of $\mathcal{F}$.

The pair
$((\mathcal{F}_{m})_{m\in\mathbb{Z}},\operatorname{Rees}(\mathcal{G}_{\lambda},\lambda)\times^{P_{\lambda}}G)$
is an element in $\operatorname{Filt}(\mathcal{F},\sigma)$. Conversely, it
follows from our discussion above that every element of
$\operatorname{Filt}(\mathcal{F},\sigma)$ arises this way. This is our sought
after describing the filtrations of the $\rho$-sheaf $(\mathcal{F},\sigma)$.
We state this formally with a definition and a proposition.

######  Definition 3.15.

Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. A weighted parabolic
reduction of $(\mathcal{F},\sigma)$ consists of a pair
$(\lambda,\mathcal{G}_{\lambda})$, where

  1. (a)

Implicit in the notation we have the choice of (any) big open subset $U\subset
X_{K}$ such that $\mathcal{F}|_{U}$ is locally-free. We denote by
$\mathcal{G}$ the $G_{K}$-bundle on $U$ corresponding to
$(\mathcal{F},\sigma)$.

  2. (b)

$(\lambda,\mathcal{G}_{\lambda})$ is a weighted parabolic reduction of
$\mathcal{G}$.

The filtration
$\mathcal{F}_{m}\vcentcolon=\widetilde{\mathcal{F}^{\prime}}_{m}\cap\mathcal{F}^{\vee\vee}$
of $\mathcal{F}$ described in the discussion above is called the underlying
filtration associated to the parabolic reduction
$(\lambda,\mathcal{G}_{\lambda})$. We say that
$(\lambda,\mathcal{G}_{\lambda})$ is nondegenerate if $\lambda$ is not the
trivial cocharacter.

Now we record the description of filtrations of $\rho$-sheaves obtained from
the discussion above.

######  Proposition 3.16.

Suppose that $G_{K}$ is split. Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on
$X_{K}$. Then every filtration in $\operatorname{Filt}(\mathcal{F},\sigma)$ is
of the form
$((\mathcal{F}_{m})_{m\in\mathbb{Z}},\operatorname{Rees}(\mathcal{G}_{\lambda},\lambda)\times^{P_{\lambda}}G)$,
where $(\lambda,\mathcal{G}_{\lambda})$ is a weighted parabolic reduction of
$(\mathcal{F},\sigma)$ with underlying associated filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$.

Moreover the filtration
$((\mathcal{F}_{m})_{m\in\mathbb{Z}},\operatorname{Rees}(\mathcal{G}_{\lambda},\lambda)\times^{P_{\lambda}}G)$
is nondegenerate if and only if $(\lambda,\mathcal{G}_{\lambda})$ is
nondegenerate. ∎

We can also give a concrete description of graded points of the stack
$\operatorname{Bun}_{\rho}(X)$. A graded $\rho$-sheaf over $X_{K}$ is defined
to be a morphism
$\left[\,\operatorname{Spec}(K)/\mathbb{G}_{m}\,\right]\rightarrow\operatorname{Bun}_{\rho}(X)$.
By unraveling the definitions, this notion admits the following concrete
description. A graded $\rho$-sheaf on $X_{K}$ consists of a $\rho$-sheaf
$(\mathcal{F},\sigma)$ on $X_{K}$, plus a $\mathbb{Z}$-grading
$\mathcal{F}=\bigoplus_{m\in\mathbb{Z}}\overline{\mathcal{F}}_{m}$ of the
torsion-free sheaf $\mathcal{F}$ such that each summand
$\overline{\mathcal{F}}_{m}$ is also torsion-free. We get a well-defined
$\mathbb{G}_{m}$-action on the scheme of $G$-reductions
$\operatorname{Red}_{G}(\mathcal{F})$ and we impose the condition that the
section $\sigma$ is $\mathbb{G}_{m}$-invariant. Hence we can think of
morphisms
$g:\left[\,\operatorname{Spec}(K)/\mathbb{G}_{m}\,\right]\rightarrow\operatorname{Bun}_{\rho}(X)$
concretely as a graded pair
$\left(\bigoplus_{m\in\mathbb{Z}}\overline{\mathcal{F}}_{m},\sigma\right)$.

We end this section by explaining a concrete description of the associated
graded of a filtration. Suppose that $G_{K}$ is split. Let
$(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. Let
$f:\Theta_{K}\rightarrow\operatorname{Bun}_{\rho}(X)$ be a filtration of
$(\mathcal{F},\sigma)$. By Proposition 3.16 $f$ comes from a weighted
parabolic reduction $(\lambda,\mathcal{G}_{\lambda})$ with underlying
associated filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$. We can restrict
$f$ to the point $0\in\Theta_{K}$ in order to obtain a graded $\rho$-sheaf
$f|_{[0/\,\mathbb{G}_{m}]}:\left[\,\operatorname{Spec}(K)/\mathbb{G}_{m}\,\right]\rightarrow\operatorname{Bun}_{\rho}(X)$.
The underlying graded torsion-free sheaf of $f|_{[0/\,\mathbb{G}_{m}]}$ is the
associated graded
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$.

We can describe the $G$-reduction over $U$ of the $\rho$-sheaf
$f|_{[0/\,\mathbb{G}_{m}]}$ as follows. Let $P_{\lambda}=L_{\lambda}\ltimes
U_{\lambda}$ be the Levi decomposition defined by $\lambda$. Then over the big
open subset $U$, the corresponding $G$-bundle is the one obtained by applying
extension of structure group under the composition
$P_{\lambda}=L_{\lambda}\ltimes U_{\lambda}\twoheadrightarrow
L_{\lambda}\hookrightarrow G_{K}$ to the $P_{\lambda}$-bundle
$\mathcal{G}_{\lambda}$.

##  4\. $\Theta$-stratification on $\operatorname{Bun}_{\rho}(X)$

In this section we define a polynomial numerical invariant $\nu$ on the stack
$\operatorname{Bun}_{\rho}(X)$. This invariant $\nu$ can be described in terms
of reduced Hilbert polynomials. We prove that $\nu$ induces a
$\Theta$-stratification, as in [HL14, Defn. 2.1.2] [HLHJ21, Defn. 2.23].

###  4.1. The numerical invariant

We define a family of line bundles $L_{n}$ on $\operatorname{Bun}_{\rho}(X)$
following [HLHJ21, §2].

######  Definition 4.1.

Let $n\in\mathbb{Z}$. We define $M_{n}$ to be the line bundle on
$\operatorname{Bun}_{\rho}(X)$ determined as follows. Let
$T\in\operatorname{Aff}_{k}$ and let
$f:T\rightarrow\operatorname{Bun}_{\rho}(X)$ be a morphism, represented by a
$\rho$-sheaf $(\mathcal{F},\sigma)$ on $X_{T}$. Let $\pi_{T}:X_{T}\to T$
denote the structure morphism. Then, we set

$f^{*}\,M_{n}\vcentcolon=\det\,\left(R\pi_{T\,*}\left(\mathcal{F}(n)\right)\right)$

Moreover, we define
$b_{d}\vcentcolon=\bigotimes_{i=0}^{d}M_{i}^{(-1)^{i}\binom{d}{i}}$. We set
$L_{n}$ to be the rational line bundle given by

$f^{*}\,L_{n}\vcentcolon=f^{*}\,M_{n}\otimes(f^{*}\,b_{d})^{-\otimes\,\overline{p}_{\mathcal{F}}(n)}$

######  Remark 4.2.

$L_{n}$ is an element of the rational Picard group
$\operatorname{Pic}(\operatorname{Bun}_{\rho}(X))\otimes\mathbb{Q}$ of the
stack. This is what we mean by rational line bundle. We will henceforth omit
the adjective “rational” and refer to $L_{n}$ as a line bundle. This should
not cause confusion, as all the arguments will apply to the rational Picard
group in the same way as for the usual Picard group.

$L_{n}$ is the restriction under the forgetful map
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}^{tf}(X)$ of the
line bundle $L_{n}$ on $\operatorname{Coh}^{tf}(X)$ defined in [HLHJ21, §2].
See the discussion therein for more details on this family of line bundles.

Next, we define a rational quadratic norm on graded $\rho$-sheaves (see
[HLHJ21, §2.5, §4.1] for more details on rational quadratic norms).

######  Definition 4.3.

We define the rational quadratic norm $b$ on graded points of
$\operatorname{Bun}_{\rho}(X)$ as follows. Let
$g:\left[\operatorname{Spec}(K)/\,\mathbb{G}_{m}\right]\rightarrow\operatorname{Bun}_{\rho}(X)$
be represented by a graded $\rho$-sheaf
$\left(\bigoplus_{m\in\mathbb{Z}}\overline{\mathcal{F}}_{m},\sigma\right)$.
Then we set

$b(g):=\sum_{m\in\mathbb{Z}}m^{2}\cdot\operatorname{rk}(\overline{\mathcal{F}}_{m})$

We define the polynomial numerical invariant $\nu$ to be the one determined by
the family of line bundles $L_{n}$ and the rational quadratic norm $b$, as it
was done for the case of pure sheaves in [HLHJ21, §4.1]. For our purposes, it
is sufficient to know the value of the numerical invariant associated to a
nondegenerate $\Theta$-filtration. We proceed to describe this next.

Let $K\supset k$ be a field extension, and let $(\mathcal{F},\sigma)$ be a
$\rho$-sheaf on $X_{K}$. Let
$f:\Theta_{K}\rightarrow\operatorname{Bun}_{\rho}(X)$ be a nondegenerate
filtration of $(\mathcal{F},\sigma)$. By Proposition 3.9 this amounts to a
filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of the torsion-free sheaf
$\mathcal{F}$ satisfying some extra compatibility conditions. The restriction
$f|_{\left[0/\mathbb{G}_{m}\right]}\rightarrow\operatorname{Bun}_{\rho}(X)$
yields a graded $\rho$-sheaf on $X_{K}$, whose underlying torsion-free sheaf
is given by the associated graded
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$. Then we define
$\nu(f)$ to be the polynomial in $\mathbb{R}[n]$ defined by

(2)
$\nu(f)\vcentcolon=\frac{\text{wt}\left(L_{n}|_{0}\right)}{\sqrt{b(f|_{[0/\mathbb{G}_{m}]})}}$

The same computation as in [HLHJ21, §4.1] shows that this polynomial numerical
invariant $\nu$ admits a description in terms of reduced Hilbert polynomials.
Denote by $P_{\mathcal{O}_{X}}$ the Hilbert polynomial of the structure sheaf
of $X$ with respect to $\mathcal{O}(1)$, which is a polynomial of degree $d$.
Let $\frac{A_{d}}{d!}$ be the leading coefficient of $P_{\mathcal{O}_{X}}$.

######  Proposition 4.4.

Let $K\supset k$ be a field extension. Let $(\mathcal{F},\sigma)$ be a
$\rho$-sheaf on $X_{K}$. Choose a nondegenerate filtration
$f:\Theta_{K}\rightarrow\operatorname{Bun}_{\rho}(X)$ of
$(\mathcal{F},\sigma)$. Suppose that $f$ has underlying associated filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$. Then, we have

(3)
$\nu(f)=\sqrt{A_{d}}\,\cdot\,\frac{\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}-\overline{p}_{\mathcal{F}})\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})}{\sqrt{\sum_{m\in\mathbb{Z}}m^{2}\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})}}$

###### Proof.

The value of the denominator $\sqrt{b(f|_{[0/\mathbb{G}_{m}]})}$ follows by
definition. The numerator has been computed in [HLHJ21, §4.1]. The extra
factor $\sqrt{A_{d}}$ appears because we define the rank to be the dimension
of the fiber of $\mathcal{F}$ at the generic point, as opposed to the rank
defined in [HLHJ21, §2.3] using the leading term of the Hilbert polynomial. ∎

Given a weighted parabolic reduction $f=(\lambda,\mathcal{G}_{\lambda})$ of a
$\rho$-sheaf $(\mathcal{F},\sigma)$, we denote by $\nu(f)$ the value of the
numerical invariant for the $\Theta$-filtration obtained by applying the Rees
construction to $f$. If $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ is the underlying
associated filtration of $(\lambda,\mathcal{G}_{\lambda})$, then the same
formula (3) applies.

###  4.2. Affine Grassmannians for $\rho$-sheaves

The setup for this subsection is as follows. Let $S$ be a quasicompact
$k$-scheme. Let $\mathcal{F}$ be a family of torsion-free sheaves of rank $r$
on $X_{S}$. Suppose that we are given an effective Cartier divisor
$D\hookrightarrow X_{S}$ that is flat over $S$. Set $Q=X_{S}\setminus D$ (note
that this is not a big open subset). The paper [HLHJ21] defines a notion of
affine Grassmannian $\operatorname{Gr}_{X_{S},D,\mathcal{F}}$. We recall the
definition and some properties proven in [HLHJ21, §3].

######  Definition 4.5.

$\operatorname{Gr}_{X_{S},D,\mathcal{F}}$ is the functor from
$\operatorname{Aff}_{S}^{\,\,op}$ to sets given as follows. For every affine
scheme $T$ in $\operatorname{Aff}_{S}$, the set
$\operatorname{Gr}_{X_{S},D,\mathcal{F}}(T)$ consists of equivalence classes
of pairs $(\mathcal{E},\theta)$, where

  1. (1)

$\mathcal{E}$ is a family of torsion-free sheaves on $X_{T}$.

  2. (2)

$\theta$ is morphism $\theta:\mathcal{F}|_{X_{T}}\rightarrow\mathcal{E}$ such
that the restriction $\theta|_{Q_{T}}$ is an isomorphism.

Two such pairs $(\mathcal{E}_{1},\theta_{1})$ and
$(\mathcal{E}_{2},\theta_{2})$ are considered equivalent if there is an
isomorphism $\mathcal{E}_{1}\xrightarrow{\sim}\mathcal{E}_{2}$ that identifies
the morphisms $\theta_{1}$ and $\theta_{2}$.

Let $j$ be the open immersion $j:Q\hookrightarrow X_{S}$. Choose
$T\in\operatorname{Aff}_{S}$ and
$(\mathcal{E},\psi)\in\operatorname{Gr}_{X_{S},D,\mathcal{F}}(T)$. The unit
$\mathcal{E}\rightarrow j_{T\,*}\,j_{T}^{*}\,\mathcal{E}$ is a monomorphism.
Hence we can use the isomorphism $j_{T*}(j_{T})^{*}\theta$ in order to view
$\mathcal{E}$ as a subsheaf of $j_{T\,*}\,j_{T}^{*}\,\mathcal{F}|_{X_{T}}$. In
particular we see that $\theta$ is automatically injective, and we have
$\mathcal{F}|_{X_{T}}\subset\mathcal{E}\subset
j_{T\,*}\,j_{T}^{*}\,\mathcal{F}|_{X_{T}}$. Note that there is a increasing
sequence of sheaves

$\mathcal{F}|_{X_{T}}\subset\mathcal{F}|_{X_{T}}(D_{T})\subset\mathcal{F}_{X_{T}}(2D_{T})\subset\cdots\subset\mathcal{F}|_{X_{T}}(nD_{T})\subset\cdots$

We can use this to define a truncated version of
$\operatorname{Gr}_{X,D,\mathcal{E}}$.

######  Definition 4.6.

Let $N$ be a positive integer. $\operatorname{Gr}^{\leq N}_{X,D,\mathcal{F}}$
is the subfunctor of $\operatorname{Gr}^{\leq N}_{X_{S},D,\mathcal{F}}$
defined as follows. For all $T$ in $\operatorname{Aff}_{T}$, we set

$\operatorname{Gr}^{\leq
N}_{X,D,\mathcal{F}}(T)\;:=\;\left\\{\begin{matrix}\;\;\;\text{pairs
$(\mathcal{E},\theta)$ in $\operatorname{Gr}_{X_{S},D,\mathcal{F}}(T)$ such
that}\;\;\;\\\
\text{$\mathcal{E}\subset\mathcal{F}|_{X_{T}}(ND_{T})$}\end{matrix}\right\\}$

For any rational polynomial $P\in\mathbb{Q}[n]$, we denote by
$\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq N,P}$ the open and closed
subfunctor of $\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq N}$ consisting of
points $(\mathcal{E},\theta)$ where $\mathcal{E}$ has Hilbert polynomial $P$
on every fiber.

######  Proposition 4.7 ([HLHJ21, Prop. 3.13]).

We have
$\operatorname{Gr}_{X_{S},D,\mathcal{F}}=\underset{N>0}{\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits}\;\operatorname{Gr}^{\leq{N}}_{X_{S},D,\mathcal{F}}$
as presheaves on $\operatorname{Aff}_{S}$. For each $N\geq 0$,

$\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq
N}=\bigsqcup_{P\in\mathbb{Q}[n]}\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq
N,P}$

Each $\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq N,P}$ is represented by a
projective scheme of finite presentation over $S$. This induces a presentation
of $\operatorname{Gr}_{X_{S},D,\mathcal{F}}$ as an ind-projective ind-scheme
over $S$.

Suppose now that we have a section
$\sigma:Q\rightarrow\operatorname{Red}_{G}(\mathcal{F})$ defined over $Q$. We
use this datum to define a version of the affine Grassmannian for
$\rho$-sheaves.

######  Definition 4.8.

The affine Grassmannian $\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}$ is
the functor from $\left(\operatorname{Aff}_{S}\right)^{op}$ into sets defined
as follows. For each affine scheme $T$ in $\operatorname{Aff}_{S}$,
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}(T)$ is the set of equivalence
classes of the following data

  1. (1)

A $\rho$-sheaf $(\mathcal{E},\zeta)$ on $X_{T}$.

  2. (2)

A morphism $\theta:\mathcal{F}|_{X_{T}}\rightarrow\mathcal{E}$ such that the
restriction $\theta|_{Q_{T}}$ is an isomorphism. Moreover, we require that the
following diagram induced by the isomorphism $\theta|_{Q_{T}}$ commutes.

${Q_{T}}$${\operatorname{Red}_{G}(\mathcal{E}|_{Q_{T}})}$${\operatorname{Red}_{G}(\mathcal{F}|_{Q_{T}})}$$\scriptstyle{\zeta|_{Q_{T}}}$$\scriptstyle{\sigma|_{Q_{T}}}$$\xrightarrow{\;\;\sim\;\;}$

Two elements above are considered equivalent if there is an isomorphism of
$\rho$-sheaves that is compatible with the morphism $\theta$.

There is a forgetful morphism
$\operatorname{Gr}_{{X_{S},D,\mathcal{F},\sigma}}\rightarrow\operatorname{Gr}_{X_{S},D,\mathcal{F}}$
defined by $(\mathcal{E},\zeta,\theta)\mapsto(\mathcal{E},\theta)$.

######  Proposition 4.9.

The forgetful morphism witnesses
$\operatorname{Gr}_{{X_{S},D,\mathcal{F},\sigma}}$ as a closed sub ind-scheme
of $\operatorname{Gr}_{X_{S},D,\mathcal{F}}$.

Before proving Proposition 4.9, we first state a couple of lemmas.

######  Lemma 4.10.

Let $Y\longrightarrow S$ be a proper morphism of finite presentation. Let
$\mathcal{G}$ and $\mathcal{F}$ be $\mathcal{O}_{Y}$-modules. Suppose that
$\mathcal{F}$ is finitely presented and $S$-flat. Fix a morphism
$\sigma:\mathcal{G}\rightarrow\mathcal{F}$. Let
$Z^{\mathcal{G},\mathcal{F}}_{\sigma}$ be the functor from $k$-schemes to sets
that sends a $k$-scheme $T$ to

$Z_{\sigma}(T)\vcentcolon=\;\left\\{\begin{matrix}\text{morphisms
$g:T\rightarrow S$ such that}\\\
\text{$(\text{id}_{Y}\times_{S}g)^{*}\sigma=0$}\end{matrix}\right\\}$

Then $Z$ is represented by a closed subscheme of $S$.

###### Proof.

By [Sta21, Tag 08K6], there is a separated $S$-scheme
$\operatorname{Hom}(\mathcal{G},\mathcal{F})$ parametrizing morphisms
$\mathcal{G}\to\mathcal{F}$. The two morphisms $\sigma$ and $0$ yield two
sections
$f_{\sigma},f_{0}:S\rightarrow\operatorname{Hom}(\mathcal{O}_{Y},\mathcal{F})$.
The functor $Z_{\sigma}$ is represented by the locus where these two morphisms
$f_{\sigma},f_{0}$ agree. This is a closed subscheme of $S$, because
$\operatorname{Hom}(\mathcal{G},\mathcal{F})\rightarrow S$ is separated. ∎

Let $S$ be a $k$-scheme. Suppose that $p:Y\rightarrow X_{S}$ is an affine
morphism of finite presentation. Let $s:Q\rightarrow Y$ be a section of $p$
defined over the open complement $Q=X_{S}\backslash D$.

######  Definition 4.11.

We define $\operatorname{Sect}^{Y}_{s}$ to be the functor from
$(\operatorname{Aff}_{S})^{op}$ to sets given as follows. For
$T\in\operatorname{Aff}_{S}$, let $s_{T}:Q_{T}\rightarrow Y_{T}$ and
$p_{T}:Y_{T}\rightarrow X_{T}$ denote the base-changes. We define

$\operatorname{Sect}^{Y}_{s}(T)\;\vcentcolon=\;\left\\{\begin{matrix}\;\;\;\text{sections
$f:X_{T}\rightarrow Y_{T}$ of $p_{T}$}\;\;\;\\\ \;\;\text{such that
$f|_{Q_{T}}=s_{T}$}\;\;\;\end{matrix}\right\\}$

######  Lemma 4.12.

The functor $\operatorname{Sect}^{Y}_{s}$ is represented by a closed subscheme
of $S$.

###### Proof.

After passing to an affine open cover, we can assume that
$S=\operatorname{Spec}(R)$. Since everything is of finite presentation, we can
use standard approximation results to reduce to the case when $R$ is
Noetherian. So $X_{S}$ is Noetherian, and
$Y=\underline{\operatorname{Spec}}_{X_{S}}(\mathcal{A})$ for some finitely
generated $\mathcal{O}_{X_{S}}$-algebra $\mathcal{A}$.

We can choose a finite open cover $\\{V_{i}\\}$ of $X_{S}$ and a finite set of
sections on each $V_{i}$ that generate $\mathcal{A}|_{V_{i}}$ as a
$\mathcal{O}_{V_{i}}$-algebra. By [Sta21, Tag 01PF] there exists a coherent
subsheaf $\mathcal{F}\subset\mathcal{A}$ that contains all of these sections.
Serre vanishing implies that exists some positive integers $N,m$ and a
surjective map of sheaves $\mathcal{O}_{X_{S}}^{\oplus
m}\twoheadrightarrow\mathcal{F}(N)$. Twisting by $\mathcal{O}_{X}(-N)$, we get
a surjection $\mathcal{O}_{X_{S}}(-N)^{\oplus
m}\twoheadrightarrow\mathcal{F}$. This induces a surjective map of
$\mathcal{O}_{X_{S}}$-algebras
$\operatorname{Sym}\left(\mathcal{O}_{X_{S}}(-N)^{\oplus
m}\right)\twoheadrightarrow\mathcal{A}$.

If we set $\mathcal{E}\vcentcolon=\mathcal{O}_{X_{S}}(-N)^{\oplus m}$, then we
have a closed immersion
$\iota:Y\hookrightarrow\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)$.
Let us denote by
$s^{\prime}:Q\rightarrow\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)$
the composition of $s:Q\rightarrow Y$ with the closed immersion $\iota$. There
is a natural map of functors
$\operatorname{Sect}^{Y}_{s}\rightarrow\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}$
given by composition with $\iota$.

The lemma is implied by the following two claims:

  1. (i)

The functor
$\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}$
is represented by a closed subscheme of $S$.

  2. (ii)

The map
$\operatorname{Sect}^{Y}_{s}\rightarrow\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}$
is a closed immersion.

For claim (i), note that $s^{\prime}$ is just an element of
$\Gamma(Q,\mathcal{E})$. Since
$j_{*}\,j^{*}\,\mathcal{E}=\underset{m\in\mathbb{Z}}{\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits}\;\mathcal{E}(mD)$,
there exists a positive integer $M$ such that $s^{\prime}$ extends to a
section $\sigma\in\Gamma(X_{S},\mathcal{E}(MD))$. Consider the composition
$\overline{\sigma}:\mathcal{O}_{X_{S}}\xrightarrow{\sigma}\mathcal{E}(MD)\twoheadrightarrow\mathcal{E}(MD)\,/\,\mathcal{E}$.
The functor
$\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}$
can be seen to be isomorphic to the functor $Z_{\overline{\sigma}}$, as in
Lemma 4.10. The coherent sheaf $\mathcal{E}(MD)\,/\,\mathcal{E}$ is $S$-flat
by the slicing criterion for flatness [Sta21, Tag 046Y]. Hence Lemma 4.10
implies that $Z_{\overline{\sigma}}$ is represented by a closed subscheme of
$S$.

We now proceed to prove claim (ii). Let $T$ be a $S$-scheme. For our purposes
we can assume that $T$ is affine and Noetherian. Choose
$T\rightarrow\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}$
represented by a section $f$ of the structure morphism
$\underline{\operatorname{Spec}}_{X_{T}}\left(\operatorname{Sym}\,\mathcal{E}_{T}\right)\rightarrow
X_{T}$. We want to show that the fiber product
$\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}\times_{\operatorname{Sect}^{Y}_{s}}T$
is represented by a closed subscheme of $T$. For any scheme $W$, we have

$\displaystyle\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}\times_{\operatorname{Sect}^{Y}_{s}}T\,(W)=\left\\{\begin{matrix}\text{morphisms
$h:W\rightarrow T$ such that}\\\ \;\;\text{$(\text{id}_{X_{T}}\times
h)^{*}f:X_{W}\rightarrow\underline{\operatorname{Spec}}_{X_{W}}\left(\operatorname{Sym}\,\mathcal{E}_{W}\right)$}\;\;\\\
\;\;\text{factors through $Y_{W}$}\;\;\end{matrix}\right\\}$

Define $B$ to be the fiber product

${B}$${X_{T}}$${Y_{T}}$${\underline{\operatorname{Spec}}_{X_{T}}\left(\operatorname{Sym}\,\mathcal{E}_{T}\right)}$$\hookrightarrow$$\scriptstyle{f}$$\scriptstyle{\iota\times\text{id}_{T}}$

The morphism $m:B\hookrightarrow X_{T}$ is a closed immersion. For any scheme
$W$, the $W$-points of the fiber product can be alternatively described by

$\displaystyle\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X_{S}}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}\times_{\operatorname{Sect}^{Y}_{s}}T\,(W)=\left\\{\begin{matrix}\text{morphisms
$h:W\rightarrow T$ such that}\\\ \;\;\text{$(\text{id}_{X_{T}}\times
h)^{*}m:B_{W}\hookrightarrow X_{W}$}\;\text{is an
isomorphism}\end{matrix}\right\\}$

Let us denote by $\varphi:\mathcal{J}\hookrightarrow\mathcal{O}_{X_{T}}$ the
the inclusion of the $\mathcal{O}_{X_{T}}$-sheaf of ideals $\mathcal{J}$
corresponding to the closed subscheme $B\hookrightarrow X_{T}$. The fiber
product
$\operatorname{Sect}^{\underline{\operatorname{Spec}}_{X}\left(\operatorname{Sym}\,\mathcal{E}\right)}_{s^{\prime}}\times_{\operatorname{Sect}^{Y}_{s}}T$
is isomorphic to $Z_{\varphi}$. Hence it is represented by a closed subscheme
of $T$ by Lemma 4.10. ∎

###### Proof of Proposition 4.9.

Let $T\in\operatorname{Aff}_{S}$ and let
$T\rightarrow\operatorname{Gr}_{X_{S},D,\mathcal{F}}$ be a morphism
represented by a pair $(\mathcal{E},\theta)$. By Definition 4.8 the
restriction $\theta|_{Q_{T}}$ over the open subset $Q_{T}$ is an isomorphism.
This yields an identification
$\operatorname{Red}_{G}(\mathcal{F})|_{Q_{T}}\xrightarrow{\sim}\operatorname{Red}_{G}(\mathcal{E})|_{Q_{T}}$.
We have a section $s$ induced by $\sigma$ as follows

$s:Q_{T}\xrightarrow{\sigma|_{Q_{T}}}\operatorname{Red}_{G}(\mathcal{F})|_{Q_{T}}\xrightarrow{\sim}\operatorname{Red}_{G}(\mathcal{E})|_{Q_{T}}$

After unraveling the definitions, it can be seen that the fiber product
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}\times_{\operatorname{Gr}_{X_{S},D,\mathcal{F}}}T$
is naturally isomorphic to
$\operatorname{Sect}^{\operatorname{Red}_{G}(\mathcal{E})}_{s}$. By Lemma
4.12, this is represented by a closed subscheme of $T$. ∎

Let $P\in\mathbb{Q}[n]$ be a rational polynomial. For each $N\geq 0$ we define
the subfunctors

$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}^{\leq
N}\vcentcolon=\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}\cap\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq
N}$ $\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}^{\leq
N,P}\vcentcolon=\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}\cap\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq
N,P}$

Both of these are closed sub ind-schemes of
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}$. Propositions 4.7 and 4.9
imply that
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}=\underset{N>0}{\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits}\;\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}^{\leq
N}$. Furthermore each $\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}^{\leq
N,P}$ is represented by a scheme that is projective and of finite presentation
over $S$.

There is a natural forgetful morphism
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}\rightarrow\operatorname{Bun}_{\rho}(X)$
that takes a point $(\mathcal{E},\zeta,\theta)$ in
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}$ and forgets the isomorphism
$\theta$. We can use this map to pullback the line bundle $L_{n}$ to
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}$. We will abuse notation and
denote this restriction by $L_{n}$ as well. The following corollary shows that
the family $L_{n}^{\vee}$ of dual line bundles on
$\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}$ is “asymptotically ample”

######  Corollary 4.13.

Let $N\in\mathbb{Z}_{\geq 0}$ and $P\in\mathbb{Q}[n]$. There exists some
integer $m\gg 0$ such that for all $n\geq m$, the restriction
$L_{n}^{\vee}|_{\operatorname{Gr}_{X_{S},D,\mathcal{F},\sigma}^{\leq N,P}}$ is
$S$-ample.

###### Proof.

This follows from Proposition 4.9 and [HLHJ21, Prop. 3.19], which states that
$L_{n}^{\vee}$ is eventually ample on
$\operatorname{Gr}_{X_{S},D,\mathcal{F}}^{\leq N,P}$. ∎

###  4.3. Monotonicity of the numerical invariant

In this subsection we prove that the invariant $\nu$ on
$\operatorname{Bun}_{\rho}(X)$ is strictly $\Theta$-monotone and strictly
$S$-monotone. We recall what these notions mean, see also [HL14, §5] or
[HLHJ21, §2.5] for more details.

We need some setup first. Let $R$ be a complete discrete valuation ring over
$k$. Choose a uniformizer $\pi$ of $R$. We define
$Y_{\Theta_{R}}\vcentcolon=\mathbb{A}^{1}_{R}$ equipped with the
$\mathbb{G}_{m}$-action that gives $t$ weight $-1$. We have
$\Theta_{R}=\left[\,\mathbb{A}^{1}_{R}/\,\mathbb{G}_{m}\,\right]$. Note that
$\mathbb{A}^{1}_{R}$ contains a unique $\mathbb{G}_{m}$-invariant closed point
cut out by the ideal $(t,\pi)$. We will denote this point by $(0,0)$.

Define
$Y_{\overline{ST}_{R}}\vcentcolon=\operatorname{Spec}\left(\,R[t,s]/(st-\pi)\,\right)$.
We equip $Y_{\overline{ST}_{R}}$ with the $\mathbb{G}_{m}$-action that gives
$s$ weight $1$ and $t$ weight $-1$. The isomorphism class of the
$\mathbb{G}_{m}$-scheme $Y_{\overline{ST}_{R}}$ is independent of the choice
of uniformizer $\pi$. Define
$\overline{ST}_{R}\vcentcolon=[\,Y_{\overline{ST}_{R}}/\,\mathbb{G}_{m}\,]$.
We will denote by $(0,0)$ the unique $\mathbb{G}_{m}$-fixed closed point in
$Y_{\overline{ST}_{R}}$. We have that $(0,0)$ is cut out by the ideal $(s,t)$.

For any field $\kappa$ and integer $a\geq 1$, we denote by
$\mathbb{P}^{1}_{\kappa}[a]$ the $\mathbb{G}_{m}$-scheme
$\mathbb{P}^{1}_{\kappa}$ equipped with the $\mathbb{G}_{m}$-action determined
by the equation $t\cdot[x:y]=[t^{-a}x:y]$. We set $0=[0:1]$ and
$\infty=[1:0]$.

With all this notation in place, we are ready to define the monotonicity
properties that we will prove.

######  Definition 4.14.

A polynomial numerical invariant $\nu$ on a stack $\mathcal{M}$ is strictly
$\Theta$-monotone (resp. strictly $S$-monotone) if the following condition
holds.

Let $R$ be any complete discrete valuation ring and set $\mathfrak{X}$ to be
$\Theta_{R}$ (resp. $\overline{ST}_{R}$). Choose a map
$\varphi:\mathfrak{X}\setminus(0,0)\rightarrow\mathcal{M}$. Then, after maybe
replacing $R$ with a finite DVR extension, there exists a reduced and
irreducible $\mathbb{G}_{m}$-equivariant scheme $\Sigma$ with maps
$f:\Sigma\rightarrow Y_{\mathfrak{X}}$ and
$\widetilde{\varphi}:\left[\Sigma/\,\mathbb{G}_{m}\right]\rightarrow\mathcal{M}$
such that

  1. (M1)

The map $f$ is proper, $\mathbb{G}_{m}$-equivariant and its restriction over
$Y_{\mathfrak{X}}\setminus(0,0)$ induces an isomorphism
$f:\,\Sigma_{Y_{\mathfrak{X}}\setminus(0,0)}\xrightarrow{\sim}Y_{\mathfrak{X}}\setminus(0,0)$.

  2. (M2)

The following diagram commutes

${\left[\left(\Sigma_{Y_{\mathfrak{X}}\setminus(0,0)}\right)/\,\mathbb{G}_{m}\right]}$${\mathfrak{X}\setminus(0,0)}$${\mathcal{M}}$$\scriptstyle{\widetilde{\varphi}}$$\scriptstyle{f}$$\scriptstyle{\varphi}$

  3. (M3)

Let $\kappa$ be a finite extension of the residue field of $R$. For any $a\geq
1$ and any finite $\mathbb{G}_{m}$-equivariant morphism
$\mathbb{P}^{1}_{\kappa}[a]\to\Sigma_{(0,0)}$, we have
$\nu\left(\;\widetilde{\varphi}|_{\left[\infty/\mathbb{G}_{m}\right]}\;\right)\geq\nu\left(\;\widetilde{\varphi}|_{\left[0/\mathbb{G}_{m}\right]}\;\right)$.

Before we proceed with the proof of monotonicity, we need we record a rational
filling condition for the stack $\operatorname{Bun}_{\rho}(X)$ in the spirit
of [HLHJ21, §4.2]. Let $Y$ denote either $Y_{\Theta_{R}}$ or
$Y_{\overline{ST}_{R}}$ as described above.

######  Proposition 4.15 ([HLHJ21, Lem. 4.7(i)]).

Let $(\mathcal{E},\zeta)$ be a $\mathbb{G}_{m}$-equivariant $\rho$-sheaf on
$X_{Y\setminus(0,0)}$. Then there exists

  1. (1)

A $\mathbb{G}_{m}$-equivariant family $\mathcal{F}$ of torsion-free sheaves on
$X_{Y}$.

  2. (2)

A relative effective Cartier divisor $D\hookrightarrow X_{Y}$ that is stable
under the action of $\mathbb{G}_{m}$.

  3. (3)

A $\mathbb{G}_{m}$-equivariant morphism
$\theta:\mathcal{F}|_{X_{Y\setminus(0,0)}}\rightarrow\mathcal{E}$ such that
the restriction
$\theta|_{(X_{Y\setminus(0,0)})\setminus(D_{Y\setminus(0,0)})}$ is an
isomorphism.

∎

We are now ready to prove the main proposition of this subsection.

######  Proposition 4.16.

The invariant $\nu$ on the stack $\operatorname{Bun}_{\rho}(X)$ is both
strictly $\Theta$-monotone and strictly $S$-monotone.

###### Proof.

The argument is a variation of the one in [HLHJ21, §4.3] using our version of
the affine Grassmannian. We let $Y=Y_{\Theta_{R}}$ or $Y_{\overline{ST}_{R}}$.
Suppose that we are given a morphism
$f:\left[\left(\,Y\setminus(0,0)\,\right)/\,\mathbb{G}_{m}\right]\rightarrow\operatorname{Bun}_{\rho}(X)$.
This consists of the data of a $\rho$-sheaf $(\mathcal{E},\zeta)$ on
$X_{Y\setminus(0,0)}$ that is $\mathbb{G}_{m}$-equivariant with respect to the
action of $\mathbb{G}_{m}$ on $Y\setminus(0,0)$. By Proposition 4.15, there is
a $\mathbb{G}_{m}$-equivariant torsion-free sheaf $\mathcal{F}$ on $X_{Y}$, a
$\mathbb{G}_{m}$-stable $Y$-flat effective Cartier divisor $D\hookrightarrow
X_{Y}$ and a $\mathbb{G}_{m}$-equivariant monomorphism
$\theta:\mathcal{F}|_{X_{(Y\setminus(0,0))}}\rightarrow\mathcal{E}$ such that
$\theta$ is an isomorphism away from $D$.

Set $Q\vcentcolon=X_{Y}\setminus D$. The isomorphism
$\theta|_{Q_{Y\setminus(0,0)}}$ induces a $\mathbb{G}_{m}$-equivariant
identification
$\operatorname{Red}_{G}(\mathcal{E})|_{Q_{Y\setminus(0,0)}}\xrightarrow{\sim}\operatorname{Red}_{G}(\mathcal{F})|_{Q_{Y\setminus(0,0)}}$.
We have a $\mathbb{G}_{m}$-equivariant section induced by $\zeta$:

$\sigma_{Q_{Y\setminus(0,0)}}:Q_{Y\setminus(0,0)}\xrightarrow{\zeta|_{Q_{Y\setminus(0,0)}}}\operatorname{Red}_{G}(\mathcal{E})|_{Q_{Y\setminus(0,0)}}\xrightarrow{\sim}\operatorname{Red}_{G}(\mathcal{F})|_{Q_{Y\setminus(0,0)}}$

The local ring $\mathcal{O}_{Y,(0,0)}$ at $(0,0)$ has depth $2$. For any point
$u$ in the fiber $Q_{(0,0)}$, the local ring $\mathcal{O}_{Q,u}$ is flat over
$\mathcal{O}_{Y,(0,0)}$. It follows that $\mathcal{O}_{Q,u}$ has depth at
least $2$ for all $u\in Q_{(0,0)}$. The same type of argument as in Lemma 2.2
(c) shows that the $\mathbb{G}_{m}$-equivariant section
$\sigma_{Q_{Y\setminus(0,0)}}$ extends uniquely to a
$\mathbb{G}_{m}$-equivariant section
$\sigma:Q\rightarrow\operatorname{Red}_{G}(\mathcal{F})$. We now have all of
the necessary data to form the affine grassmannian
$\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}$. Since everything is
$\mathbb{G}_{m}$-equivariant, the ind-scheme
$\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}$ acquires a natural
$\mathbb{G}_{m}$-action.

The data of the $\rho$-sheaf $(\mathcal{E},\zeta)$ and the monomorphism
$\theta$ is by definition a $\mathbb{G}_{m}$-equivariant section
$s:Y\setminus(0,0)\rightarrow\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}$
of the structure morphism
$\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}\rightarrow Y$. We can
summarize everything in the following diagram, where all of the maps are
$\mathbb{G}_{m}$-equivariant:

${\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}}$${\operatorname{Bun}_{\rho}(X)}$${Y\setminus(0,0)}$${Y}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{s}$$\xhookrightarrow{\;\;\;\;\;\;\;\;\;\;}$

By construction, the composition $\operatorname{Forget}\,\circ\,s$ is the
original $\mathbb{G}_{m}$-equivariant morphism
$f:Y\setminus(0,0)\rightarrow\operatorname{Bun}_{\rho}(X)$.

It follows from the definition of
$\operatorname{Gr}_{X_{Y},D,\mathcal{F}}^{\leq N,P}$ that the
$\mathbb{G}_{m}$-action on $\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}$
preserves each stratum $\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq
N,P}$. Since $Y$ is affine and connected, the map
$s:Y\setminus(0,0)\rightarrow\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}$
factors through $\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq N,P}$
for some $N\gg 0$ and $P\in\mathbb{Q}[x]$. Therefore, we have the following
commutative diagram with $\mathbb{G}_{m}$-equivariant morphisms.

${\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq
N,P}}$${\operatorname{Bun}_{\rho}(X)}$${Y\setminus(0,0)}$${Y}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{s}$$\xhookrightarrow{\;\;\;\;\;\;\;\;\;\;}$

Let $\Sigma\subset\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq N,P}$
denote the scheme theoretic image of $Y\setminus(0,0)$ in
$\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq N,P}$. Note that
$\Sigma$ is a reduced $\mathbb{G}_{m}$-scheme with a natural structure
morphism to $Y$. The map $\Sigma\rightarrow Y$ is projective, because
$\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq N,P}$ is projective over
$Y$ and $\Sigma$ is a closed subscheme of
$\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq N,P}$. By construction
the morphism $\Sigma\rightarrow Y$ is $\mathbb{G}_{m}$-equivariant and
restricts to an isomorphism over the open $Y\setminus(0,0)$.

The composition
$\Sigma\rightarrow\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq
N,P}\xrightarrow{\operatorname{Forget}}\operatorname{Bun}_{\rho}(X)$ restricts
to the original $\mathbb{G}_{m}$-equivariant morphism
$f:Y\setminus(0,0)\rightarrow\operatorname{Bun}_{\rho}(X)$ over the open
subset $Y\setminus(0,0)\,\subset\,\Sigma$. We therefore obtain a morphism
$\widetilde{f}:\left[\Sigma/\,\mathbb{G}_{m}\right]\rightarrow\operatorname{Bun}_{\rho}(X)$
satisfying condition (M2) in Definition 4.14.

We are left to check condition (M3) in Definition 4.14. Choose an integer
$a\geq 1$. Consider $\mathbb{P}^{1}_{\kappa}$ equipped with the
$\mathbb{G}_{m}$-action defined by $t\cdot[x:y]=[t^{-a}x:y]$. Suppose that we
are given a finite $\mathbb{G}_{m}$-equivariant morphism
$\mathbb{P}^{1}_{\kappa}\to\Sigma_{(0,0)}$. By Corollary 4.13, there exists
some $M\gg 0$ such that for all $n\geq M$ the restriction
$L_{n}^{\vee}|_{\operatorname{Gr}_{X_{Y},D,\mathcal{F},\sigma}^{\leq N,P}}$ is
$Y$-ample. Therefore the restriction $L_{n}|_{\mathbb{P}^{1}_{\kappa}}$ is
anti-ample for $n\gg 0$. This means that
$L_{n}|_{\mathbb{P}^{1}_{\kappa}}\cong\mathcal{O}_{\mathbb{P}^{1}_{\kappa}}(-N_{n})$
for some $N_{n}>0$. Note that $\mathcal{O}_{\mathbb{P}_{\kappa}^{1}}(-N_{n})$
admits a unique $\mathbb{G}_{m}$-equivariant structure up to twisting by a
character, and we have
$\text{wgt}\,\mathcal{O}_{\mathbb{P}^{1}_{\kappa}}(-N_{n})|_{\infty}\geq\text{wgt}\,\mathcal{O}_{\mathbb{P}^{1}_{\kappa}}(-N_{n})|_{0}$.
By definition, we have

$\nu\left(\widetilde{f}|_{\left[\infty/\mathbb{G}_{m}\right]}\right)=\frac{\operatorname{wgt}\,L_{n}|_{\infty_{i}}}{\sqrt{b\left(\widetilde{f}|_{\left[\infty_{i}/\mathbb{G}_{m}\right]}\right)}}$

$\nu\left(\widetilde{f}|_{\left[0/\mathbb{G}_{m}\right]}\right)=\frac{\operatorname{wgt}\,L_{n}|_{0_{i}}}{\sqrt{b\left(\widetilde{f}|_{\left[0_{i}/\mathbb{G}_{m}\right]}\right)}}$

We have just shown that
$\operatorname{wgt}\,L_{n}|_{\infty}\geq\operatorname{wgt}\,L_{n}|_{0}$ for
$n\gg 0$. Hence, we will be done if we can prove
$b\left(\widetilde{f}|_{\left[\infty/\mathbb{G}_{m}\right]}\right)=b\left(\widetilde{f}|_{\left[0/\mathbb{G}_{m}\right]}\right)$.
Let $\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m,0}$ (resp.
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m,\infty}$) denote the underlying
graded torsion-free sheaves of
$\widetilde{f}|_{\left[0/\mathbb{G}_{m}\right]}$ (resp.
$\widetilde{f}|_{\left[\infty/\mathbb{G}_{m}\right]}$). The norm $b$ only
depends on the ranks $\text{rk}(\mathcal{F}_{m,0})$ (resp.
$\text{rk}(\mathcal{F}_{m,\infty})$) of each graded piece. Hence it suffices
to show that $\text{rk}(\mathcal{F}_{m,0})=\text{rk}(\mathcal{F}_{m,\infty})$
for all $m\in\mathbb{Z}$. By construction, the graded $\rho$-sheaves
$\widetilde{f}|_{\left[0/\mathbb{G}_{m}\right]}$ and
$\widetilde{f}|_{\left[\infty/\mathbb{G}_{m}\right]}$ come from the affine
Grassmannian $\text{Gr}_{X_{Y},D,\mathcal{F},\sigma}$. More precisely, away
from the divisor $D$, the underlying graded sheaves
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m,0}|_{X_{\kappa}\setminus
D_{\kappa}}$ and
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m,0}|_{X_{\kappa}\setminus
D_{\kappa}}$ are both isomorphic to the graded sheaf
$\mathcal{F}|_{X_{(0,0)}\setminus D_{(0,0)}}$ after scaling each weight by
$a$. Since the rank is determined at the generic point of $X_{\kappa}$, this
shows that $\text{rk}(\mathcal{F}_{m,0})=\text{rk}(\mathcal{F}_{m,\infty})$
for all $m$, as desired. ∎

###  4.4. HN boundedness

We start by recalling the definition of HN boundedness (see also [HL14, §5] or
[HLHJ21, §2.5]).

######  Definition 4.17 (HN Boundedness).

Let $\mathcal{M}$ be an algebraic stack locally of finite type over $k$. Let
$\nu$ be a polynomial numerical invariant on $\mathcal{M}$. We say that $\nu$
satisfies the HN boundedness condition if the following is true.

Choose an algebraically closed field extension $K\supset k$. Let $S$ be a
scheme of finite type over $K$ and let $g:S\rightarrow\mathcal{M}$ be a
morphism. Then, there exists a quasicompact open substack
$\mathcal{U}_{S}\subset\mathcal{M}$ such that the following holds: For all
closed points $s\in S$ and all nondegenerate filtrations
$f:\Theta\rightarrow\mathcal{M}$ of the point $g(s)$ with $\nu(f)>0$, there
exists another filtration $f^{\prime}$ of $g(s)$ satisfying
$\nu(f^{\prime})\geq\nu(f)$ and $f^{\prime}|_{0}\in\mathcal{U}_{S}$.

In plain words, this says that for all bounded families and all destabilizing
filtrations $f$ of an element in the family, we can always find a more
destabilizing filtration $f^{\prime}$ such that the associated graded
$f^{\prime}|_{0}$ lies on some fixed bounded family.

In the case of $\operatorname{Bun}_{\rho}(X)$ we can reduce the proof of HN
boundedness to a more concrete statement. This is the content of the following
lemma.

######  Lemma 4.18.

In order to show that $\nu$ satisfies HN boundedness on the stack
$\operatorname{Bun}_{\rho}(X)$, it is sufficient to prove the following
statement.

(*) Let $K\supset k$ be an algebraically closed field extension of $k$. Let
$S$ be a scheme of finite type over $K$. Let
$g:S\rightarrow\operatorname{Bun}_{\rho}(X)$ be a morphism represented by a
$\rho$-sheaf $(\mathcal{F},\sigma)$ on $X_{S}$. Then, there exists a real
number $C_{S}$ such that the following holds: For all closed points $s\in S$
and all nondegenerate weighted parabolic reductions
$f=(\lambda,\mathcal{G}_{\lambda})$ of the point
$(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$ with $\nu(f)>0$, there exists another
weighted parabolic reduction
$f^{\prime}=(\lambda^{\prime},\mathcal{G}_{\lambda^{\prime}})$ of
$(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$ with underlying associated filtration
$(\mathcal{F}^{\prime}_{m})_{m\in\mathbb{Z}}$ such that the following are
satisfied.

  1. (1)

$\nu(f^{\prime})\geq\nu(f)$.

  2. (2)

We have
$\widehat{\mu}_{d-1}(\mathcal{F}^{\prime}_{m}/\mathcal{F}^{\prime}_{m+1})\geq
C_{S}$ for all $m\in\mathbb{Z}$.

###### Proof.

We can replace $k$ by $K$ in order to assume that the ground field is
algebraically closed. Then, all reductive groups are automatically split. By
Proposition 3.16, all filtrations of $(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$
arise from weighted parabolic reductions of
$(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$. Let $\mathfrak{S}$ denote the set of
all pairs $(s,f^{\prime})$, where $s\in S$ is a closed point and $f^{\prime}$
is a weighted parabolic reduction of $(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$
satisfying $(2)$ above. It suffices to show that the set of associated graded
points $\mathfrak{S}_{2}\vcentcolon=\left\\{\;f^{\prime}|_{0}\;\text{ for
$(s,f^{\prime})\in\mathfrak{S}$}\;\;\right\\}$ is contained in a quasicompact
open substack of $\operatorname{Bun}_{\rho}(X)$.

Recall that we have a forgetful morphism
$\operatorname{Forget}:\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$
into the stack of rank $r$ torsion-free sheaves on $X$. By Proposition 3.6,
the morphism $\operatorname{Forget}$ is of finite type. Hence it is sufficient
to prove that the set
$\mathfrak{S}_{3}\vcentcolon=\left\\{\;\;\operatorname{Forget}(f^{\prime}|_{0})\;\text{
for $(s,f^{\prime})\in\mathfrak{S}$}\;\;\right\\}$ is contained in a
quasicompact open substack of $\operatorname{Coh}_{r}^{tf}(X)$. Let
$\mathfrak{S}^{\prime}$ denote the set of pairs
$(s,(\mathcal{F}_{m})_{m\in\mathbb{Z}})$, where $s\in S$ is a closed point and
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ is a decreasing $\mathbb{Z}$-filtration
of the sheaf $\mathcal{F}|_{X_{s}}$ such that
$\widehat{\mu}_{d-1}(\mathcal{F}_{m}/\mathcal{F}_{m+1})\geq C_{S}$ for all
$m\in\mathbb{Z}$. It is sufficient to prove that the set of sheaves

$\mathfrak{S}^{\prime}_{1}=\left\\{\;\;\;\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}\;\text{
\; \; \; for $(s,(\mathcal{F}_{m})_{m\in\mathbb{Z}})\in\mathfrak{S}^{\prime}$
}\;\;\;\right\\}$

is contained in a quasicompact open substack of
$\operatorname{Coh}_{r}^{tf}(X)$. This follows from [HLHJ21, Lemma 5.4]. ∎

To prove HN boundedness for $\operatorname{Bun}_{\rho}(X)$ in Proposition
4.26, we need to understand how the numerical invariant changes for parabolic
subgroups associated to different filtrations. We will use Lemma 4.18. After
replacing $k$ with the extension $K$, we can assume that the ground field is
algebraically closed (in particular all tori are automatically split). Fix a
$\rho$-sheaf $(\mathcal{F},\sigma)$ on $X$. Let $j:U\hookrightarrow X$ be a
big open subset such that $\mathcal{F}|_{U}$ is locally-free. We will develop
some setup and prove some preliminary lemmas before the proof of HN
boundedness.

Let $Z$ denote the maximal central torus of $G$. Let $X_{*}(Z)$ denote the
group of cocharacters of $Z$. We denote by $X^{*}(Z)$ the dual group of
characters of $Z$. Set
$X_{*}(Z)_{\mathbb{R}}\vcentcolon=X_{*}(Z)\otimes\mathbb{R}$ and similarly
$X^{*}(Z)_{\mathbb{R}}\vcentcolon=X^{*}(Z)\otimes\mathbb{R}$. We fix once and
for all a basis $z_{1},\ldots,z_{h}$ of $X_{*}(Z)$.

Let $f=(\lambda,\mathcal{G}_{\lambda})$ be a weighted parabolic filtration of
$(\mathcal{F},\sigma)$, and let $P_{\lambda}\subset G$ be the parabolic
subgroup associated to $\lambda$. Choose a maximal torus $T$ of $P_{\lambda}$
containing the image of the cocharacter $\lambda$. Set
$T^{\prime}:=T\cap[G,G]$, the intersection of $T$ with the derived subgroup
$[G,G]$ of $G$. Denote the groups of cocharacters of $T$ and $T^{\prime}$ by
$X_{*}(T)$ and $X_{*}(T^{\prime})$ respectively. Observe that there are
natural inclusions $X_{*}(Z)_{\mathbb{R}}\hookrightarrow
X_{*}(T)_{\mathbb{R}}$ and $X_{*}(T^{\prime})_{\mathbb{R}}\hookrightarrow
X_{*}(T)_{\mathbb{R}}$. These induce a direct sum decomposition
$X_{*}(T)_{\mathbb{R}}=X_{*}(Z)_{\mathbb{R}}\oplus
X_{*}(T^{\prime})_{\mathbb{R}}$.

Choose a Borel subgroup $B\supset T$ contained in $P_{\lambda}$. This yields a
set of simple roots
$\Delta=\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{l}\\}\subset
X_{*}(T^{\prime})_{\mathbb{R}}^{\vee}$, where $l$ is the rank of $G$, which
form a basis for $X^{*}(T^{\prime})_{\mathbb{R}}$. Denote by
$\omega_{i}^{\vee}\in X_{*}(T^{\prime})_{\mathbb{R}}$ the fundamental coweight
dual to the simple root $\alpha_{i}$ and define the ordered basis
$\mathcal{B}$ of the vector space $X_{*}(T)_{\mathbb{R}}$ as

$\mathcal{B}=(z_{1},z_{2},\ldots,z_{h},\omega_{1}^{\vee},\omega_{2}^{\vee},\ldots,\omega^{\vee}_{l})$

This induces a dual basis
$\mathcal{B}^{\vee}=(z_{1}^{\vee},z_{2}^{\vee},\ldots,z_{h}^{\vee},\alpha_{1},\alpha_{2},\ldots,\alpha_{l})$
on the dual vector space $X^{*}(T)_{\mathbb{R}}$.

The torus $T\subset G$ acts on the representation $V$. Choose a decomposition
of $V$ as a direct sum of one-dimensional weight spaces

$V=\bigoplus_{i=1}^{r}V_{\chi_{i}}$

where $(\chi_{i})_{i=1}^{r}$ is an ordered $r$-tuple of characters in
$X^{*}(T)_{\mathbb{R}}$ with some possible repetitions.

######  Definition 4.19.

Using coordinates in the dual basis $\mathcal{B}^{\vee}$, we denote by $M$ the
$(h+l)\times r$-matrix with coefficients in $\mathbb{Q}$ corresponding to the
tuple $(\chi_{i})_{i=1}^{r}$.

The matrix $M$ does not depend on the choice of torus $T$ and Borel subgroup
$B$ (once we fix the basis of $X_{*}(Z)$), because all torus-Borel pairs
$T\subset B$ are conjugate in $G$.

The parabolic $P_{\lambda}$ determines a subset $I_{P_{\lambda}}\subset\Delta$
consisting of those simple roots $\alpha_{i}\in\Delta$ such that the root
group $U_{-\alpha_{i}}$ corresponding to $-\alpha_{i}$ is not contained in
$P_{\lambda}$ (e.g. we have $I_{B}=\Delta$ and $I_{G}=\emptyset$). Let
$J_{P_{\lambda}}\subset\\{1,2,\ldots,l\\}$ be the set of indexes such that
$I_{P_{\lambda}}=\\{\alpha_{j}\;\mid\;j\in J_{P_{\lambda}}\\}$.

######  Definition 4.20.

We define $\mathcal{C}_{\lambda}$ to be the closed $\mathbb{R}$-span of the
cone of $P_{\lambda}$-dominant coweights in $X_{*}(T)_{\mathbb{R}}$, with the
origin removed:

$\mathcal{C}_{\lambda}=\left(\sum_{i=1}^{h}\mathbb{R}z_{i}+\sum_{j\in
J_{P_{\lambda}}}\mathbb{R}^{\geq 0}\omega^{\vee}_{j}\right)\setminus\\{0\\}$

For any cocharacter $\delta\in\mathcal{C}_{\lambda}\cap X_{*}(T)$, let
$P_{\delta}$ be its associated parabolic subgroup. Note that
$P_{\delta}\supset P_{\lambda}$. To every such $\delta$ we can associate a
weighted parabolic reduction $(\delta,\mathcal{G}_{\delta})$, where
$\mathcal{G}_{\delta}$ is the $P_{\delta}$-bundle obtained from the
$P_{\lambda}$-bundle $\mathcal{G}_{\lambda}$ via extension of structure group.
The evaluation of the numerical invariant $\nu$ on these weighted parabolic
reductions $(\delta,\mathcal{G}_{\delta})$ defines a function

$\nu:\mathcal{C}_{\lambda}\cap
X_{*}(T)\rightarrow\mathbb{R}[n],\;\;\;\delta\mapsto\nu(\delta,\mathcal{G}_{\delta}).$

By the formula in Proposition 4.4, the polynomial
$\nu(\delta,\mathcal{G}_{\delta})$ has degree less than or equal to $d-1$.

######  Definition 4.21.

We define $\nu_{d-1}:\mathcal{C}_{\lambda}\cap X_{*}(T)\rightarrow\mathbb{R}$
to be the function that associates to each $\delta$ the (leading) degree
$(d-1)$-coefficient of the polynomial $\nu(\delta,\mathcal{G}_{\delta})$.

We denote by $L_{\lambda}$ the unique Levi subgroup of the parabolic
$P_{\lambda}$ containing the maximal torus $T$, as described in Subsection
3.3. Let $Z_{\lambda}$ be the maximal central torus of $L_{\lambda}$. For each
$i$, let $\chi_{i}^{\lambda}$ be the restriction of the character $\chi_{i}$
to $X_{*}(Z_{\lambda})$. Set
$V_{\chi^{\lambda}_{i}}=\bigoplus_{\\{j\,\mid\,\chi^{\lambda}_{j}=\chi^{\lambda}_{i}\\}}V_{\chi^{\lambda}_{j}}$.
We have a $X^{*}(Z_{\lambda})$-grading on the vector space $V$ with weight
decomposition:

$V=\bigoplus_{\\{[\chi^{\lambda}_{i}]\\}}V_{\chi^{\lambda}_{i}}$

Here the index set $\\{[\chi^{\lambda}_{i}]\\}$ runs over the set of all
restrictions $\chi^{\lambda}_{i}\in X^{*}(Z_{\lambda})$ without repetitions,
and $Z_{\lambda}$ acts on $V_{\chi^{\lambda}_{i}}$ with weight
$\chi^{\lambda}_{i}$. Note that each $V_{\chi_{i}^{\lambda}}$ is a
representation of $L_{\lambda}$, because $Z_{\lambda}$ is central in
$L_{\lambda}$. Let $\mathcal{G}_{L_{\lambda}}$ denote the $L_{\lambda}$-bundle
on $U$ obtained from $\mathcal{G}_{\lambda}$ via the natural quotient morphism
$P_{\lambda}\twoheadrightarrow L_{\lambda}$. Let
$\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}V_{\chi_{i}^{\lambda}}$ denote
the associated vector bundle on $U$.

######  Definition 4.22.

We define $\mathcal{E}_{\chi_{i}^{\lambda}}$ to be the reflexive extension
$j_{*}\left(\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}V_{\chi_{i}^{\lambda}}\right)$
on $X$.

We will denote by $H$ the hyperplane class in $X$ corresponding to the ample
line bundle $\mathcal{O}(1)$, and denote by
$c_{1}(\mathcal{E}_{\chi_{i}^{\lambda}})$ the first Chern class of
$\mathcal{E}_{\chi_{i}^{\lambda}}$.

######  Lemma 4.23.

Set $c_{i}\vcentcolon=\frac{c_{1}(\mathcal{E}_{\chi^{\lambda}_{i}})\cdot
H^{d-1}}{\operatorname{rk}(\mathcal{E}_{\chi^{\lambda}_{i}})}$ and
$c\vcentcolon=\frac{c_{1}(\mathcal{F})\cdot H^{d-1}}{r}$. Then, we have

(4)