# A guide to moduli theory beyond GIT

Tomás L. Gómez  Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás
Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain<EMAIL_ADDRESS>,  Andres Fernández Herrero  Mathematics Hall MC 4404, 2990 Broadway, New
York, N.Y. 10027, USA<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>Dedicated to Peter E. Newstead, on his $80^{th}$ birthday

(Date: February 1 2023)

###### Abstract.

In this survey we provide an overview of some recent developments in the
construction of moduli spaces using stack-theoretic techniques. We will also
explain the analogue of Harder-Narasimhan stratifications for general stacks,
known as $\Theta$-stratifications. As an application of the ideas exposed
here, we address the moduli problem of principal bundles over higher
dimensional projective varieties, as well as its different compactifications
by the so-called principal $\rho$-sheaves. We construct a stratification by
instability types whose lower strata admits a proper good moduli space of
“Gieseker semistable” objects and a new Gieseker-type Harder-Narasimhan
filtration for these objects. Detailed proofs of the latter results will
appear elsewhere.

###### Key words and phrases:

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

###### 2020 Mathematics Subject Classification:

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

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

##  1\. Introduction

The construction and study of moduli spaces has been one of the driving forces
in algebraic geometry for decades. After the algebraic construction of the
Picard scheme by Grothendieck, the study of the moduli of vector bundles came
next. Not only has Peter Newstead been one of the pioneers in this study since
his early papers in the 60’s, but by founding the group VBAC (Vector Bundles
on Algebraic Curves), he has been pivotal in building a large and successful
community dedicated to this and related areas. It is an honour for us to
dedicate this survey to him.

Since its inception by Mumford, Geometric Invariant Theory (GIT) [MFK94] has
been a powerful way to construct moduli spaces for many moduli problems. When
performing GIT, we do not attempt to classify all possible objects in a given
moduli problem, but rather only those declared to be semistable. After adding
the additional data of a “framing”, these objects are parametrized by a scheme
called the parameter space. Mumford’s GIT provides a way to define the
quotient of this scheme by the action of an algebraic group encoding the
framing, revisiting Hilbert’s $14^{th}$ problem, to obtain a moduli scheme
parametrizing semistable objects.

The intrinsic point of view in the Beyond GIT program developed in [HL21] does
not focus on the subclass of semistable objects but on the whole moduli stack.
A generalization of the Hilbert-Mumford criterion leads to a new stacky
numerical criterion through test stacks taking the role of former one-
parameter subgroups. This makes the notion of semistability naturally appear
in the stack, avoiding the tedious checking that semistable objects correspond
to GIT semistable ones and vice-versa. Thanks to the recent paper [AHLH19],
there are some intrinsic conditions that can be checked locally in an
algebraic stack in order to show that it admits a separated good moduli space
in the sense of Alper [Alp13].

Unstable objects usually come in a natural way with a maximal degeneration (a
stacky Harder-Narasimhan filtration). The theory of $\Theta$-stratifications
in [HL21] can be used to find such maximal degenerations without having to
consider non-canonically a GIT problem sufficiently large to contain the
required unstable object as in [GSZ15]. This opens the door to understanding
Harder-Narasimhan filtrations also in an intrinsic way, directly from the
whole stack.

Harder-Narasimhan filtrations (also known as canonical filtrations) were first
considered in [HN75] for vector bundles on curves. Shatz studied its behaviour
in families in [Sha77]. In higher dimensions we have Gieseker semistablity,
and we have Gieseker-Harder-Narasimahn filtrations for torsion free sheaves.
Nitsure used them to produce schematic stratifications in [Nit11] (see also
[GN14] for $\Lambda$-modules), which give rise to algebraic stacks
corresponding to each Harder-Narasimhan type, thus defining a filtration of
the stack of pure coherent sheaves by locally closed substacks.

As an application of this theory, we address the moduli problem of principal
bundles over higher dimensional projective varieties, as well as its different
compactifications by the so-called principal $\rho$-sheaves. We construct in
[GHZ21] a stratification (the $\Theta$-stratification of a given numerical
invariant), whose lower open strata admit proper good moduli spaces (as in
[Alp13]) of “Gieseker semistable” objects [Gie77]. This gives an intrinsic
stack-theoretic construction to the already known moduli space of principal
bundles on higher dimensional varieties [Sch02, GS05, GLSS08, GLSS10], as well
as its compactifications.

For principal bundles, a Harder-Narasimhan filtration is usually called
canonical reduction. They were announced by Ramanathan in [Ram79] and have
been constructed over curves by Behrend [Beh91] and Biswas-Holla [BH04]. A
reduction to a parabolic subgroup is also defined in [AB83] by a different
method, and it is proved in [AAB02] to coincide with the canonical filtration.
Gurjar and Nitsure shows that this reduction also gives a schematic
stratification (i.e., a stratification of the stack) [GN14, GN18]. For
principal bundles in positive characteristics, Gurjar and Nitsure [GN16] show
that there are algebraic stacks corresponding to each HN-type, and these are
radicial over the algebraic stack of all principal bundles. If the Behrend
conjecture holds then these stacks are shown to define a stratification of the
stack of all principal bundles (see also [Hei08a]). However, over higher
dimension varieties, where a polynomial Gieseker stability condition is used
to construct the compactification of the moduli space, canonical reductions
for principal bundles in the literature only take into account the leading
coefficients of the reduced Hilbert polynomial (analogous to slope stability
as described in [Mum63a]). There has been partial attempts to construct a
filtration that accounts for all of the terms in the Hilbert polynomial in
lower rank [Zam16]. On the other hand, in [BZ16] it was shown that the notion
of Gieseker Harder-Narasimhan parabolic reduction cannot be related to the
Gieseker-Harder-Narasimhan filtration of the underlying vector bundles for
faithful representations, which meant that the main strategies in the
literature (e.g. [AAB02]) could not possibly apply if we try to account for
all terms of the Hilbert polynomial.

An important outcome of the results in [GHZ21] is a new notion of a Gieseker-
Harder-Narasimhan filtration for principal $\rho$-sheaves, which is
constructed by iterating a filtration coming from the $\Theta$-stratification
(called the leading term HN filtration). The Gieseker Harder-Narasimhan
filtration thus obtained consists of a parabolic reduction of the underlying
rational principal bundle that measures instability by considering all degree
terms of the numerical polynomials. Detailed proofs of the results contained
in this survey will appear elsewhere.

######  Notation 1.1.

For this survey we work over a fixed ground field $k$, which we assume to be
of characteristic $0$ unless explicitly stated. We denote by $\text{Sch}_{k}$
the category of schemes over $k$. We will sometimes work instead over the
smaller category $\text{Aff}_{k}$ of affine schemes over $k$. For any
$k$-schemes $X$ and $S$, we will write $X_{S}$ to denote the fiber product
$X\times S$, which is a product over $k$ unless otherwise stated.

##  2\. Intrinsic approach to moduli problems

###  2.1. Algebraic stacks

Moduli theory rests on the insight that many classification problems in
algebraic geometry can be parametrized by schemes. An example of this is the
projective space $\mathbb{P}^{n}_{k}$, which classifies line bundles along
with the choice of $n+1$ generating sections. In order to naturally deal with
the algebraic structure of such classification problems, the Grothendieck
school of algebraic geometry initiated the functorial point of view on such
moduli problems.

We can view any scheme $X$ as a functor $h_{X}$ from $(\text{Aff}_{k})^{op}$
into sets via the Yoneda embedding. We say that functors of the form $h_{X}$
are representable by schemes. We will equip $\text{Sch}_{k}$ and
$\text{Aff}_{k}$ with the additional structure of the Grothendieck topology
generated by fppf (finitely presented faithfully flat) covers. All
representable functors satisfy descent with respect to this fppf topology.

Equipped with the functorial point of view, Grothendieck proved the
representability of some of the central moduli problems in algebraic geometry:
the Quot functor and the Picard functor. We refer the reader to the articles
by Nitsure and Kleiman in [FGI+05] for a nice exposition of these
achievements. While the definition of the Quot functor intuitively captures
the sought after moduli problem, surprises arise when one tries to parametrize
line bundles on a given projective variety $X$. The natural functor
classifying isomorphism classes of line bundles does not satisfy fppf descent,
and therefore cannot be represented by a scheme. However, Grothendieck proved
that the fppf sheafification $\text{Pic}(X)$ of the natural functor is indeed
represented by a scheme.

When the variety $X$ is proper but not projective, Artin showed the fppf
sheafification $\text{Pic}(X)$ is represented by an algebraic space [Art69,
Thm 7.3]. Algebraic spaces form a subcategory of all fppf sheaves. This
subcategory provides a more robust setting to parametrize moduli problems,
while remaining fundamentally geometric at the same time. We refer to [Knu71],
[Sta23] for a detailed treatments of the theory of algebraic spaces. Artin’s
result on the representability of $\operatorname{Pic}(X)$ is one of the many
achievements that hinted at the usefulness of enlarging the category of
schemes in order to study moduli problems.

A more serious issue arises when one tries to generalize the Picard scheme to
the setting of higher rank vector bundles on a given projective variety $X$.
Even in the case when $X$ is a curve, it becomes impossible to parametrize the
moduli problem of all vector bundles in terms of a separated scheme. The main
feature of this moduli problem is the existence of large nontrivial
automorphisms of the objects, which makes the condition of descent for
isomorphism classes a more delicate affair. One approach, pioneered by Mumford
when $X$ is a smooth projective curve, is to restrict to a special class of
reasonably “rigid” vector bundles called stable vector bundles [Mum63a].
Alternatively, if one wants to be able to parametrize all vector bundles, then
it becomes necessary to account for their automorphisms in order to naturally
deal with the descent condition. This is elegantly encoded in the notion of an
algebraic stack, which is another geometric enlargement of the category of
schemes originally introduced by Deligne and Mumford [DM69].

For the rest of the survey we will assume that the reader has some familiarity
with the theory of algebraic stacks. For an informal introduction we refer the
reader to [G0́1] [Hei10b]; on the other hand we recommend [CMW18] [Alp23] for
more detailed developments of the theory. Some standard references are
[LMB00], [Ols16] and [Sta23].

We will favor the point of view of stacks as pseudofunctors valued in
groupoids, which we believe is more intuitive for moduli problems. There
should be no issue in translating things into the theoretically cleaner
language of fibered categories thanks to [FGI+05, Part 1, §3.1.3].

Recall that a stack $\mathcal{M}$ is a pseudofuctor from
$(\text{Sch}_{k})^{op}$ into groupoids that satisfies fppf descent. We say
that a stack is algebraic if it admits a smooth morphism $f:U\to\mathcal{M}$
representable by algebraic spaces such that the source $U$ is a scheme. Such
morphism $f$ is called an atlas. Many of the properties of algebraic stacks
(locally of finite type, quasi-compact, smooth, normal) can be encoded in
terms of properties of some atlas.

Just as in the case of schemes, algebraic stacks are closed under the
operation of fiber product. One can define the diagonal of a stack in a
similar way as in the case of schemes. The main difference is that in the case
of algebraic stacks the diagonal is not a monomorphism. Instead, the fibers of
the diagonal encode automorphisms of the corresponding points of the stack.

Throughout this survey we will only work with stacks that are locally of
finite type over $k$. Moreover we shall restrict our attention to stacks with
affine diagonal.

###  2.2. Good moduli spaces

Many moduli problems in algebraic geometry, such as the moduli of vector
bundles on a smooth projective curve, are parametrized by algebraic stacks
[Hei10b]. As in the case of the Picard scheme or the substack of stable vector
bundles, it is often possible to “rigidify” the functor to remove the
automorphisms, thus obtaining a functor valued in sets that is representable
by an algebraic space. Such algebraic space is called the moduli space
associated to the stack.

When we are working with a stack $\mathcal{M}$ with finite inertia, the
theorem of Keel and Mori [KM97] [Sta23, Tag 0DUT] provides a general procedure
to obtain a “rigidification” in the form of an algebraic space $M$ called the
coarse moduli space as in the following.

######  Definition 2.1.

A morphism $f:{\mathcal{M}}\to M$ into an algebraic space $M$ is called a
coarse moduli space morphism (and $M$ is called a coarse moduli space) if the
following are satisfied

  1. (1)

$f$ is initial among morphisms to algebraic spaces.

  2. (2)

For any algebraically closed field $K$, the natural transformation
$f:{\mathcal{M}}(K)\to M(K)$ induces a bijection on isomorphism classes of
points.

There is a coarse moduli space morphism $\mathcal{M}\to M$ that relates both
functors. Many geometric properties of the original algebraic stack
$\mathcal{M}$ (and of the moduli problem parametrized by it) are reflected in
the algebraic space $M$, which lives in a closer setting to the more geometric
world of varieties.

When the objects parametrized by the algebraic stack $\mathcal{M}$ have
infinite automorphisms (i.e. algebraic automorphism groups of positive
dimension), then the power of the Keel-Mori theorem becomes unavailable.
Unfortunately, many natural moduli problems arising in algebraic geometry have
infinite symmetry groups, so it becomes important to develop technical tools
to deal with this type of situation. Mumford’s Geometric Invariant Theory
(GIT) [MFK94] offers a powerful device to construct a moduli space in the case
when the algebraic stack $\mathcal{M}$ is a quotient stack of the form
$\left[X/G\right]$, where $X$ is open inside a projective-over-affine variety
and $G$ is a reductive group. In this case, the properties of the
corresponding morphism $\left[X/G\right]\to M$ are captured by the notion of a
“good quotient”.

Alper’s notion of good moduli space is an elegant way to encode and generalize
Mumford’s theory to the setting of arbitrary algebraic stacks that do not
necesarily arise as quotient stacks.

######  Definition 2.2 (Good moduli space, [Alp13, Defn. 4.1]).

A quasi-compact and quasiseparated morphism $\varphi:\mathcal{M}\to M$ from an
algebraic stack $\mathcal{M}$ to an algebraic space $M$ is called a good
moduli space if both of the following properties are satisfied:

  1. (a)

The pushforward morphism
$\varphi_{*}:\operatorname{QCoh}(\mathcal{M})\to\text{QCoh}(M)$ is exact.

  2. (b)

The induced morphism of structure sheaves
$\mathcal{O}_{M}\to\varphi_{*}\mathcal{O}_{\mathcal{M}}$ is an isomorphism.

We call the algebraic space $M$ the good moduli space of $\mathcal{M}$.

######  Remark 2.3.

The notion of good moduli space becomes quite restrictive if we work over a
base of positive or mixed characteristic. Most of the classical moduli spaces
that admit a good moduli space in characteristic $0$ (such as the moduli of
semistable vector bundles on a curve) do not admit a good moduli space over a
field of positive characteristic. Instead, they admit morphisms
$\mathcal{M}\to M$ to an algebraic space satisfying a weakening of condition
(a) in Definition 2.2. A more appropriate concept, the notion of adequate
moduli space, was introduced by Alper in [Alp14] to approach moduli problems
in arbitrary characteristic.

The above Definition 2.2 is deceivingly concise; it has a lot of strong
implications which link the properties of the moduli space $M$ and the stack
$\mathcal{M}$. The first main property justifies the name “the” moduli space.

######  Theorem 2.4 ([Alp13, Thm. 6.6]).

Let $\mathcal{M}$ be a locally Noetherian algebraic stack with a good moduli
space $\phi:\mathcal{M}\to M$. Then $\phi$ is initial among all maps from
$\mathcal{M}$ into algebraic spaces. In particular the good moduli space
$\phi:\mathcal{M}\to M$ is unique up to unique isomorphism.

Another important property of good moduli spaces is that they behave well with
respect to base-change.

######  Proposition 2.5 ([Alp13, Prop. 4.7(i)]).

Let $\mathcal{M}$ be an algebraic stack with a good moduli space
$\phi:\mathcal{M}\to M$. For any morphism of algebraic spaces $X\to M$, the
base-change $\phi_{X}:\mathcal{M}\times_{M}X\to X$ is a good moduli space for
the fiber product $\mathcal{M}\times_{M}X$.

We finally summarize some of the important properties of good moduli spaces in
the following

######  Proposition 2.6.

Let $\phi:\mathcal{M}\to M$ be a good moduli space. Then the following hold.

  1. (i)

$\phi$ is surjective and universally closed [Alp13, Thm. 4.16(i)+(ii)].

  2. (ii)

If $\phi$ has affine diagonal, then $\phi$ is of finite type [AHR20, Thm.
A.1].

  3. (iii)

The fibers of $\phi$ are geometrically connected, each $\phi$-fiber contains a
unique closed point [Alp13, Thm. 4.16(vii) + Prop. 9.1].

  4. (iv)

If $\mathcal{M}$ is locally Noetherian, then so is $M$, and the push-forward
$\phi_{*}$ preserves coherence of sheaves [Alp13, Thm. 4.16(x)].

  5. (v)

If $\mathcal{M}$ is of finite type over an excellent scheme $S$, then $M$ is
also of finite type over $S$ [Alp13, Thm. 4.16(xi)].

In addition, there are some results that relate the singularities of a stack
and the singularities of its good moduli space. We would like to highlight the
following recent generalization of a classical theorem of Boutot.

######  Theorem 2.7 ([BGLM21, Thm. 5]).

Let $\mathcal{M}$ be a smooth algebraic stack with affine diagonal over a
field $k$ with good moduli space $\phi:\mathcal{M}\to M$. If $M$ is
quasiprojective, then $M$ is of klt type.

We end this section by providing some examples of good moduli spaces.

######  Example 2.8 (Affine modulo linearly reductive).

Let $X=\text{Spec}(A)$ be an affine variety equipped with the action of a
linearly reductive group $G$ over $k$. We consider the quotient stack
$\left[X/G\right]$. We denote by $A^{G}\subset A$ the subring of invariants.
There is a morphism $\phi:\left[X/G\right]\to\text{Spec}(A^{G})$ which
exhibits $\text{Spec}(A^{G})$ as a good moduli space for $\left[X/G\right]$.
Indeed, let us check both properties of a good moduli space morphism.

  1. (a)

The abelian category of quasicoherent sheaves $\text{QCoh}(\left[X/G\right])$
on the stack $\left[X/G\right]$ is equivalent to the category of
$G$-equivariant $A$-modules. The push-forward $\phi_{*}(-)$ takes a
$G$-equivariant $A$-module to the module of $\phi_{*}(M)=M^{G}\subset M$ of
$G$-invariants, viewed as a module over $A^{G}$. Since the algebraic group $G$
is assumed to be linearly reductive, the functor of taking $G$-invariants
$(-)^{G}$ is exact, and therefore $\varphi_{*}(-)$ is exact.

  2. (b)

From the description of $\phi_{*}(-)$ in (a) above, it follows that
$\varphi_{*}(\mathcal{O}_{\left[X/G\right]})=A^{G}=\mathcal{O}_{\text{Spec}(A^{G})}$.

Thanks to the étale slice theorem for algebraic stacks proved by Alper, Hall
and Rydh [AHR19], algebraic stacks that admit good moduli spaces are étale
locally of the form $\left[\text{Spec}(A)/G\right]$ as in the example above.
Therefore the example serves as an excellent local picture to keep in mind
when dealing with good moduli spaces.

The other familiar example of good moduli space comes from GIT.

######  Example 2.9 (GIT, [Alp13, §13.5]).

Let $G$ be a linearly reductive group over a field $k$. Let $X$ be a
projective-over-affine scheme over $k$ equipped with an action of $G$. Let
$\mathcal{L}$ be an ample $G$-equivariant line bundle on $X$. Let
$X^{\mathcal{L}{\operatorname{-}}\rm{ss}}\subset X$ denote the open
$G$-subscheme of $\mathcal{L}$-semistable points. Then the natural morphism
$\phi:[X^{\mathcal{L}{\operatorname{-}}\rm{ss}}/G]\to\text{Proj}\left(\bigoplus_{n\geq
0}H^{0}(X,\mathcal{L}^{\otimes n})^{G}\right)$ is a good moduli space for the
stack $[X^{\mathcal{L}{\operatorname{-}}\rm{ss}}/G]$.

Note that Example 2.8 is a special case of Example 2.9, where we take the
equivariant line bundle to be the structure sheaf ${\mathcal{O}}_{X}$, so that
all points are semistable.

We should stress that not all stacks admit good moduli spaces. Having a good
moduli space imposes some necessary conditions on the stack $\mathcal{M}$. For
example, it forces every closed point of $\mathcal{M}$ to have linearly
reductive stabilizers. This leads to an important question: what properties of
an algebraic stack guarantee that it admits a good moduli space? In the next
subsection we shall provide an answer to this question by explaining a set of
intrinsic conditions [AHLH19] that are necessary and sufficient for a stack
over $k$ to admit a separated good moduli space.

###  2.3. Intrinsic construction of moduli spaces

One of the most useful foundational results in the theory of algebraic spaces
and algebraic stacks is Artin’s theorem [Art74] [Sta23, Tag 07SZ]. This
theorem provides a set of intrinsic local criteria (now known as the Artin
criteria) which guarantee that a given functor or pseudofunctor is represented
by an algebraic space or an algebraic stack. Artin’s criteria are one of the
main tools in the construction of an algebraic stack that parametrizes a given
moduli problem.

If we want to “rigidify” the moduli problem and look for the construction of
good moduli spaces, then classically the main tool available is the theorem of
Keel and Mori in the case when the moduli problem has finite inertia. One can
first apply Artin’s criteria to show representability by an algebraic stack,
and then use the Keel-Mori theorem to construct the coarse space.

In the case when the moduli problem has infinite automorphisms, a recent
breakthrough paper of Alper, Halpern-Leistner and Heinloth [AHLH19] provides a
set of intrinsic local conditions for an algebraic stack to admit a separated
good moduli space.

The main strategy for the construction of such a space is conceptually simple.
First, the étale slice theorem for stacks due to Alper, Hall and Rydh [AHR20,
AHR19] states that, under certain hypotheses, one can present an algebraic
stack $\mathcal{M}$ étale locally as a quotient stack of the form
$\left[\text{Spec}(A)/G\right]$, where $G$ is a linearly reductive group. We
know how to construct moduli spaces for such stacks, by Example 2.8. Then one
would hope to be able to glue such moduli spaces in order to obtain a good
moduli space for the stack $\mathcal{M}$. The insight of [AHLH19] is that
there are some intrinsic properties of the stack $\mathcal{M}$ that guarantee
that: (1) the hypotheses of the slice theorem are satisfied; and (2) one can
glue the good moduli spaces for the local models in order to obtain a
separated good moduli space for $\mathcal{M}$.

This provides an intrinsic way of constructing moduli spaces associated to
moduli problems in algebraic geometry. One can first apply the Artin criteria
to prove that the natural groupoid-valued pseudofunctor is represented by an
algebraic stack. Then, using the criteria for the existence of good moduli
spaces in [AHLH19], one can construct the sought-after moduli space.

This strategy has been useful in cases where the techniques of GIT are not
available. One such example is the construction of the moduli of Bridgeland
semistable objects in the derived category of a smooth projective variety
[AHLH19, §7], where the stack $\mathcal{M}$ is not known to be a quotient
stack. Another example is the moduli of $K$-polystable Fano varieties
[ABHLX20, BLX22], where the line bundle inducing semistability is not ample on
the natural parameter space given by the Hilbert scheme. More recently, the
same techniques have also been applied to obtain intrinsic constructions of
moduli of quiver representations [BDF+22]. We would also like to mention the
construction of moduli spaces of objects in abelian categories, treated in
[AHLH19]. Another important question is the projectivity of the moduli spaces
obtained via intrinsic methods [Taj21, ABB+22, BDF+22]; we will not treat this
aspect of the theory in this survey.

To end this section, we would like to describe more carefully the criteria in
[AHLH19]. They consist of filling conditions with respect to some morphisms
from test stacks, reminiscent of the valuative criterion for properness in
classical algebraic geometry.

We start by introducing the relevant test stacks. Let $\mathbb{G}_{m}$ denote
the multiplicative group, whose group of $S$-points is the group of invertible
functions $H^{0}({\mathcal{O}}_{S}^{\times})$ for any $k$-scheme $S$. We let
$\mathbb{G}_{m}$ act on the affine space
$\mathbb{A}^{1}_{k}=\text{Spec}(k[t])$ linearly such that the weight of the
coordinate $t$ is $-1$.

######  Definition 2.10 (stack $\Theta$).

We denote by $\Theta$ the quotient stack
$\left[\mathbb{A}^{1}_{k}/\mathbb{G}_{m}\right]$.

Notice that the stack $\Theta$ has two $k$-points: the closed point $0$ with
stabilizer $\mathbb{G}_{m}$ corresponding to the origin in
$\mathbb{A}^{1}_{k}$, and the open point $1$ with trivial stabilizer
corresponding the open orbit in $\mathbb{A}^{1}_{k}$. If $R$ is a discrete
valuation ring over $k$ with uniformizer $\varpi$, then the base-change
$\Theta_{R}:=\Theta\times_{\text{Spec(k)}}\text{Spec}(R)$ has a unique closed
point $(0,0)$ corresponding to the simultaneous vanishing locus of $t$ and
$\varpi$ inside $\mathbb{A}^{1}_{R}=\text{Spec}(R[t])$.

######  Definition 2.11 ($\Theta$-reductivity).

An algebraic stack $\mathcal{M}$ over $k$ is said to be $\Theta$-reductive if
for all discrete valuation rings $R$ over $k$ and all morphisms
$f:\Theta_{R}\setminus(0,0)\to\mathcal{M}$, there is a unique dashed arrow as
below making the diagram commutative

${\Theta_{R}\setminus(0,0)}$${\Theta_{R}}$${\mathcal{M}}$$\scriptstyle{f}$$\hookrightarrow$

We continue with a choice of discrete valuation ring $R$ over $k$ with
uniformizer $\varpi$.

######  Definition 2.12 (stack $\overline{ST}_{R}$).

We define the quotient stack

$\overline{ST}_{R}:=\left[\text{Spec}(R[t,s]/(st-\varpi))/(\mathbb{G}_{m})_{R}\right],$

where
$(\mathbb{G}_{m})_{R}=\mathbb{G}_{m}\times_{\text{Spec}(k)}\text{Spec}(R)$
acts with weight $-1$ on $t$ and with weight $1$ on $s$.

The stack $\overline{ST}_{R}$ has a unique closed point$(0,0)$ corresponding
to the simultaneuous vanishing locus of $s,t,\varpi$ in
$\text{Spec}(R[t,s]/(st-\varpi))$.

######  Definition 2.13 ($S$-completeness).

An algebraic stack $\mathcal{M}$ over $k$ is defined to be $S$-complete if for
all discrete valuation rings $R$ over $k$ and all morphisms
$f:\overline{ST}_{R}\setminus(0,0)\to\mathcal{M}$, there is a unique dashed
arrow as below making the diagram commutative

${\overline{ST}_{R}\setminus(0,0)}$${\overline{ST}_{R}}$${\mathcal{M}}$$\scriptstyle{f}$$\hookrightarrow$

We refer the reader to [AHLH19, §3.5] for some intuition behind the filling
condition in the definition above. We are now ready to state one of the main
theorems in [AHLH19].

######  Theorem 2.14 ([AHLH19, Thm. A]).

Let $\mathcal{M}$ be an algebraic stack with affine diagonal and of finite
type over a field $k$ of characteristic $0$. Then $\mathcal{M}$ admits a
separated good moduli space if and only if it is $\Theta$-reductive and
$S$-complete.

######  Remark 2.15.

The pursue of the right analogue of Theorem 2.14 in positive or mixed
characteristic is a question that is being actively investigated.

Our main concern now is to be able to identify moduli stacks that are
$\Theta$-reductive and $S$-complete. Unfortunately, many algebraic stacks
arising in moduli theory (such as the moduli stack of vector bundles on a
curve) are not $\Theta$-reductive or $S$-complete. This is to be expected; we
know from the work of Mumford that in order to construct a moduli space for
vector bundles it is necessary to restrict to those vector bundles that are
(semi)stable. This phenomenon is fairly common in moduli theory. It is often
impossible to parametrize all objects in the moduli problem; instead one
generally restricts their attention to a subset of objects satisfying a
stability condition, which are “rigid” enough to admit a parametrization in
terms of a moduli space.

Hence we put ourselves in the following situation: we start with a “large”
stack $\mathcal{M}$ parametrizing all objects for a given moduli problem. The
main question is: can we identify a reasonable open substack
$\mathcal{M}^{\rm{ss}}\subset\mathcal{M}$ of “semistable objects” such that
$\mathcal{M}^{\rm{ss}}$ admits a good moduli space? In the next subsection we
provide an answer to this question via the notion of $\Theta$-stability.

###  2.4. $\Theta$-stability

In the setting of GIT, Mumford introduced a notion of semistability depending
on the choice of an equivariant line bundle. Suppose that $X$ is a projective
variety over $k$ equipped with an action of a reductive group $G$. Given a
$G$-equivariant ample line bundle $\mathcal{L}$ on $X$, Mumford defined a
$G$-stable open subset $X^{\rm{ss}}\subset X$ called the semistable locus. He
showed that the stack $\left[X^{\rm{ss}}/G\right]$ admits a projective good
moduli space. Hence GIT provides a powerful tool to identify appropriate open
substacks $\left[X^{\rm{ss}}/G\right]\subset\left[X/G\right]$ depending on an
equivariant line bundle $\mathcal{L}\in\operatorname{Pic}^{G}(X)$, which
corresponds to a line bundle on the stack $\left[X/G\right]$.

The notion of $\Theta$-stability, introduced by Halpern-Leistner [HL21] and
Heinloth [Hei17], generalizes the notion of GIT stability to the setting of
arbitrary algebraic stacks. It is inspired by the Hilbert-Mumford criterion
[MFK94, Chpt. 2.1], which describes the semistable locus in GIT in terms of
one-parameter subgroups of the reductive group $G$. In the case of a more
general algebraic stack, the one-parameter subgroups are replaced by the
notion of a $\Theta$-filtration. This uses the test stack $\Theta$ introduced
in Definition 2.10.

######  Definition 2.16 ($\Theta$-Filtration).

Let $\mathcal{M}$ be an algebraic stack over $k$. Let $K\supset k$ be a field
over $k$, and let $p:\text{Spec}(K)\to\mathcal{M}$ be a $K$-point in
$\mathcal{M}(K)$. A $\Theta$-filtration of $p$ consists of the data of a
morphism $f:\Theta_{K}\to\mathcal{M}$ along with an isomorphism
$\Theta|_{1_{K}}\simeq p$.

######  Remark 2.17.

In [HL21] a $\Theta$-filtration is simply called a filtration. We stress the
name $\Theta$-filtration to clearly distinguish this intrinsic notion of
filtration from the many other filtrations appearing in this survey.

Fix a line bundle $\mathcal{L}$ on the stack $\mathcal{M}$. For any filtration
$f:\Theta_{K}\to\mathcal{M}$ of a point $p\in\mathcal{M}(K)$, the
$0_{K}$-fiber of the pullback $f^{*}(\mathcal{L})$ acquires a natural action
of the stabilizer $(\mathbb{G}_{m})_{K}$. We denote by
$\text{wt}\left(f^{*}(\mathcal{L})|_{0_{K}}\right)\in\mathbb{Z}$ the weight of
the $\mathbb{G}_{m}$ action on the one dimensional $0_{K}$-fiber.

######  Definition 2.18 ($\Theta$-stability).

Let $\mathcal{M}$ be an algebraic stack over $k$, and let $\mathcal{L}$ be a
line bundle on $\mathcal{M}$. Fix an algebraically closed field $K\supset k$
and a geometric point $p:\text{Spec}(K)\to\mathcal{M}$. We say that $p$ is
$\Theta$-semistable (with respect to $\mathcal{L}$) if for all
$\Theta$-filtrations $f:\Theta_{K}\to\mathcal{M}$ of the point $p$ we have
$\text{wt}\left(f^{*}(\mathcal{L})|_{0_{K}}\right)\leq 0$.

The set of $\Theta$-semistable geometric points with respect to a line bundle
$\mathcal{L}$ yields a subset of the topological space of the stack
$|\mathcal{M}|$ (as in [Sta23, Tag 0DQM]). We say that the $\Theta$-semistable
locus is open if this subset is open. In this case the $\Theta$-semistable
points consist exactly of the geometric points of an open substack, which we
denote by
$\mathcal{M}^{\mathcal{L}{\operatorname{-}}\rm{ss}}\subset\mathcal{M}$ (or
just by $\mathcal{M}^{\rm{ss}}$, if the choice of the line bundle is
implicit).

######  Example 2.19 (Stability for vector bundles, [Hei17]).

Let $\operatorname{Bun}_{n}(C)$ denote the stack of rank $n$ vector bundles on
a smooth projective curve $C$ over $k$. There is a line bundle
$L_{det}\in\operatorname{Pic}(\operatorname{Bun}_{n}(C))$ called the
determinant line bundle, defined as follows.

By definition, there is a universal rank $n$ vector bundle
$\mathcal{E}_{univ}$ on the product $\operatorname{Bun}_{n}(C)\times C$. The
derived push-forward $R(\text{pr}_{1})_{*}\mathcal{E}_{univ}$ under the first
projection $\text{pr}_{1}:\operatorname{Bun}_{n}(C)\times
C\to\operatorname{Bun}_{n}(C)$ is a perfect complex on
$\operatorname{Bun}_{n}(C)$, since the morphism $\text{pr}_{1}$ is proper,
schematic and flat. We set $L_{det}$ to be the determinant
$L_{det}=\text{det}(R(\text{pr}_{1})_{*}\mathcal{E}_{univ})$ of the perfect
complex $R(\text{pr}_{1})_{*}\mathcal{E}_{univ}$.

In [Hei17] it is proven that the semistable locus in
$\operatorname{Bun}_{n}(C)$ with respect to the line bundle $L_{det}$ is open
and consists of the slope semistable vector bundles in the sense of Mumford.
We already know by GIT that this substack of semistable vector bundles
$\operatorname{Bun}_{n}(C)^{\rm{ss}}$ admits a good moduli space.

Now equipped with the notion of $\Theta$-semistability, we have new tools to
identify open substacks inside $\mathcal{M}$ that have a chance at admitting
good moduli spaces. The question becomes: how can we check the sufficient
existence criteria ($\Theta$-reductivity and $S$-completeness) for the stack
of $\Theta$-semistable points $\mathcal{M}^{\rm{ss}}\subset\mathcal{M}$? We
will answer this question later in this survey; this is Halpern-Leistner’s
Theorem 2.30 for intrinsic GIT. In order to state this theorem, we will need
to gain some understanding of the complement of the $\Theta$-semistable locus
$\mathcal{M}\setminus\mathcal{M}^{\rm{ss}}$, which is called the
$\Theta$-unstable locus. And to acquire knowledge about the unstable locus
and, as a consequence, gain insight on the $\Theta$-semistable locus, Halpern-
Leistner [HL21] introduced the concept of $\Theta$-stratifications, to be
explained in the next section.

###  2.5. $\Theta$-stratifications

In this section we explain the notions of $\Theta$-stratifications and
numerical invariants introduced [HL21]. This aims to generalize to the setting
of stacks the stratifications from classical GIT [Sha77, Kem78, Hes81, Nes84,
Kir84, Nit11], which have also been translated to non-algebraic gauge
theoretical frameworks [BT05]. For simplicity we work with an algebraic stack
$\mathcal{M}$ with affine diagonal and locally of finite type over a field $k$
of characteristic $0$.

We denote by $\text{Filt}(\mathcal{M})=\text{Map}(\Theta,\,\mathcal{M})$ the
stack of $\Theta$-filtrations. It is represented by an algebraic stack locally
of finite type over $k$ [HL21, Prop. 1.1.2]. There is a morphism
$\text{ev}_{1}:\text{Filt}(\mathcal{M})\to\mathcal{M}$ given by evaluating at
$1\in\Theta$.

######  Definition 2.20 ($\Theta$-stratum, [HL21, Defn. 2.1.1]).

Let $\mathcal{X}$ be an open substack of $\mathcal{M}$. A $\Theta$-stratum of
$\mathcal{X}$ is a union of connected components of $\text{Filt}(\mathcal{X})$
such that the restriction of $\text{ev}_{1}$ is a closed immersion.

We can think of a $\Theta$-stratum as a closed substack of $\mathcal{X}$ that
is identified with some connected components of $\text{Filt}(\mathcal{X})$.

######  Definition 2.21 ($\Theta$-stratification, [HL21, Defn. 2.1.2]).

A $\Theta$-stratification of $\mathcal{M}$ consists of a collection of open
substacks $(\mathcal{M}_{\leq c})_{c\in\Gamma}$ indexed by a totally ordered
set $\Gamma$. We require the following conditions to be satisfied

  1. (1)

$\mathcal{M}_{\leq c}\subset\mathcal{M}_{\leq c^{\prime}}$ for all
$c<c^{\prime}$.

  2. (2)

$\mathcal{M}=\bigcup_{c\in\Gamma}\mathcal{M}_{\leq c}$.

  3. (3)

For all $c$, there exists a $\Theta$-stratum
$\mathfrak{S}_{c}\subset\text{Filt}(\mathcal{M}_{\leq c})$ of
$\mathcal{M}_{\leq c}$ such that

$\mathcal{M}_{\leq
c}\setminus\text{ev}_{1}(\mathfrak{S}_{c})=\bigcup_{c^{\prime}<c}\mathcal{M}_{\leq
c^{\prime}}.$

  4. (4)

For every point $p\in\mathcal{M}$, the set
$\left\\{c\in\Gamma\,\mid\,p\in\mathcal{M}_{\leq c}\right\\}$ has a minimal
element.

A useful way to define $\Theta$-stratifications is via the use of numerical
invariants in the sense of [HL21, Defn. 0.0.3]. Let us give in this paragraph
a simplified account of how this works for intuition; a more precise
mathematical discussion will follow afterwards. A numerical invariant yields a
locally constant function $\nu$ on the topological space
$|\text{Filt}(\mathcal{M})|$ of the stack of $\Theta$-filtrations. This
function is taken to be valued in a totally ordered group $\Gamma$ (usually
$\Gamma=\mathbb{R}$ or more generally $\Gamma=\mathbb{R}[n]$ for polynomial
numerical invariants). For any given geometric point $p$ of the stack
$\mathcal{M}$, we can consider the subspace
$|\text{Filt}(\mathcal{M},p)|\subset|\text{Filt}(\mathcal{M})|$ consisting of
the images of all $\Theta$-filtrations of $p$ (by forgetting the
identification $f(1)\simeq p$). It is often the case that the function $\nu$
attains a maximum on $|\text{Filt}(\mathcal{M},p)|$, which roughly measures
the “instability” of the point $p$. For each $c\in\Gamma$, we can then define
$|\mathcal{M}_{\leq c}|\subset|\mathcal{M}|$ to be the set of geometric points
$p$ such that the maximum of $\nu$ on $|\text{Filt}(\mathcal{M},p)|$ (i.e. the
“degree of instability”) is at most $c$. Under favorable circumstances, the
subspace $|\mathcal{M}_{\leq c}|\subset|\mathcal{M}|$ is actually the set of
geometric points of an open substack $\mathcal{M}_{\leq c}\subset\mathcal{M}$,
and we obtain a $\Theta$-stratification of $\mathcal{M}$.

For the rest of this section we shall give a more mathematically precise
account of the previous paragraph. We shall restrict ourselves with numerical
invariants valued in the group $\Gamma=\mathbb{R}[n]$ of polynomials in the
variable $n$ with coefficients in $\mathbb{R}$. This is sufficient for many
applications, such as in the case of Gieseker stability discussed in later
sections.

######  Definition 2.22 (Total order on polynomials).

For any two polynomials $p(n),q(n)\in\mathbb{R}[n]$, we write $p(n)\leq q(n)$
if there exists some $N$ such that $p(m)\leq q(m)$ for all $m\in\mathbb{R}$
with $m>N$.

The relation “$\leq$” in Definition 2.22 equips $\mathbb{R}[n]$ with the
structure of a totally ordered abelian group.

######  Definition 2.23 (Non-degenerate graded points).

For $q\geq 1$, let
$B\mathbb{G}^{q}_{m}\vcentcolon=\left[\operatorname{Spec}(k)/\mathbb{G}^{q}_{m}\right]$.
Let $K\supset k$ be a field. We say that a morphism
$g:B(\mathbb{G}^{q}_{m})_{K}\rightarrow\mathcal{M}$ is non-degenerate if the
induced homomorphism
$\gamma:(\mathbb{G}_{m}^{q})_{K}\rightarrow\text{Aut}(g(\operatorname{Spec}(K)))$
has finite kernel.

We note that the data of a morphism
$B(\mathbb{G}^{q}_{m})_{K}\to{\mathcal{M}}$ as in the definition above is
equivalent to a pair $(p,\gamma)$ of a point $p\in{\mathcal{M}}(K)$ and a
homomorphism $\gamma:(\mathbb{G}_{m}^{q})_{K}\to\text{Aut}(p)$.

A numerical invariant is an assignment of certain scale-invariant functions to
each non-degenerate graded point
$g:B(\mathbb{G}^{q}_{m})_{K}\rightarrow\mathcal{M}$. A more formal definition
follows.

######  Definition 2.24 (Polynomial numerical invariant).

A polynomial numerical invariant $\nu$ on the stack $\mathcal{M}$ is an
assignment defined as follows. Let $K\supset k$ be a field and let
$p\in\mathcal{M}(K)$. Let
$\gamma:(\mathbb{G}_{m}^{q})_{K}\rightarrow\text{Aut}(p)$ be a homomorphism
with finite kernel. The polynomial numerical invariant $\nu$ assigns to this
data a scale-invariant function
$\nu_{\gamma}:\mathbb{R}^{q}\setminus\\{0\\}\rightarrow\mathbb{R}[n]$ such
that:

  1. (1)

$\nu_{\gamma}$ is unchanged under field extensions $K\subset K^{\prime}$.

  2. (2)

$\nu$ is locally constant in algebraic families. In other words, choose any
scheme $T$, a morphism $\xi:T\rightarrow\mathcal{M}$, and a homomorphism
$\gamma:(\mathbb{G}_{m}^{q})_{T}\rightarrow\text{Aut}(\xi)$ of $T$-group
schemes with finite kernel. Then as we vary $t\in T$, we require that the
function $\nu_{\gamma_{t}}$ is locally constant in $T$.

  3. (3)

Given a homomorphism
$\phi:(\mathbb{G}^{w}_{m})_{K}\rightarrow(\mathbb{G}^{q}_{m})_{K}$ with finite
kernel, the function $\nu_{\gamma\circ\phi}$ is the restriction of
$\nu_{\gamma}$ along the inclusion
$\mathbb{R}^{w}\hookrightarrow\mathbb{R}^{q}$ induced by $\phi$.

A useful way to construct polynomial numerical invariants on a stack
$\mathcal{M}$ is to use line bundles on $\mathcal{M}$, which are the analogues
of equivariant polarizations in the classical setting of GIT. Fix a sequence
of rational line bundles $(\mathcal{L}_{n})_{n\in\mathbb{Z}}$, where each
$\mathcal{L}_{n}$ is in the rational Picard group
$\operatorname{Pic}(\mathcal{M})\otimes_{\mathbb{Z}}\mathbb{Q}$ of the stack.
For each morphism $g:(B\mathbb{G}^{q}_{m})_{K}\rightarrow\mathcal{M}$, the
pullback line bundle $g^{*}(\mathcal{L}_{n})$ amounts to a rational character
in $X^{*}(\mathbb{G}^{q}_{m})\otimes_{\mathbb{Z}}\mathbb{Q}$. Under the
natural identification
$X^{*}(\mathbb{G}^{q}_{m})\otimes_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{Q}^{q}$,
we can interpret this as a $q$-tuple of rational numbers
$(w_{n}^{(i)})_{i=1}^{q}$, which we call the weight of
$g^{*}(\mathcal{L}_{n})$.

In practice we can often choose the line bundles
$(\mathcal{L}_{n})_{n\in\mathbb{Z}}$ in such a way that, for each fixed $i$,
there is a polynomial $w^{(i)}(T)\in\mathbb{R}[T]$ such that the weight
$w_{n}^{(i)}$ is $w^{(i)}(n)$. If this is the case, then for every $g$ we can
define an $\mathbb{R}$-linear function $L_{g}:\mathbb{R}^{q}\to\mathbb{R}[n]$
given by

$L_{g}\left((r_{i})_{i=1}^{q}\right)=\sum_{i=1}^{q}r_{i}\cdot w^{(i)}(n).$

In order to obtain a scale invariant $\nu$, we use a rational quadratic norm
on graded points as in [HL21, Defn. 4.1.12], which is the analogue of a Weyl
invariant quadratic form in the classical setting of GIT.

######  Definition 2.25 (Rational quadratic norm on graded points).

A rational quadratic norm $b$ on graded points of the stack $\mathcal{M}$ is
an assignment defined as follows. Let $K\supset k$ be a field and let
$p\in\mathcal{M}(K)$. Let
$\gamma:(\mathbb{G}_{m}^{q})_{K}\rightarrow\text{Aut}(p)$ be a homomorphism
with finite kernel. Then $b$ assigns to this data a positive definite
quadratic norm $b_{\gamma}:\mathbb{R}^{q}\to\mathbb{R}$ with rational
coefficients such that:

  1. (1)

$b_{\gamma}$ is unchanged under field extensions $K\subset K^{\prime}$.

  2. (2)

$b$ is locally constant in algebraic families. In other words, choose any
scheme $T$, a morphism $\xi:T\rightarrow\mathcal{M}$ and a homomorphism
$\gamma:(\mathbb{G}_{m}^{q})_{T}\rightarrow\text{Aut}(\xi)$ of $T$-group
schemes with finite kernel. Then as we vary $t\in T$, we require that the
function $b_{\gamma_{t}}$ is locally constant in $T$.

  3. (3)

Given a homomorphism
$\phi:(\mathbb{G}^{w}_{m})_{K}\rightarrow(\mathbb{G}^{q}_{m})_{K}$ with finite
kernel, the norm $b_{\gamma\circ\phi}$ is the restriction of $b_{\gamma}$
along the inclusion $\mathbb{R}^{w}\hookrightarrow\mathbb{R}^{q}$ induced by
$\phi$.

######  Construction 2.26.

Given a sequence of rational line bundles $(\mathcal{L}_{n})_{n\in\mathbb{Z}}$
as above and a rational quadratic norm on graded points $b$, we define a
polynomial numerical invariant $\nu$ as follows. For all non-degenerate
$g:(B\mathbb{G}^{q}_{m})_{K}\rightarrow\mathcal{M}$ with corresponding
morphism
$\gamma:(\mathbb{G}^{q}_{m})_{K}\to\text{Aut}(g(\operatorname{Spec}(K)))$, we
set

$\nu_{\gamma}(\vec{r})=\frac{L_{g}(\vec{r})}{\sqrt{b_{\gamma}(\vec{r})}}$

where $\overrightarrow{r}=(r_{1},\ldots,r_{q})$.

We next explain how a polynomial numerical invariant $\nu$ can be used top
define a $\Theta$-stratification on $\mathcal{M}$ (see [HL21, §4.1] for more
details). We say that a $\Theta$-filtration
$f:\Theta_{K}\rightarrow\mathcal{M}$ is non-degenerate if the restriction
$f|_{0}:[0/\,(\mathbb{G}_{m})_{K}]\to\mathcal{M}$ is non-degenerate. We regard
$\nu$ as a function on the set of non-degenerate $\Theta$-filtrations by
defining $\nu(f)\vcentcolon=\nu_{f|_{0}}(1)\in\mathbb{R}[n]$.

######  Definition 2.27 (Semistability for numerical invariants).

Given a polynomial numerical invariant $\nu$ on $\mathcal{M}$ and a point
$p\in|\mathcal{M}|$, we say that $p$ is $\nu$-semistable if all non-degenerate
$\Theta$-filtrations $f$ with $f(1)\simeq p$ satisfy $\nu(f)\leq 0$. Otherwise
we say that $p$ is $\nu$-unstable.

For any $\nu$-unstable point $p$, we set $M^{\nu}(p)$ to be the supremum of
$\nu(f)$ over all $\Theta$-filtrations $f$ with $f(1)\simeq p$ (if such
supremum exists). If $p$ is $\nu$-semistable, then by convention we set
$M^{\nu}(p)=0$. For any $c\in\mathbb{R}[n]_{\geq 0}$, let $\mathcal{M}_{\leq
c}$ be the set of all points $p$ satisfying $M^{\nu}(p)\leq c$, and we let
${\mathcal{M}}^{\nu{\operatorname{-}}\rm{ss}}:={\mathcal{M}}_{\leq 0}$ denote
the set of $\nu$-semistable points. If this is an open stratification of the
stack coming from a $\Theta$-stratification, then we say that $\nu$ defines a
$\Theta$-stratification.

######  Remark 2.28.

Generally one only considers non-degenerate $\Theta$-filtrations, because
these are the ones relevant for stability. Hence the adjective “non-
degenerate” is usually omitted.

######  Definition 2.29 ($\nu$-Harder-Narasimhan filtration).

For any $\nu$-unstable point $p$, a $\Theta$-filtration $f$ of $p$ is called a
$\nu$-Harder-Narasimhan filtration, abbreviated $\nu$-HN filtration, if
$\nu(f)=M^{\nu}(p)$.

If $\nu$ defines a $\Theta$-stratification, then by [HL21, Lemma 2.1.4] every
$\nu$-unstable point admits a $\nu$-Harder-Narasimhan filtration that is
unique up to pre-composing with a ramified covering $\Theta\to\Theta$.

We should remark that not all numerical invariants $\nu$ coming from sequences
of line bundles and rational quadratic norms define $\Theta$-stratifications.
In the next section we discuss some criteria that guarantee the existence of
such a $\Theta$-stratification and, in addition, imply that the open
semistable locus $\mathcal{M}^{\nu{\operatorname{-}}\rm{ss}}$ admits a good
moduli space.

###  2.6. Monotonicity and intrisic moduli theory

In this section we explain Halpern-Leistner’s intrinsic GIT theory [HL21, §5].
We keep our assumption that $\mathcal{M}$ is an algebraic stack with affine
diagonal and locally of finite type over a field $k$ of characteristic $0$.

A natural question to ask is: when does a polynomial numerical invariant $\nu$
define a $\Theta$-stratification as described in the previous section? Another
thing we would like to know is whether the locus
$\mathcal{M}^{\nu{\operatorname{-}}\rm{ss}}$ of $\nu$-semistable objects
admits a good moduli space. A general strategy to address such questions
proceeds in three steps:

  1. (Step 1)

Show that the polynomial numerical invariant $\nu$ satisfies some monotonicity
conditions (as in Definition 2.33).

  2. (Step 2)

Prove that $\nu$ satisfies the HN boundedness condition (as in Definition
2.34).

  3. (Step 3)

Check that the semistable locus $\mathcal{M}^{\nu{\operatorname{-}}\rm{ss}}$
is a disjoint union of quasi-compact open substacks.

The following theorem describes this strategy more precisely.

######  Theorem 2.30 (Intrinsic GIT, [HL21, Theorem B]).

Let $\mathcal{M}$ be an algebraic stack with affine diagonal and locally of
finite type over a field $k$ of characteristic $0$. Let $\nu$ be a polynomial
numerical invariant on $\mathcal{M}$ defined by a sequence of rational line
bundles and a norm on graded points, as explained in the Construction 2.26.
Then,

  1. (1)

If $\nu$ is strictly $\Theta$-monotone (Definition 2.33) and satisfies the HN
boundedness condition (Definition 2.34), then it defines a
$\Theta$-stratification of $\mathcal{M}$.

  2. (2)

Assume that $\nu$ satisfies the conditions in (1). In addition, suppose that
$\nu$ is strictly $S$-monotone (Definition 2.33) and that the semistable locus
$\mathcal{M}^{\nu{\operatorname{-}}\rm{ss}}$ can be written as a disjoint
union of quasi-compact open substacks. Then
$\mathcal{M}^{\nu{\operatorname{-}}\rm{ss}}$ has a separated good moduli
space.

  3. (3)

If all the conditions in (2) are satisfied and if $\mathcal{M}$ satisfies the
existence part of the valuative criterion for properness for complete discrete
valuation $k$-algebras [Sta23, Tag 0CLK], then each quasi-compact open and
closed substack of $\mathcal{M}^{\nu{\operatorname{-}}\rm{ss}}$ admits a good
moduli space that is proper over $k$.

Recall that, in Mumford’s GIT, the quotient of a projective variety with an
ample linearization is automatically projective. In many applications of GIT,
this is the way in which properness of the moduli space is proved. But in
Theorem 2.30 we see that, in order to prove properness, we have to show a
valuative criterion. The prototype of such criteria is Langton’s semistable
reduction theorem for torsion free sheaves [Lan75]. There has been a lot of
work extending this to principal bundles in different contexts (cf. [BS02,
BP03, Bal07, Hei08b, Hei10a]).

We end this subsection by explaining each of the hypotheses that one needs to
check in Theorem 2.30. We need some setup first.

######  Notation 2.31 (“Rigidified” $\Theta$ and $\overline{ST}_{R}$).

Let $R$ be a complete discrete valuation ring over $k$. Choose a uniformizer
$\varpi$ 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 $(t,\varpi)$. We will denote this point by $(0,0)$.

We define
$Y_{\overline{ST}_{R}}\vcentcolon=\operatorname{Spec}\left(\,R[t,s]/(st-\varpi)\,\right)$.
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 $\varpi$. 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 $(s,t)$.

######  Notation 2.32 (Weighted lines).

Let $\kappa\supset k$ be a field and let $a\geq 1$ be an integer. 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
$\infty=[1:0]$.

In the following definition we describe a simplified version of the
monotonicity conditions in [HL21, §5].

######  Definition 2.33 (Monotonicity).

A polynomial numerical invariant $\nu$ on $\mathcal{M}$ 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. $\overline{ST}_{R}$), and $Y_{\mathfrak{X}}$ as in
Notation 2.31. 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 scheme $\Sigma$ equipped with a $\mathbb{G}_{m}$-action along with
$\mathbb{G}_{m}$-equivariant 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
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 $\kappa$ denote 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
$\nu\left(\;\widetilde{\varphi}|_{\left[\infty/\mathbb{G}_{m}\right]}\;\right)>\nu\left(\;\widetilde{\varphi}|_{\left[0/\mathbb{G}_{m}\right]}\;\right)$.

######  Definition 2.34 (HN Boundedness).

We say that a polynomial numerical invariant $\nu$ satisfies the HN
boundedness condition if the following is always satisfied:

(HNB) Let $T$ be an affine Noetherian scheme, equipped with morphism
$g:T\rightarrow\mathcal{M}$. Then there exists a quasi-compact open substack
$\mathcal{U}_{T}\subset\mathcal{M}$ such that the following holds. For all
geometric points $t\in T$ with residue field $\kappa(t)$ and all
$\Theta$-filtrations $f:\Theta_{\kappa(t)}\rightarrow\mathcal{M}$ of the point
$g(t)$ with $\nu(f)>0$, there exists another filtration $f^{\prime}$ of $g(t)$
satisfying $\nu(f^{\prime})\geq\nu(f)$ and
$f^{\prime}|_{0}\in\mathcal{U}_{T}$.

This says that for the purposes of maximizing $\nu(f)$ among all
$\Theta$-filtrations of points in a bounded family (i.e. finding $\nu$-HN
filtrations as in Definition 2.29), it suffices to consider only those
$\Theta$-filtrations $f$ such that the associated graded $f|_{0}$ lies in some
other (possibly larger) bounded family.

##  3\. Infinite dimensional GIT for torsion-free sheaves

In this section we illustrate the previous notions with one of the main
examples in moduli theory: the moduli problem of torsion-free sheaves. This
moduli space was constructed using GIT by Mumford [Mum63b] (stable vector
bundles) and Seshadri (compactification with semistable vector bundles)
[Ses67] over curves, Gieseker [Gie77] for surfaces, later Maruyama [Mar78] and
Simpson [Sim94] in higher dimension.

###  3.1. Torsion-free sheaves

Let $X$ be a smooth projective variety of dimension $d$ over the field $k$,
and fix an ample line bundle $\mathcal{O}(1)$ on $X$. Let $T$ be a $k$-scheme
and let $\mathcal{F}$ be a $\mathcal{O}_{X_{T}}$-module. We say that
$\mathcal{F}$ is a relative torsion-free sheaf of rank $r$ on $X_{T}$ if it is
$T$-flat, finitely presented, and for all points $t\in T$ the fiber
$\mathcal{F}|_{X_{t}}$ is torsion-free of rank $r$ (i.e. the dimension of the
generic fiber is $r$).

Let us present the stack of coherent sheaves that is relevant for this moduli
problem.

######  Definition 3.1 ([GHZ21, Defn. 2.9]).

Denote by $\operatorname{Coh}_{r}^{tf}(X)$ the pseudofunctor from
$\left(\operatorname{Aff}_{k}\right)^{op}$ into groupoids such that, for any
affine scheme $T\in\operatorname{Aff}_{k}$, it gives

$\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]$

The stack $\operatorname{Coh}_{r}^{tf}(X)$ is an open substack of
$\operatorname{Coh}_{r}(X)$, the algebraic stack of coherent sheaves in $X$,
and contains the open substack $\operatorname{Bun}_{r}(X)$ of rank $r$ vector
bundles on $X$.

Denote by $P_{\mathcal{F}}$ the Hilbert polynomial of a torsion-free sheaf
$\mathcal{F}$ on $X$ with respect to the line bundle $\mathcal{O}(1)$. This is
a polynomial of degree $d$ in the variable $n$

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

with rational coefficients $a_{i}$ (see [HL97, Lem. 1.2.1]). We define the
reduced Hilbert polynomial as
$\overline{p}_{\mathcal{F}}\vcentcolon=\frac{1}{a_{d}}P_{\mathcal{F}}$. For
each $0\leq i\leq d-1$, define the $i^{th}$ slope of $\mathcal{F}$ as
$\widehat{\mu}_{i}(\mathcal{F})\vcentcolon=\frac{a_{i}}{a_{d}}$, where
$\widehat{\mu}_{d-1}(\mathcal{F})=\frac{a_{d-1}}{a_{d}}$ is the usual slope of
$\mathcal{F}$.

We define a family of line bundles on $\operatorname{Coh}_{r}^{tf}(X)$ as
follows.

######  Definition 3.2 ([GHZ21, Defn. 4.1]).

Given a $k$-scheme $T$, let $\pi_{T}:X_{T}\rightarrow T$ the structure
morphism. Let $f:T\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ be a morphism
represented by a rank $r$ torsion-free sheaf $\mathcal{F}$ of dimension $d$ on
$X_{T}$. For each integer $n$, define a line bundle $f^{*}M_{n}$ on $T$ by

$f^{\ast}M_{n}:=\det R_{\pi_{T}\ast}(\mathcal{F}(n)).$

The line bundles $f^{*}M_{n}$ for all
$f:T\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ are compatible with pullbacks
and yield a line bundle $M_{n}$ on $\operatorname{Coh}_{r}^{tf}(X)$. We set
$b_{d}:=\bigotimes_{i=0}^{d}M_{i}^{(-1)^{d-i}\binom{d}{i}}$, and we define

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

The following step in the construction is to define $\Theta$-filtrations of
the stack of torsion-free coherent sheaves, as in Definition 2.16. A
$\Theta$-filtration $f:\Theta_{K}\to\operatorname{Coh}_{r}^{tf}(X)$ of a
$K$-point of the stack $\mathcal{F}\in\operatorname{Coh}_{r}^{tf}$ consists of

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

By using the Rees construction [HL21, Prop. 1.0.1] we can rewrite the
definition of $\Theta$-filtration in a much more explicit way.

######  Proposition 3.3 ([GHZ21, Prop. 3.8]).

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

  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.

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
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$.

For each $\Theta$-filtration the restrictions $f^{*}L_{n}$ of the line bundles
are given by $\mathbb{G}_{m}$-equivariant line bundles on
$\mathbb{A}^{1}_{\mathbb{K}}$ and are determined by the fiber $L_{n}|_{0}$
considered as a $1$-dimensional representation of $\mathbb{G}_{m}$. The
integer classifying the character of $\mathbb{G}_{m}$ associated to this
representation is called the weight $\text{wt}(L_{n}|_{0})$.

On the other hand, $M_{n}|_{0}$ is isomorphic to
$\det\left(R_{\pi_{K}\ast}(\widetilde{\mathcal{F}}|_{0})(n)\right)$, where the
graded sheaf is
$\widetilde{\mathcal{F}}|_{0}=\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$.
Given that $\mathbb{G}_{m}$ acts on $\mathcal{F}_{m}/\mathcal{F}_{m+1}$ with
weight $m$, we get

$\text{wt}(M_{n}|_{0})=\sum_{m\in\mathbb{Z}}\left(P_{\mathcal{F}_{m}}(n)-P_{\mathcal{F}_{m+1}}(n)\right)\cdot
m$

and

$\text{wt}(b_{d}|_{0})=\sum_{m\in\mathbb{Z}}\left(\operatorname{rk}_{\mathcal{F}_{m}}-\operatorname{rk}_{\mathcal{F}_{m+1}}\right)\cdot
m.$

Therefore,

$\text{wt}(L_{n}|_{0})=\text{wt}(M_{n}|_{0})-\overline{p}_{\mathcal{F}}(n)\cdot\text{wt}(b_{d}|_{0})=$
$\sum_{m\in\mathbb{Z}}m\cdot\left(\overline{p}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}(n)-\overline{p}_{\mathcal{F}}(n)\right)\cdot\operatorname{rk}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}.$

In order to set the polynomial numerical invariant as in Definition 2.24, we
define a rational quadratic norm on the graded points of
$\operatorname{Coh}^{tf}_{r}(X)$ (c.f. [HLHJ21, Section 4.1]). For simplicity
we will focus only on $B\mathbb{G}_{m}^{q}$-points with $q=1$.

If $g:(B\mathbb{G}_{m})_{K}\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ is a
$\mathbb{Z}$-graded torsion-free sheaf
$\overline{F}=\bigoplus_{m\in\mathbb{Z}}\overline{F}_{m}$ of rank $r$ on
$X_{K}$, define $b_{g}$ to be the positive definite rational quadratic form on
$\mathbb{R}$ given by

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

which is uniquely determined by its value in $v=1$.

######  Definition 3.4 ([HLHJ21, Defn. 4.3]).

Let $f:\Theta_{K}\rightarrow\operatorname{Coh}^{tf}_{r}(X)$ be a
$\Theta$-filtration corresponding to the $\mathbb{Z}$-filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$. We define the polynomial numerical
invariant $\nu(f)$ to be

(3.1)
$\nu(f):=\frac{\text{wt}(L_{n}|_{0})}{\sqrt{b_{f|_{0}}(1)}}=\frac{\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}-\overline{p}_{\mathcal{F}})\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})}{\sqrt{\sum_{m\in\mathbb{Z}}m^{2}\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})}}\,.$

###  3.2. Affine grassmannians, monotonicity and good moduli space

Affine grassmannians are the tool developed in [HLHJ21] to show the
monotonicity properties (Definition 2.33) required to prove Theorem 2.30. The
idea is to build an algebraic counterpart to infinite-dimensional Atiyah-Bott-
Donaldson symplectic reduction.

We can define a fibered category $\operatorname{Coh}^{tf}_{r,rat}(X)$ whose
objects are pairs $(D,\mathcal{F})$ of a divisor and torsion-free sheaf, and a
morphism is a pair

$(i:D_{1}\hookrightarrow D_{2},\psi:\mathcal{F}_{2}\to\mathcal{F}_{1})$

where $i$ is an inclusion and $\psi$ is a homomorphism which is an isomorphism
when restricted to the complement of $D_{2}$ (notice that this is not fibered
in groupoids). This is called the stack of rational maps, see [HLHJ21, section
3.2] for more details. We have a natural morphism
$\operatorname{Coh}^{tf}_{r}(X)\rightarrow\operatorname{Coh}^{tf}_{r,rat}(X)$,
sending a torsion free sheaf $\mathcal{F}$ to the pair
$(\emptyset,\mathcal{F})$. The fibers of this morphism are ind-projective ind-
schemes. We call them affine Grassmannians, in analogy with the notion
appearing in the moduli of vector bundles over curves. Changing the
trivialization over the open subset, we can heuristically think of this
morphism as being a local presentation of the stack
$\operatorname{Coh}^{tf}_{r}(X)$ as the quotient of infinite-dimensional
spaces, the ind-schemes, by the action of an infinite-dimensional group of
rational gauge transformations.

This, together with the fact that the family of line bundles defined in the
previous section is asymptotically ample on each projective stratum, will be
the key to show monotonicity in Theorem 2.30. In the following, $\mathcal{F}$
will denote a family of torsion-free sheaves of rank $r$ on $X_{S}$, where $S$
is a quasi-compact $k$-scheme. Fix an effective Cartier divisor
$D\hookrightarrow X_{S}$ flat over $S$ and set $Q=X_{S}\setminus D$.

######  Definition 3.5 (cf. [HLHJ21, Defn. 3.12]).

We define $\operatorname{Gr}_{X_{S},D,\mathcal{F}}$ to be the functor from
$\operatorname{Aff}_{S}^{\,\,op}$ to sets such that 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 rank $r$ 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.

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

We remark that the fiber over $(D,\mathcal{F}$) of the natural morphism
$\operatorname{Coh}^{tf}_{r}(X)\rightarrow\operatorname{Coh}^{tf}_{r,rat}(X)$
is $\operatorname{Gr}_{X_{S},D,\mathcal{F}}$.

Now let $j$ be the open immersion $j:Q\hookrightarrow X_{S}$ and choose
$T\in\operatorname{Aff}_{S}$ and an element
$(\mathcal{E},\theta)\in\operatorname{Gr}_{X_{S},D,\mathcal{F}}(T)$. Using
that the unit $\mathcal{E}\rightarrow j_{T\,*}\,j_{T}^{*}\,\mathcal{E}$ is a
monomorphism we can use the isomorphism $j_{T*}(j_{T})^{*}\theta$ to
understand $\mathcal{E}$ as a subsheaf of
$j_{T\,*}\,j_{T}^{*}\,\mathcal{F}|_{X_{T}}$ and have
$\mathcal{F}|_{X_{T}}\subset\mathcal{E}\subset
j_{T\,*}\,j_{T}^{*}\,\mathcal{F}|_{X_{T}}$. We use the increasing sequence of
sheaves

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

to define the truncated subfunctor $\operatorname{Gr}^{\leq
N}_{X,D,\mathcal{F}}\subset\operatorname{Gr}_{X,D,\mathcal{F}}$, where

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

Recall that a strict ind-scheme is a colimit of closed immersions.

######  Proposition 3.6 ([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 $\operatorname{Aff}_{S}$) and, for each $N\geq 0$,

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

where 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
strict ind-scheme over $S$.

There is a natural forgetful morphism
$\operatorname{Gr}_{X_{S},D,\mathcal{F}}\rightarrow\operatorname{Coh}_{r}^{tf}(X)$
sending $(\mathcal{E},\theta)$ to $\mathcal{E}$. We denote again by $L_{n}$,
$b_{d}$ and $M_{n}$ the line bundles on
$\operatorname{Gr}_{X_{S},D,\mathcal{F}}$ pulled back from those on
$\operatorname{Coh}_{r}^{tf}(X)$ defined before, by abuse of notation.

######  Proposition 3.7 ([HLHJ21, Prop. 3.19]).

There exists some integer $m\gg 0$, depending on $N\in\mathbb{Z}_{\geq 0}$ and
$P\in\mathbb{Q}[n]$, 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.

Using the affine grassmannians above and the rational filling condition proven
in [HLHJ21, Lem. 4.7], one can show that the stack
$\operatorname{Coh}^{tf}_{r}(X)$ satisfies the two monotonicity conditions in
Definition 2.33. This type of argument was called “Infinite dimensional GIT”
in [HLHJ21]. We refer the reader to the introduction in [HLHJ21] for a more
detailed explanation of this argument, and how the affine grassmannian comes
in.

######  Theorem 3.8 ([HLHJ21, Thm. 4.8]).

The polynomial numerical invariant in Definition 3.4 is strictly
$\Theta$-monotone and strictly $S$-monotone on the stack
$\operatorname{Coh}^{tf}_{r}(X)$.

Recall the notion of HN boundedness in Definition 2.34. The following theorem
was proven by a concrete optimization argument in [HLHJ21].

######  Theorem 3.9 ([HLHJ21, Prop. 5.5] ).

The polynomial numerical invariant in Definition 3.4 on the stack
$\operatorname{Coh}^{tf}_{r}(X)$ satisfies the HN boundedness condition.

Using the $\Theta$-monotonicity condition satisfied by the stack of torsion-
free coherent sheaves in Theorem 3.8, plus the HN boundedness condition of the
polynomial numerical invariant $\nu$ in Theorem 3.9, it follows that there
exists a $\Theta$-stratification on $\operatorname{Coh}^{tf}_{r}(X)$
(Definition 2.16), as stated by the Intrinsic GIT Theorem 2.30(1).

######  Theorem 3.10 ([HLHJ21, Thm. 5.11]).

The polynomial numerical invariant $\nu$ defines a $\Theta$-stratification on
the stack $\operatorname{Coh}^{tf}_{r}(X)$.

From the polynomial numerical invariant in Definition 3.4 we can derive a
definition of $\nu$-stability by Definition 2.27 in terms of
$\Theta$-filtrations or, by Proposition 3.3, $\mathbb{Z}$-filtrations. Recall
the total order on polynomials in Definition 2.22.

######  Definition 3.11 ($\nu$-semistability).

Let $K\supset k$ be an algebraically closed field extension and let
$\mathcal{F}$ be a torsion-free coherent sheaf on $X_{K}$. We say that
$\mathcal{F}$ is $\nu$-semistable if for all $\mathbb{Z}$-filtrations
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ we have

$\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}-\overline{p}_{\mathcal{F}})\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})\leq
0$

Otherwise, $\mathcal{F}$ is called $\nu$-unstable.

In [HLHJ21, Prop. 5.16] it is shown that a torsion-free coherent sheaf
${\mathcal{F}}$ is $\nu$-semistable if and only if it is Gieseker semistable
as in [Gie77, Mar78, Sim94]. Therefore the stack intrinsic definition of
$\nu$-stability for this polynomial numerical invariant happens to coincide
with the usual polynomial Gieseker stability for torsion-free sheaves the in
literature.

Using the definition of $\Theta$-stratification, the substack of semistable
torsion-free sheaves of rank $r$, denoted by
$\operatorname{Coh}^{tf}_{r}(X)^{\nu{\operatorname{-}}\rm{ss}}$, is an open
substack of $\operatorname{Coh}^{tf}_{r}(X)$. Given a rational polynomial
$P\in\mathbb{Q}[n]$, we denote by
$\operatorname{Coh}^{tf}_{r}(X)_{P}^{\nu{\operatorname{-}}\rm{ss}}$ the open
and closed substack of
$\operatorname{Coh}^{tf}_{r}(X)^{\nu{\operatorname{-}}\rm{ss}}$ parametrizing
sheaves whose Hilbert polynomial is $P$.

######  Proposition 3.12 ([HLHJ21, Prop. 5.18]).

The substack
$\operatorname{Coh}^{tf}_{r}(X)_{P}^{\nu{\operatorname{-}}\rm{ss}}$ is quasi-
compact.

Finally, we gather Theorem 3.10, the $S$-monotonicity property shown in
Theorem 3.8 and last Proposition 3.12 to prove the existence of a separated
good moduli space by Theorem 2.30 (2). Together with the proof of the
existence part of the valuative criterion for properness in [HLHJ21, Prop.
5.20] we obtain an intrinsic construction of a proper good moduli space of
torsion-free sheaves as in Theorem 2.30 (3).

######  Theorem 3.13 ([HLHJ21, Thm. 5.21]).

The stack $\operatorname{Coh}^{tf}_{r}(X)_{P}^{\nu{\operatorname{-}}\rm{ss}}$
admits a proper good moduli space.

###  3.3. Leading term and Gieseker-Harder-Narasimhan filtrations

Let $K$ be an algebraically closed field and let $\mathcal{F}$ be an unstable
torsion-free coherent sheaf on $X_{K}$. Recall from Definition 2.29 that a
$\Theta$-filtration $f$, with associated underlying filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$, is called a $\nu$-Harder-Narasimhan
filtration if the polynomial numerical invariant $\nu(f)$ achieves its maximal
value. Having a $\Theta$-stratification on $\operatorname{Coh}^{tf}_{r}(X)$
allows us to define a unique $\nu$-HN filtration maximizing $\nu$, as seen at
the end of Section 2.5, for any field valued point of the stack.

######  Proposition 3.14.

Let $K\supset k$ be an arbitrary field extension and let $\mathcal{F}$ be an
unstable torsion-free coherent sheaf on $X_{K}$. Then, $\mathcal{F}$ admits a
$\nu$-Harder-Narasimhan filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ which
is uniquely determined up to scaling all of the indexes by a constant rational
number. We call this filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ the
leading term HN filtration of $\mathcal{F}$.

Depending on whether we use slope semistability by Mumford [Mum63a], or
Gieseker semistability [Gie77, Sim94], we can consider two different
filtrations, the slope Harder-Narasimhan filtration (slope HN) and the
Gieseker Harder-Narasimhan filtration (Gieseker HN). Other intermediate
polynomial stability conditions can be considered as in [HL97, Thm. 1.6.7].

The leading term HN filtration of an unstable torsion-free coherent sheaf
$\mathcal{F}$ from Proposition 3.14 corresponds to an intermediate Harder-
Narasimhan filtration where the sequence of Hilbert polynomials of the
quotients find their differences not at the top-degree coefficient (as in the
slope HN filtration), nor at any degree coefficient (as in the Gieseker HN
filtration), but at an intermediate degree coefficient, from where the
filtration takes its name. Let us explain this situation.

Start with the Gieseker HN filtration of $\mathcal{F}$ (c.f. [HL97, Thm.
1.3.4]):

$0=\mathcal{F}^{\text{HN}}_{0}\subset\mathcal{F}^{\text{HN}}_{1}\subset\mathcal{F}^{\text{HN}}_{2}\subset\cdots\subset\mathcal{F}^{\text{HN}}_{n}=\mathcal{F}$

which verifies

$\overline{p}_{\mathcal{F}_{1}^{\text{HN}}}>\overline{p}_{\mathcal{F}_{2}^{\text{HN}}/\mathcal{F}_{1}^{\text{HN}}}>\cdots>\overline{p}_{\mathcal{F}_{n}^{\text{HN}}/\mathcal{F}_{n-1}^{\text{HN}}}$

There exists a minimal index $i$ such that for all $h>i$ we have that all
$h^{th}$ slopes are equal

$\widehat{\mu}_{h}(\mathcal{F}_{1}^{\text{HN}})=\widehat{\mu}_{h}(\mathcal{F}_{2}^{\text{HN}}/\mathcal{F}_{1}^{\text{HN}})=\cdots=\widehat{\mu}_{h}(\mathcal{F}_{n}^{\text{HN}}/\mathcal{F}_{n-1}^{\text{HN}}),$

but $i$ is the largest index for which we find at least one strict inequality
in the sequence of $i^{th}$ slopes:

$\widehat{\mu}_{i}(\mathcal{F}_{1}^{\text{HN}})\geq\widehat{\mu}_{i}(\mathcal{F}_{2}^{\text{HN}}/\mathcal{F}_{1}^{\text{HN}})\geq\cdots\geq\widehat{\mu}_{i}(\mathcal{F}_{n}^{\text{HN}}/\mathcal{F}_{n-1}^{\text{HN}}).$

Define indexes $h_{0}<h_{1}<h_{2}<\cdots<h_{j}=n$ for the jumps of the
$\widehat{\mu}_{i}$ sequence: let $h_{0}=0$ and

$\\{h_{m}\\}_{m=1}^{j}=\left\\{\;1\leq l\leq
n\;\left|\;\widehat{\mu}_{i}(\mathcal{F}_{l+1}^{\text{HN}}/\mathcal{F}_{l}^{\text{HN}})<\widehat{\mu}_{i}(\mathcal{F}_{l}^{\text{HN}}/\mathcal{F}_{l-1}^{\text{HN}})\right.\right\\}$

and set $\mathcal{F}^{\text{l-term}}_{k}=\mathcal{F}^{\text{HN}}_{h_{k}}$.

The filtration

$0=\mathcal{F}^{\text{l-term}}_{0}\subset\mathcal{F}^{\text{l-term}}_{1}\subset\mathcal{F}^{\text{l-term}}_{2}\subset\cdots\subset\mathcal{F}^{\text{l-term}}_{j}=\mathcal{F}$

is called the unweighted leading term filtration of $\mathcal{F}$. We call $i$
the leading term index, which coincides with the degree of the polynomial
$\nu(f)$ has degree $i$, and call $\widehat{\mu}_{i}(\mathcal{F})$ the leading
slope. Observe that, in the case that $\mathcal{F}$ is slope unstable, then
$i=d-1$ and hence the leading term HN filtration coincides with the slope HN
filtration.

######  Definition 3.15 ([HLHJ21, Thm. 5.11] [GHZ21, Defn. 9.2]).

Let $\mathcal{F}$ be a $\nu$-unstable torsion-free coherent sheaf. If the
unweighted leading term HN filtration for $\mathcal{F}$ is given by

$0=\mathcal{F}^{\text{l-term}}_{0}\subset\mathcal{F}^{\text{l-term}}_{1}\subset\mathcal{F}^{\text{l-term}}_{2}\subset\cdots\subset\mathcal{F}^{\text{l-term}}_{j}=\mathcal{F}$

with leading slope $\widehat{\mu}_{i}(\mathcal{F})$, we define the (weighted)
leading term HN filtration $(\mathcal{F}^{\text{Can}})_{m\in\mathbb{Z}}$ as
follows (up to scaling of the weights).

Set $\mu=\widehat{\mu}_{i}(\mathcal{F})$,
$\mu_{k}=\widehat{\mu}_{i}\left(\mathcal{F}^{\text{l-term}}_{k}/\mathcal{F}^{\text{l-term}}_{k-1}\right)$.
Choose a positive rational number $L$ such that $L\cdot\mu\in\mathbb{Z}$ and
$L\cdot\mu_{k}\in\mathbb{Z}$ for all $k$. Then,

  1. (1)

$\mathcal{F}^{\text{Can}}_{m}=0$ for $m>L(\mu_{1}-\mu)$.

  2. (2)

Let $1\leq k\leq j-1$. For each $L(\mu_{k}-\mu)\geq m>L(\mu_{k+1}-\mu)$, we
set $\mathcal{F}^{\text{Can}}_{m}=\mathcal{F}^{\text{l-term}}_{k}$.

  3. (3)

$\mathcal{F}^{\text{Can}}_{m}=\mathcal{F}$ for $L(\mu_{j}-\mu)\geq m$.

This is precisely the leading term HN filtration where the $k^{th}$ jump
happens at the integer $L(\mu_{k}-\mu)$.

The weighted leading term HN filtration recovers a destabilizing HN-type
filtration at the maximal degree-coefficient of the Hilbert polynomial, the
leading term, yielding a sequence of decreasing leading slopes. If we take the
graded object of the filtration and iterate the process we will check all
possible instability terms at all lower degrees in the polynomial. This means
that we will arrive at the classical Gieseker HN filtration at the end. Let us
explain this procedure (c.f. [GHZ21, Prop. 9.3]).

Denote the (weighted) leading term HN filtration in Definition 3.15 by
$\left({}^{(1)}\mathcal{F}_{m}\right)_{m\in\mathbb{Z}}$ and the leading term
index by $i_{1}$. Define the $1^{st}$ associated graded Levi sheaf as

${\rm
gr}_{1}(\mathcal{F})=\left(\mathcal{F}_{i}^{l-term}/\mathcal{F}_{i-1}^{l-term}\right)_{i=1}^{j_{1}}$

The recursion step consists of taking the leading term HN filtration of this
graded object, which amounts to taking the leading term HN filtration for each
$\mathcal{F}_{i}^{l-term}/\mathcal{F}_{i-1}^{l-term}$, with
$i=1,2,\ldots,j_{1}$. Note that all leading term indices are strictly smaller
than $i_{1}$.

This yields a lexicographic filtration indexed by $\mathbb{Z}^{2}$, denoted by
$\left({}^{(2)}\mathcal{F}_{\overrightarrow{m}}\right)_{\overrightarrow{m}\in\mathbb{Z}^{2}}$,
with jumps at the pairs

$\overrightarrow{m}_{1}>\overrightarrow{m}_{2}>\cdots>\overrightarrow{m}_{j_{2}}$

This way, the underlying unweighted filtration reads

$0\subset^{(2)}\mathcal{F}_{\overrightarrow{m}_{1}}\subset^{(2)}\mathcal{F}_{\overrightarrow{m}_{2}}\subset\cdots\subset^{(2)}\mathcal{F}_{\overrightarrow{m}_{j_{2}}}=\mathcal{F}$

and satisfies that the Hilbert polynomials of the graded pieces
${}^{(2)}\mathcal{F}_{\overrightarrow{m}_{i}}/^{(2)}\mathcal{F}_{\overrightarrow{m}_{i-1}}$
are decreasing but their differences occur, at most, at a degree
$i_{2}<i_{1}$, $i_{2}$ being the leading term index of this second leading
term HN filtration.

Continuing this process (i.e. taking the leading term HN filtration of the
graded object while possible), we arrive to a $q^{th}$ filtration
$\left({}^{(q)}\mathcal{F}_{\overrightarrow{m}}\right)_{\overrightarrow{m}\in\mathbb{Z}^{q}}$
with unweighted filtration

$0\subset^{(q)}\mathcal{F}_{\overrightarrow{m}_{1}}\subset^{(q)}\mathcal{F}_{\overrightarrow{m}_{2}}\subset\cdots\subset^{(q)}\mathcal{F}_{\overrightarrow{m}_{j_{q}}}=\mathcal{F}$

whose graded pieces forming the $q^{th}$ associated Levi sheaf ${\rm
gr}_{q}(\mathcal{F})=\left({}^{(q)}\mathcal{F}_{\overrightarrow{m}_{i}}/^{(q)}\mathcal{F}_{\overrightarrow{m}_{i-1}}\right)$
do not admit any further filtering by any lower leading degree: they are
Gieseker semistable. Therefore this final filtration is, indeed, the Gieseker
HN filtration that we find in the literature [Sim94, HL97].

##  4\. $G$-bundles on higher dimensional varieties

In this section we expose the theory of the “Beyond GIT” program for the
moduli problem of $G$-bundles on higher dimensional varieties, providing a
stack-intrinsic construction of the moduli space and a new notion of a
Gieseker Harder-Narasimhan filtration. A more extensive treatment can be found
in [GHZ21].

For the rest of this section $X$ will denote a smooth projective variety of
dimension $d$ over an algebraically closed field $k$ of characteristic $0$.

###  4.1. Principal $G$-bundles

Let $G$ be an algebraic reductive group. Let
$\rho:G\hookrightarrow\operatorname{GL}(V)$ be a faithful representation. A
principal $G$-bundle $P$ on a scheme $X$ induces a vector bundle
${\mathcal{F}}:=P(V)$ of rank equal to $\dim V$. Using the converse
construction, we want to describe a principal bundle as a pair consisting of a
vector bundle and a reduction of structure group.

Let $S$ be an affine $k$-scheme. Given a vector bundle ${\mathcal{F}}$ on
$X_{S}$ of rank $\dim V$, we define the associated principal
$\operatorname{GL}(V)$-bundle

$P_{\operatorname{GL}}={\underline{\operatorname{Iso}}}({\mathcal{F}},V\otimes{\mathcal{O}}_{X_{S}}),$

with the right action of $\operatorname{GL}(V)$ defined as the post-
composition with the inverse of $h\in\operatorname{GL}(V)$:

$(\beta,h)\mapsto\beta\cdot h:=h^{-1}\circ\beta,$

where we take the inverse to obtain an action on the right.

Define the scheme of reductions as the quotient

${\rm
Red_{G}({\mathcal{F}})=\underline{\operatorname{Iso}}({\mathcal{F}},V\otimes{\mathcal{O}}_{X_{S}})/G}\longrightarrow
X_{S},$

where $G$ acts on the right via the representation $\rho$. A reduction of
structure group is given by a section of
$\operatorname{Red}_{G}({\mathcal{F}})$ over $X_{S}$.

If $\dim X>1$, then the moduli space of vector bundles does not satisfy the
existence part of the valuative criterion for properness in general, because a
family of vector bundles might degenerate to a torsion-free sheaf. Likewise,
if we consider principal $G$-bundles we have to allow for more general objects
to appear in the moduli problem. These will be called principal
$\rho$-sheaves.

Before considering the general case, let us have a look at a simpler example.
A subset $U\subset X_{S}$ is called big if for any point $s\in X$, the fiber
$U_{s}\subset X_{s}=X$ is an open set with codimension in $X$ at least two.

######  Example 4.1.

If $G={\rm Sp}(r)$ is the symplectic group and $\rho$ is the standard
representation, then a reduction is the same thing as a homomorphism
$\varphi:{\mathcal{F}}\otimes{\mathcal{F}}\rightarrow{\mathcal{O}}_{X_{S}}$
which is bilinear, skew-symmetric and non-degenerate. In this case, a
$\rho$-sheaf on $X$ will be a pair

$({\mathcal{F}},\varphi:{\mathcal{F}}\otimes{\mathcal{F}}\rightarrow{\mathcal{O}}_{X}),$

where ${\mathcal{F}}$ is a torsion-free sheaf and $\varphi$ is a skew-
symmetric bilinear form, which is non-degenerate when restricted to the open
set $U_{{\mathcal{F}}}$ where $\mathcal{F}$ is locally free. Equivalently, a
$\rho$-sheaf is a principal ${\rm Sp}(r)$-bundle $P$ on a big open set
$U\subset X$, a torsion-free sheaf ${\mathcal{F}}$ on $X$, and an isomorphism
between the vector bundle associated to $P$ and the restriction
${\mathcal{F}}|_{U}$.

######  Remark 4.2.

If we want our definition of stability to reduce to the definition of
Ramanathan when $\dim X=1$ [Ram75], then we have to require that the
representation $\rho:G\to\operatorname{GL}(V)$ is central (meaning that the
representation $\rho$ sends the identity component of the center of $G$ to the
center of $\operatorname{GL}(V)$). Otherwise, for noncentral $\rho$ our
definition of semistability is stronger than Ramanathan’s, see [GHZ21, Example
10.10] for an explanation. Notice that if $G$ is semisimple, then $\rho$ is
automatically central. On the other hand, $\rho$ cannot be central if the
dimension of the center of $G$ is larger than one; this is why we consider
representations into products of general linear groups in subsection 4.4.

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

The representation $\rho:G\hookrightarrow\operatorname{GL}(V)$ gives a left
action of $G$ on $V$, hence a left action on $\det(V)$, and therefore
$\det(V)$ acquires the structure of a right $k[G]$-comodule:

(4.1) $\det(V)\longrightarrow\det(V)\otimes k[G].$

Explicitly, $v\mapsto v\otimes\det(\rho(g))$.

Using the inverse $v\mapsto g^{-1}v$ we obtain a right action of $G$ on $V$.
This right action of $G$ on $V$ induces on $V$ a left $k[G]$-comodule
structure $V\to k[G]\otimes V$. We equip $V^{\vee}$ with the dual right
$k[G]$-comodule structure:

(4.2) $c:V^{\vee}\longrightarrow V^{\vee}\otimes k[G].$

Explicitly, if $\\{e_{i}\\}$ is a basis of $V$ and $\\{e^{i}\\}$ is the dual
basis of $V^{\vee}$, then

$c(e^{i})=\sum_{j}e^{j}\otimes(\rho(g)^{-1})^{i}{}_{j}$

(note that $A\mapsto(A^{-1})^{i}{}_{j}$ is an element of
$k[\operatorname{GL}(V)]$). The value that it takes on an element
$A\in\operatorname{GL}(V)$ is the $(i,j)$-entry of the inverse of $A$,
expressed in the basis $\\{e_{i}\\}$.

These right $k[G]$-comodule structures (4.1) (4.2) induce a right
$k[G]$-comodule structure on

(4.3)
$E_{V}({\mathcal{F}}):=\operatorname{Sym}^{\bullet}({\mathcal{F}}\otimes_{k}V^{\vee})\otimes_{{\mathcal{O}}_{X_{S}}}\operatorname{Sym}^{\bullet}(\det({\mathcal{F}})^{\vee}\otimes_{k}\det(V)).$

Therefore, the scheme
$H({\mathcal{F}},V):=\underline{\operatorname{Spec}}_{X_{S}}(E_{V}({\mathcal{F}}))$
(which is affine over $X_{S}$ and of finite presentation) has a right action
by $G$. Consider the GIT quotient

$H({\mathcal{F}},V){/\\!\\!/}G=\underline{\operatorname{Spec}}_{X_{S}}(E_{V}({\mathcal{F}})^{G}),$

where $E_{V}({\mathcal{F}})^{G}$ is the subalgebra of $G$-invariants.

The restriction $H:=H({\mathcal{F}},V)|_{U}$ to the open set $U\subset X_{S}$
where ${\mathcal{F}}$ is locally free is

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

parametrizing pairs $(\alpha:{\mathcal{F}}|_{U}\to
V\otimes_{k}{\mathcal{O}}_{U},\beta:\det({\mathcal{F}}|_{U})^{\vee}\to\det(V^{\vee})\otimes_{k}{\mathcal{O}}_{U}$.
The equation $\det(\alpha)\beta=1$ cuts out a closed subscheme which is
isomorphic to

$\underline{\operatorname{Iso}}({\mathcal{F}}|_{U},V\otimes_{k}{\mathcal{O}}_{U}).$

Furthermore, the equation $\det(\alpha)\beta=1$ is $G$-invariant, so it also
cuts out a closed subscheme $(H{/\\!\\!/}G)_{0}\subset H{/\\!\\!/}G$ which is
isomorphic to

$\underline{\operatorname{Iso}}({\mathcal{F}}|_{U},V\otimes_{k}{\mathcal{O}}_{U})/G=\operatorname{Red}_{G}({\mathcal{F}}|_{U}).$

######  Definition 4.3 ([GHZ21, Defn. 2.14]).

Let ${\mathcal{F}}$ be a torsion-free sheaf on $X$. The scheme of reductions
$\operatorname{Red}_{G}({\mathcal{F}})$ is defined to be the scheme theoretic
closure of $\operatorname{Red}_{G}({\mathcal{F}}|_{U})$ in
$H({\mathcal{F}},V){/\\!\\!/}G$.

The scheme $\operatorname{Red}_{G}({\mathcal{F}})\to X_{S}$ is affine of
finite type and allows to give a definition of principal $\rho$-sheaf.

######  Definition 4.4 ([GHZ21, Defn. 3.1]).

A principal $\rho$-sheaf on $X_{S}$ is a pair $({\mathcal{F}},\sigma)$ where

  1. (1)

${\mathcal{F}}$ is a torsion-free sheaf on $X_{S}$.

  2. (2)

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

We have the following Hartogs-like result (see [GHZ21, Lem. 2.2]).

######  Proposition 4.5 ([GHZ21, Prop. 2.18]).

Let $U\subset X_{S}$ be any big open set where ${\mathcal{F}}$ is locally free
(note that such $U$ always exists [GHZ21, Lemma 2.5]). There is a bijection
between the sections of the morphism $\operatorname{Red}_{G}({\mathcal{F}})\to
X_{S}$ and the sections on the restriction
$\operatorname{Red}_{G}({\mathcal{F}}|_{U})\to U$.

This allows us to give an equivalent definition of $\rho$-sheaf.

######  Lemma 4.6.

Giving a principal $\rho$-sheaf on $X_{S}$ is the same thing as giving a
triple $(P,{\mathcal{F}},\psi)$ where

  1. (1)

${\mathcal{F}}$ is a torsion-free sheaf on $X_{S}$, and $U$ is a big open
subset where ${\mathcal{F}}$ is locally free (we may take $U$ to be the
maximal such open).

  2. (2)

$P$ is a principal $G$-bundle on $U$.

  3. (3)

$\psi$ is an isomorphism between the vector bundle $P(\rho)$ associated to
$\rho$, and the restriction ${\mathcal{F}}|_{U}$

$\psi:P(\rho)\cong{\mathcal{F}}|_{U}.$

Denote by $\operatorname{Bun}_{\rho}(X)$ the stack of principal $\rho$-sheaves
on $X$.

######  Definition 4.7.

The algebraic stack of principal $\rho$-sheaves on $X$ is the pseudofunctor
defined by

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

There is a forgetful morphism
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ which
is schematic, affine and of finite type ([GHZ21, Prop 3.6]). This allows to
relate the constructions for $\rho$-sheaves to those of section 3 for torsion-
free coherent sheaves.

###  4.3. Moduli space of $\rho$-sheaves

Here we introduce a polynomial numerical invariant on the stack
$\operatorname{Bun}_{\rho}(X)$ of principal $\rho$-sheaves, which yields the
notion of Gieseker stability. Once we prove that the requirements of Theorem
2.30 are met, it will be shown that the substack of semistable $\rho$-sheaves
admits a good moduli space.

Let $\lambda:\mathbb{G}_{m}\to G$ be a one-parameter subgroup . We define a
parabolic subgroup $P_{\lambda}\subset G$ associated to $\lambda$ by

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

whose Levi subgroup is

$L_{\lambda}=\left\\{\,g\in G\,\mid\,\lim\limits_{x\to
0}\,\lambda(x)\,g\,\lambda(x)^{-1}=g\,\right\\}.$

Composing with the representation $\rho$ we obtain a parabolic subgroup of
$\operatorname{GL}(V)$:

$P^{\operatorname{GL}(V)}_{\lambda}=\left\\{\,g\in\operatorname{GL}(V)\,\mid\,\lim\limits_{x\to
0}\,\rho\circ\lambda(x)\,g\,(\rho\circ\lambda(x))^{-1}\;\text{exists}\,\right\\}$

and it is clear that $P_{\lambda}=P^{\operatorname{GL}(V)}_{\lambda}\cap G$.

Let $({\mathcal{F}},\sigma)$ be a principal $\rho$-sheaf on $X$. Let $P\to
U\subset X$ be the associated principal $G$-bundle on the open set $U$ where
${\mathcal{F}}$ is locally free, and let $\mathcal{G}_{\lambda}\to
U^{\prime}\subset U$ be a reduction of structure group of $P$ to $P_{\lambda}$
on a big open set $U^{\prime}\subset X$. The inclusion $P_{\lambda}\subset
P^{\operatorname{GL}(V)}_{\lambda}$ gives a principal
$P^{\operatorname{GL}(V)}_{\lambda}$-bundle $P^{\prime}$ on $U^{\prime}$,
which is a reduction of structure group to the parabolic
$P^{\operatorname{GL}(V)}_{\lambda}$ of the principal
$\operatorname{GL}(V)$-bundle associated to the vector bundle
${\mathcal{F}}|_{U^{\prime}}$.

Giving a reduction to a parabolic of the linear group is the same thing as
giving a filtration of the vector bundle ${\mathcal{F}}|_{U^{\prime}}$, and
such a filtration extends uniquely over $X_{S}$ to a filtration of
${\mathcal{F}}$ by torsion-free sheaves (with the condition that successive
quotients of the filtration are also torsion-free). Moreover, we use the
weights of the one-parameter subgroup obtained by composition
$\rho\circ\lambda:\mathbb{G}_{m}\to\operatorname{GL}(V)$ to obtain a
$\mathbb{Z}$-weighted filtration $({\mathcal{F}}_{m})_{m\in\mathbb{Z}}$ of
${\mathcal{F}}$ as in Proposition 3.3.

Recall from Definition 2.16 that a $\Theta$-filtration of a point $p$ in a
stack ${\mathcal{M}}$ is a morphism
$f:\Theta=[\mathbb{A}^{1}/\mathbb{G}_{m}]\to{\mathcal{M}}$ together with an
isomorphism $f(1)\simeq p$. Any $\Theta$-filtrations
$f:\Theta\rightarrow\operatorname{Bun}_{\rho}(X)$ of a $\rho$-sheaf
$(\mathcal{F},\sigma)$ can be composed with the forgetful morphism
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ to
obtain a $\mathbb{Z}$-weighted filtration of $\mathcal{F}$ satisfying a series
of compatibility conditions. This way, a $\Theta$-filtration of a $\rho$-sheaf
$(\mathcal{F},\sigma)$ can be understood as a $\mathbb{Z}$-weighted filtration
$({\mathcal{F}}_{m})_{m\in\mathbb{Z}}$ of ${\mathcal{F}}$ (as in Proposition
3.3) which is associated to a reduction of structure group ([GHZ21, Prop. 3.12
and 3.16]). See [GHZ21, Sections 3.2 and 3.3] for details of this
construction.

In a similar way to section 3.1, we can define a family of line bundles
[GHZ21, Defn. 4.1] and a rational quadratic norm [GHZ21, Defn. 4.3] to define
a polynomial numerical invariant ([GHZ21, Prop. 4.4]):

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

Note that this is the same expression as in Definition 3.4 for torsion-free
sheaves, but for a $\rho$-sheaf $(\mathcal{F},\sigma)$ it accounts just for
$\Theta$-filtrations of $\operatorname{Bun}_{\rho}(X)$.

The polynomial numerical invariant $\nu$ defined on the stack
$\operatorname{Bun}_{\rho}(X)$ can be shown to be strictly $\Theta$-monotone
and strictly $S$-monotone ([GHZ21, Prop. 4.16]) and to satisfy the HN
boundedness condition ([GHZ21, Prop. 4.26]). Then it defines a
$\Theta$-stratification on $\operatorname{Bun}_{\rho}(X)$ by Theorem 2.30(1)
([GHZ21, Thm. 4.27]). This allows to define a notion of $\nu$-stability which
is an open condition.

######  Definition 4.8 ([GHZ21, Defn. 4.28]).

A principal $\rho$-sheaf $({\mathcal{F}},\sigma)$ is $\nu$-semistable if for
all one-parameter subgroups $\lambda:\mathbb{G}_{m}\to G$ and all reductions
as above, we have

$\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{{\mathcal{F}}_{m}/{\mathcal{F}}_{m+1}}-\overline{p}_{{\mathcal{F}}})\cdot\operatorname{rk}({\mathcal{F}}_{m}/{\mathcal{F}}_{m+1})\leq
0.$

This definition coincides with the semistability notion in [Sch02, GS05]. The
substack $\operatorname{Bun}_{\rho}(X)^{\nu{\operatorname{-}}\rm{ss}}$ of
semistable $\rho$-sheaves satisfies the existence part of the valuative
criterion for properness in [GHZ21, Thm. 5.8]. After fixing the degree
$\vartheta$ of a $\rho$-sheaf $(\mathcal{F},\sigma)$ ([GHZ21, Defn. 6.3]) and
the Hilbert polynomial $P$ of $\mathcal{F}$, the substacks
$\operatorname{Bun}_{\rho}(X)^{\nu{\operatorname{-}}\rm{ss},P}_{\vartheta}$
are quasi-compact ([GHZ21, Prop. 6.6], then by Theorem 2.30 (2,3) they do
admit a proper good moduli space ([GHZ21, Thm. 6.8]). This is a stack
intrinsic alternative to the constructions in [Sch02, GS05].

###  4.4. The scheme of $G$-reductions for products of representations

As we explained in Remark 4.2, if we want to consider reductive groups that
are not semisimple, we need to deal with tuples of representations. In section
4.5 we will observe that when constructing the Gieseker HN filtration, even if
we start with a semisimple group, the Levi factors we need to go through make
necessary to consider non-semisimple groups.

Let $V^{\bullet}=(V^{i}){}_{i=1}^{b}$ be a tuple of finite dimensional vector
spaces. We consider a representation into the tuple

$\rho=(\rho_{1},\ldots,\rho_{b}):G\longrightarrow\operatorname{GL}(V^{1})\times\cdots\times\operatorname{GL}(V^{b}).$

Let ${\mathcal{F}}^{\bullet}=({\mathcal{F}}^{i}){}_{i=1}^{b}$ be a tuple of
torsion-free sheaves on $X_{S}$. Using the construction (4.3), we consider the
product scheme

$H({\mathcal{F}}^{\bullet},V^{\bullet}):=\prod_{i=1}^{b}H({\mathcal{F}}^{i},V^{i}).$

The group $G$ has a right action on this product, induced by the right action
on each factor defined by $\rho_{i}:G\to\operatorname{GL}(V^{i})$. Let
$H({\mathcal{F}}^{\bullet},V^{\bullet}){/\\!\\!/}G$ be the GIT quotient. Let
$U$ be an open set where all sheaves ${\mathcal{F}}^{i}$ are locally free. On
each factor $H({\mathcal{F}}^{i}|_{U},V^{i})$ we have an equation
$\alpha_{i}\det(\beta_{i})=1$ as before, cutting out a divisor isomorphic to
the scheme of isomorphisms
$\operatorname{Iso}({\mathcal{F}}^{i}|_{U},V^{i}\otimes{\mathcal{O}}_{U})$.
Finally, we define $\operatorname{Red}_{G}({\mathcal{F}}^{\bullet})$ as the
scheme theoretic closure of the subscheme

$\prod\operatorname{Iso}({\mathcal{F}}^{i}|_{U},V^{i}\otimes{\mathcal{O}}_{U})\cong\prod(H({\mathcal{F}}^{i}|_{U},V^{i}))_{\alpha^{i}\det(\beta^{i})=1}\hookrightarrow
H({\mathcal{F}}^{\bullet},V^{\bullet}){/\\!\\!/}G.$

######  Definition 4.9 ([GHZ21, Defn. 7.1]).

A principal $\rho$-sheaf on $X_{S}$ is a pair
$({\mathcal{F}}^{\bullet},\sigma)$ where

  1. (1)

${\mathcal{F}}^{\bullet}$ is a tuple of torsion-free sheaves on $X_{S}$.

  2. (2)

$\sigma$ is a section of $\operatorname{Red}_{G}({\mathcal{F}}^{\bullet})\to
X_{S}$.

As in the case of a single representation, we denote by
$\operatorname{Bun}_{\rho}(X)$ the stack of $\rho$-sheaves.

For any big open set $U$ where all sheaves ${\mathcal{F}}^{i}$ are locally
free, the section $\sigma$ gives a reduction of structure group, from the
principal
$\operatorname{GL}(V^{1})\times\cdots\times\operatorname{GL}(V^{b})$-bundle
associated to the sum of vector bundles
${\mathcal{F}}^{1}|_{U}\oplus\cdots\oplus{\mathcal{F}}^{b}|_{U}$, to a
principal $G$-bundle $P\to U$ on $U$. Therefore, using the Hartogs-like
Proposition 4.5, we have the following

######  Lemma 4.10.

Giving a principal $\rho$-sheaf is the same thing as giving a triple
$({\mathcal{F}}^{\bullet},P,\psi^{\bullet})$ where

  1. (1)

$P$ is a principal $G$-bundle on a big open set $U$.

  2. (2)

${\mathcal{F}}^{\bullet}=({\mathcal{F}}^{i})_{i=1}^{b}$ is a tuple of $b$
torsion-free sheaves, all of them locally free on $U$.

  3. (3)

$\psi^{\bullet}=(\psi^{i})_{i=1}^{b}$ is a collection of isomorphisms

$\psi^{i}:P(\rho_{i})\cong{\mathcal{F}}^{i}|_{U}\;.$

Finally, we can extend to this case the notion of stability.

######  Definition 4.11.

A principal $\rho$-sheaf $({\mathcal{F}}^{\bullet},\sigma)$ on $X$ is
$\nu$-semistable if for all one-parameter subgroups $\lambda:\mathbb{G}_{m}\to
G$ and all reductions as described in subsection 4.3, we have

$\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{{\mathcal{F}}^{i}_{m}/{\mathcal{F}}^{i}_{m+1}}-\overline{p}_{{\mathcal{F}}^{i}})\cdot\operatorname{rk}({\mathcal{F}}^{i}_{m}/{\mathcal{F}}^{i}_{m+1})\leq
0$

###  4.5. Gieseker-Harder-Narasimhan filtration for $\rho$-sheaves

A unstable torsion-free sheaf has an associated canonical filtration, called
the Harder-Narasimhan filtration, showing that any torsion-free sheaf can be
constructed uniquely as a successive extension of semistable torsion-free
sheaves [HL97, Thm. 1.3.4]. As we mentioned in subsection 3.3, this can be the
slope HN or the Gieseker HN filtration, depending on the stability condition
we are using.

In the case of principal bundles, there is already in [Beh91] a notion of
Harder-Narasimhan reduction if we use Ramanathan’s definition of semistability
(which is the direct generalization of Mumford’s slope condition in [Mum63a]).
This is generalized to higher dimensional varieties in [AAB02]. However, if we
use Gieseker’s notion [Gie77] of semistability considering all terms of the
Hilbert polynomial, the right notion of Harder-Narasimhan filtration had not
been defined until the arrival of the “beyond GIT” theory [HL21]. The notion
of semistability for $\rho$-sheaves (Definitions 4.8 and 4.11) can be obtained
through the techniques of the “beyond GIT” program, in terms of
$\Theta$-filtrations directly from the stack picture. The advantage of this
approach is that it provides a way to define a correct Gieseker Harder-
Narasimhan filtration for moduli problems with a polynomial stability
condition.

Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf and let
$f:\Theta\rightarrow\operatorname{Bun}_{\rho}(X)$ be a $\Theta$-filtration of
$(\mathcal{F},\sigma)$. Recall from subsection 4.3 the definition of a
polynomial numerical invariant which, when considering representations into
tuples $V^{\bullet}$ as in subsection 4.4, is written as ([GHZ21, Section 7]):

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

We can rephrase (Definition 2.18) the condition of $\nu$-semistability by
saying that a $\rho$-sheaf $({\mathcal{F}},\sigma)$ is $\nu$-semistable if and
only if for all $\Theta$-filtrations $f:\Theta\to\operatorname{Bun}_{\rho}(X)$
with $f(1)\simeq({\mathcal{F}},\sigma)$ we have $\nu(f)\leq 0$.

Let $({\mathcal{F}},\sigma)$ be a $\nu$-unstable $\rho$-sheaf. Given that
$\operatorname{Bun}_{\rho}(X)$ is endowed with a $\Theta$-stratification it is
shown in [GHZ21, Prop. 4.31] that there is a unique111It is unique up to
multiplication of all the indexes $m\in\mathbb{Z}$ by a scalar. This
corresponds to reparametrization of the one-parameter subgroup by $z\mapsto
z^{a}$ $\Theta$-filtration $f$ giving the maximum value of $\nu(f)$. This is a
$\nu$-HN filtration of the point $(\mathcal{F},\sigma)$ in the stack
$\operatorname{Bun}_{\rho}(X)$ as in Definition 2.29.

Associated to this maximal $\Theta$-filtration $f$, we have a reduction of
structure group to a parabolic subgroup $P_{\lambda_{1}}\subset G$ for some
$\lambda_{1}:\mathbb{G}_{m}\to G$, and a $\mathbb{Z}$-filtration
$\left({\mathcal{F}}_{m}\right)_{m\in\mathbb{Z}}$ of ${\mathcal{F}}$
satisfying the compatibility conditions described in [GHZ21, Section 3.2]. By
the same reasons explained in subsection 3.3 this is called the leading term
HN filtration, which is the version for $\rho$-sheaves of the filtration
described there for torsion-free coherent sheaves. In order to obtain a
filtration of Gieseker type, meaning a filtration being able to compare all
terms of Hilbert polynomials and whose graded object is semistable, we need to
reproduce the algorithm described in that subsection 3.3 translated to the
stack $\operatorname{Bun}_{\rho}(X)$.

Using the projection $P_{\lambda_{1}}\to L_{\lambda_{1}}$ from the parabolic
group to its Levi factor, this reduction induces a principal
$\rho_{\lambda_{1}}$-sheaf with structure group $L_{\lambda_{1}}$, the Levi
factor of $P_{\lambda_{1}}$. The representation of this Levi is of the form

$\rho_{\lambda_{1}}:L_{\lambda_{1}}\to\operatorname{GL}(V^{1})\times\cdots\times\operatorname{GL}(V^{b}),$

and the tuple of sheaves ${\mathcal{F}}^{\prime}{}^{\bullet}$ is given by the
successive quotients of the filtration
$\left({\mathcal{F}}_{m}\right)_{m\in\mathbb{Z}}$, i.e.,
${\mathcal{F}}^{\prime}{}^{i}={\mathcal{F}}_{i}/{\mathcal{F}}_{i+1}$. However,
this Levi might not be semistable yet, therefore we need to further take the
leading term HN filtration of this $\rho_{\lambda_{1}}$-sheaf. In that case,
we obtain a new reduction to a parabolic $P_{\lambda_{2}}\subset
L_{\lambda_{1}}\subset G$ and repeat this process until we obtain a reduction
whose associated Levi sheaf is semistable. The final associated filtration
will be called the Gieseker HN filtration (see [GHZ21, Section 8.3].

###  4.6. Example of a Gieseker Harder-Narasimhan filtration

We illustrate the process of producing the Gieseker HN filtration with a
concrete numerical example.

Let $G={\rm Sp}(6)$ with $\rho:{\rm Sp}(6)\to\operatorname{GL}(6)$ the
standard representation. Let $X=\mathbb{P}^{3}$. Let $p$ and $C$ be,
respectively, a given fixed point and a fixed line in $\mathbb{P}^{3}$. Let
$\mathcal{I}_{p}$ and $\mathcal{I}_{C}$ be their respective ideal sheaves. Set

${\mathcal{F}}=\mathcal{I}_{C}\oplus(\mathcal{I}_{p}\oplus\mathcal{I}_{p})\oplus({\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X})\oplus{\mathcal{O}}_{X}$

and let $\varphi:{\mathcal{F}}\otimes{\mathcal{F}}\to{\mathcal{O}}_{X}$ be the
symplectic form defined by the matrix

$\varphi=\left(\begin{array}[]{ccccccc}0&\\!\cdots\\!&&&&1\\\ \vdots&&&&1&\\\
&&&1&&\\\ &&-1&&&\\\ &-1&&&&\vdots\\\ -1&&&&\\!\cdots\\!&0\\\
\end{array}\right)$

The Hilbert polynomials are

$\displaystyle P_{\mathcal{O}}(n)$
$\displaystyle=\frac{1}{6}(n^{3}+6n^{2}+10n+3)\,,$ $\displaystyle
P_{\mathcal{I}_{p}}(n)$ $\displaystyle=\frac{1}{6}(n^{3}+6n^{2}+10n-3)\,,$
$\displaystyle P_{\mathcal{I}_{C}}(n)$
$\displaystyle=\frac{1}{6}(n^{3}+6n^{2}-\;6n-3)\,.$

We will consider one-parameter subgroups of ${\rm Sp}(6)$ of the form

$f(t)=\operatorname{diag}\operatorname{(}t^{a},t^{b},t^{c},t^{-c},t^{-b},t^{-a})$

with $a\geq b\geq c\geq 0$. The associated filtration ${\mathcal{F}}_{m}$
satisfies $\mathcal{F}_{m}^{\perp}=\mathcal{F}_{-m+1}$, where the orthogonal
of a subsheaf $\mathcal{F}_{i}\subset\mathcal{F}$ is given by

$\mathcal{F}_{i}{}^{\perp}:=\ker\big{(}\mathcal{F}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\mathcal{F}^{\vee}\longrightarrow\mathcal{F}_{i}{}^{\vee}\big{)}.$

The bilinear form gives a reduction of structure group of
${\mathcal{F}}|_{\mathbb{P}^{3}\setminus{(C\cup p)}}$ to ${\rm Sp}(6)$, so the
data $({\mathcal{F}},\varphi)$ defines a principal $\rho$-sheaf. It can be
checked that it is $\nu$-unstable (Definition 4.8), and the one-parameter
subgroup associated to the leading term HN filtration maximizing $\nu$ is
$\lambda_{1}=\operatorname{diag}(t,1,1,1,1,t^{-1})$. The filtration of sheaves
$\left({}^{(1)}\mathcal{F}_{m}\right)_{m\in\mathbb{Z}}$ associated to
$\lambda_{1}$ is

$\overbrace{\mathcal{F}}^{\mathcal{F}_{-1}}\quad\supset\quad\overbrace{\mathcal{I}_{p}\oplus\mathcal{I}_{p}\oplus\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}_{0}}\quad\supset\quad\overbrace{\mathcal{O}}^{\mathcal{F}_{1}}$

We have the associated Levi $L_{\lambda_{1}}=\operatorname{GL}(1)\times{\rm
Sp}(4)$ and the representation $\rho_{\lambda_{1}}$, which is given by

$\begin{array}[]{rccc}\rho_{\lambda_{1}}:&\operatorname{GL}(1)\times{\rm
Sp}(4)&\longrightarrow&\operatorname{GL}(1)\times\operatorname{GL}(4)\times\operatorname{GL}(1)\\\
&(t,g)&\mapsto&(t,g,t^{-1})\\\ \end{array}.$

The $1^{st}$ associated Levi graded sheaf ${\rm gr}_{1}(\mathcal{F})$ is

$\overbrace{\mathcal{I}_{L}}^{\mathcal{F}^{-1}:=\mathcal{F}_{-1}/\mathcal{F}_{0}}\quad\oplus\quad\overbrace{\mathcal{I}_{p}\oplus\mathcal{I}_{p}\oplus\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}^{0}:=\mathcal{F}_{0}/\mathcal{F}_{-1}}\quad\oplus\quad\overbrace{\mathcal{O}}^{\mathcal{F}^{1}:=\mathcal{F}_{1}}.$

The principal $\rho_{1}$-sheaf defined by this is not $\nu$-semistable, then
we take its leading term HN filtration having associated one-parameter
subgroup

$\begin{array}[]{rccc}\lambda_{2}:&\mathbb{G}_{m}&\longrightarrow&\operatorname{GL}(1)\times{\rm
Sp(4)}\\\ &t&\mapsto&\big{(}1,(t,t,t^{-1},t^{-1})\big{)}\\\ \end{array}$

The corresponding $2^{nd}$ filtration
$\left({}^{(2)}\mathcal{F}_{(m_{1},m_{2})}\right)_{(m_{1},m_{2})\in\mathbb{Z}^{2}}$
is

(4.4)
${\mathcal{F}}\quad\supset\quad(\mathcal{I}_{p}\oplus\mathcal{I}_{p})\oplus({\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X})\oplus{\mathcal{O}}_{X}\quad\supset\quad({\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X})\oplus{\mathcal{O}}_{X}\quad\supset\quad{\mathcal{O}}_{X}$

whose $2^{nd}$ associated Levi graded sheaf ${\rm gr}_{2}(\mathcal{F})$ is

$\mathcal{I}_{C}\quad\oplus\quad(\mathcal{I}_{p}\oplus\mathcal{I}_{p})\quad\oplus\quad({\mathcal{O}}_{X}\oplus{\mathcal{O}}_{X})\quad\oplus\quad{\mathcal{O}}_{X}$

The second Levi factor $L_{\lambda_{2}}$ is
$\operatorname{GL}(1)\times\operatorname{GL}(2)$ (note that the embedding
$\operatorname{GL}(2)\subset{\rm Sp}(4)$ is given by $A\mapsto(A,A^{-1})$) and
the principal $\rho_{2}$-sheaf is $\nu$-semistable. Since this is semistable,
the process stops, and the Gieseker Harder-Narasimhan reduction has (4.4) as
associated filtration.

Acknowledgements: We thank Andrés Ibáñez Núñez, Sveta Makarova and Nitin
Nitsure, as well as the anonymous referee, for helpful comments on the
manuscript. We also thank the VBAC community lead by Peter Newstead and the
editors of this contributed volume for their continuous work.

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

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