# Walls and asymptotics for Bridgeland stability conditions on threefolds

Marcos Jardim  IMECC - UNICAMP
Departamento de Matemática
Rua Sérgio Buarque de Holanda, 651
13083-970 Campinas-SP, Brazil
ORCiD: 0000-0002-8419-7044<EMAIL_ADDRESS>and  Antony Maciocia
University of Edinburgh
School of Mathematics
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
ORCiD: 0000-0003-1422-3187<EMAIL_ADDRESS>
###### Abstract.

We consider Bridgeland stability conditions for three-folds conjectured by
Bayer–Macrì–Toda in the case of Picard rank one. We study the differential
geometry of numerical walls, characterizing when they are bounded, discussing
possible intersections, and showing that they are essentially regular. Next,
we prove that walls within a certain region of the upper half plane that
parameterized geometric stability conditions must always intersect the curve
given by the vanishing of the slope function and, for a fixed value of $s$,
have a maximum turning point there. We then use these facts to prove that
Gieseker semistability is equivalent to a strong form of asymptotic
semistability along a class of paths in the upper half plane, and show how to
find large families of walls. We illustrate how to compute all of the walls
and describe the Bridgeland moduli spaces for the Chern character $(2,0,-1,0)$
on complex projective 3-space in a suitable region of the upper half plane.

###### 2010 Mathematics Subject Classification:

14F05 (Primary) 14D20, 14J30, 18E30, 53A05 (Secondary)

##  1\. Introduction

Bridgeland’s notion of stability conditions on triangulated categories,
introduced in [9] and [10], provides a new set of tools to study moduli spaces
of sheaves on smooth projective varieties. Such tools have been successfully
applied by many authors first to the study of sheaves on surfaces, for
example, [1, 2, 4, 5, 13, 14, 15, 21, 39], and more recently on three-folds
(especially ${\mathbb{P}^{3}}$), see for instance [16, 25, 34, 36]. One way to
study moduli spaces of sheaves using Bridgeland stability spaces is to
restrict attention to the so called geometric stability conditions
parameterized by (a subset of) the upper half plane $\mathbb{H}$. Once we know
that the moduli space of Bridgeland stable objects is asymptotically given by
the Gieseker semistable moduli space along an unbounded path we can try to
locate all the points where the moduli space changes along this path (these
isolated points are called walls) and compute the change to the moduli space.
Eventually, we might reach a point where the Bridgeland space is empty and
then we can reverse our steps to reconstruct the Gieseker moduli space.

For surfaces, this is a fairly well understood process. In that case, it is
known that the geometric stability space is non-empty, that there only
finitely many walls in $\mathbb{H}$ which are nested semi-circles centered
along the horizontal axis, and that Bridgeland stability is asymptotic to
(twisted) Gieseker stability. Furthermore, there is an effective algorithm to
find all such walls for a given Chern character, and then we can carry out the
process above to recover the moduli space of semistable sheaves. One approach
to finding walls in this case is to observe that every wall for a given Chern
character $v$ intersects a special curve which we will denote $\Theta_{v}$ in
this paper, given by the vanishing locus of the slope function
$\nu_{\alpha,\beta}(v)$, and we can then restrict attention to finding walls
along $\Theta_{v}$.

The whole process becomes much more complicated for three-folds. We can still
use the 2-dimensional construction, but it does not produce full stability
conditions and it is unable to detect sufficient features of the Gieseker
moduli spaces, since the latter does not coincide, in general, with the
asymptotic moduli space.

The first step to improve this was made possible by a number of results
guaranteeing the existence of Bridgeland stability conditions on the derived
category of sheaves on different types of three-folds, based on the pioneering
work of Bayer, Macrì and Toda [7]. Their idea is to start with the surface
case and tilt again. This provides a full stability condition, and the family
of moduli spaces is considerably more refined than the one provided by the
first tilt. Even though there is no general result which shows that their
construction works for all smooth three-folds, it is known to work for a wide
variety of relevant examples: ${\mathbb{P}^{3}}$ [23], smooth quadric three-
folds [33], abelian three-folds [6, 22], Fano three-folds with Picard rank one
[18], more general Fano threefolds [8, 30] and smooth quintic three-folds
[19]. More precisely, the geometric stability conditions constructed by Bayer,
Macrì and Toda via the generalized Bogomolov–Gieseker inequality proposed in
[7] depends on three real parameters
$(\alpha,\beta,s)\in\mathbb{R}^{+}\times\mathbb{R}\times\mathbb{R}^{+}$. For
each of these we have an abelian category $\mathcal{A}^{\alpha,\beta}$ and a
slope function $\lambda_{\alpha,\beta,s}$ which allows us to test the
stability of objects of $\mathcal{A}^{\alpha,\beta}$. There are, however,
known counterexamples (see [35] and [26]) where the generalized Bogomolov
inequality fails.

The goal of this paper is to advance on the other two stages of the process
outlined above, namely the understanding of the structure of walls, and of
asymptotic stability. We only consider the case where $X$ is a smooth
projective three-fold of Picard rank $1$ over an algebraically closed field of
characteristic zero. This means that we can view our Chern classes (and their
twists) as purely numerical vectors of the form $v=(v_{0},v_{1},v_{2},v_{3})$.

In order to study walls, we start by providing a uniform way to define the
slope functions and their differences in terms of skew-symmetric functions. We
go on to consider a number of general properties of numerical $\lambda$-walls
(defined as the locus where two $\lambda$-slopes are equal) in Section 4.1.
They are, in general, quartic curves, possibly unbounded and not connected.

In our first main result, we find a simple characterization of those numerical
$\lambda$-walls which are bounded, and show that unbounded walls satisfy a
version of Bertram’s Nested Wall Theorem. Given numerical Chern characters
$v$, $u$ and $u^{\prime}$, we define
$\delta_{01}(u,v):=u_{0}v_{1}-u_{1}v_{0}$, and an equivalence relation
$u\sim_{v}u^{\prime}$ which is essentially that the $\lambda$-walls for $v$
corresponding to $u$ and $u^{\prime}$ are the same, see (34) for a precise
definition. We also remark that when $v$ is a numerical Chern character
satisfying the Bogomolov inequality $v_{1}^{2}-2v_{0}v_{2}\geq 0$ and
$v_{0}\neq 0$, the curve $\Theta_{v}$ allows us to divide the upper half plane
$\mathbb{H}$ into four regions (see Figure 1 for an example and Section 3 for
details).

######  Main Theorem 1.

Suppose $v_{0}\neq 0$ and $u\not\sim_{v}u^{\prime}$.

  1. (1)

The numerical $\lambda$-wall for $v$ corresponding to $u$ is bounded if and
only if $\delta_{01}(u,v)\neq 0$.

  2. (2)

If $\delta_{01}(u,v)=0=\delta_{01}(u^{\prime},v)$ then the numerical
$\lambda$-walls corresponding to $u$ and $u^{\prime}$ do not intersect.

  3. (3)

If $\delta_{01}(u,v)\neq 0$ and $\operatorname{ch}_{\leq
2}(u)=\operatorname{ch}_{\leq 2}(u^{\prime})$, then the numerical
$\lambda$-walls for $v$ corresponding to $u$ and $u^{\prime}$ only intersect
on $\Theta_{v}$.

  4. (4)

An unbounded numerical $\lambda$-wall for $v$ does not intersect $\Theta_{v}$,
and its unbounded connected components are contained in $R^{0}_{v}$.

The different parts of Main Theorem 1 are proved in various results contained
in Section 4.

Figure 1. The four regions of the plane as defined by the hyperbola
$\Theta_{v}$ and the vertical line $\\{\beta=\mu(v)\\}$ when $v$ is a
numerical Chern character satisfying the Bogomolov inequality
$v_{1}^{2}-2v_{0}v_{2}\geq 0$ and $v_{0}\neq 0$. In this picture, we used
$\operatorname{ch}_{\leq 2}(v)=(2,-1,-5/2)$.

The third stage, determining the asymptotics, is not well known. In [7,
Section 6], it is shown that the large volume limit as $\alpha\to\infty$ for
$(\alpha,\beta)\in\mathbb{H}$ gives a polynomial stability condition, but the
converse and other directions were not considered. However, unlike the two-
dimensional case, we cannot assume that the walls are bounded in all
directions. In fact, it is easy to check that, as $s\to 0$, the walls are
unbounded and it is theoretically possible that the number of walls is
infinite.

This means that the large volume limit is more subtle than for surfaces. We
need, therefore, to be more careful about what we mean by asymptotic
stability, which we define precisely in Definition 7.1. We use a strong form
of such asymptotic stability which effectively includes finiteness of the
number of walls for a given object. We also need to be careful to specify the
curve along which we are considering the asymptotics. To this end we introduce
the notion of unbounded $\Theta^{\pm}$ curve which is essentially a curve
which is asymptotically either to the left or to the right of all $\Theta$
curves, see Definition 5.9.

To help set this up, we also consider what would happen for surfaces in
Section 5. In our case, we look at so called $\nu$-stability for three-folds,
which mimics stability for surfaces given by the first tilt on the category of
coherent sheaves, by reproving results about the large volume limit without
the assumption that the walls are bounded.

In order to describe the asymptotics, we need to understand how the stability
of an object varies along curves. For $\nu$-stability, it turns out that
stable objects can only be destabilized once along inward moving curves (which
cross $\nu$-walls only once). It turns out that this also holds for
$\lambda$-stability outside the $\nu$-wall. See Theorem 7.8 for the details.
Accomplishing such task requires an understanding of the geometry of
$\lambda$-walls. For the $\nu$-walls, this was simple because they were
circles and the key property is that they cross the $\Theta_{v}$ curve at
their maximum. To understand the similar properties of $\lambda$-walls we need
to understand their differential properties in a similar way. We do this is
Section 6. Unlike $\nu$-walls, $\lambda$-walls need not be regular and we
carefully study when regularity fails. When $s=1/3$ it is straightforward to
prove that the walls are regular excpt if they happen to cross a very special
point (on $\Gamma_{u,1/3}$ and its $\nu$-wall) in which case there is a cusp.
For other values of $s$ it is much harder. It turns out to be easier to study
the differential properties of the 2-dimensional wall where we allow $s$ to
vary which we call $\Sigma_{u,v}$ regarded as a real algebraic quartic surface
in $\mathbb{R}^{3}$. We show that $\Sigma_{u,v}$ is regular except at some
exceptional points and for exceptional $u$ and $v$ (see Theorem 6.7 for the
details).

Moreover, there is again a special curve, here denoted $\Gamma_{v,s}$, which
is defined as the vanishing locus of the slope function
$\lambda_{\alpha,\beta,s}(v)$. When $s=1/3$, a numerical $\lambda$-wall for
$v$ crosses $\Gamma_{v,1/3}$ at is maximum point (just as a $\nu$-wall crosses
$\Theta_{v}$ at its maximum), and the associated $\nu$-wall at its minimum
point. This imposes large constraints on the possible numerical
$\lambda$-walls when $s=1/3$. We also show that for $s\geq 1/3$ any wall
existing for one value of $s$ must also exist for all $s$. When $s<1/3$ we
show that any wall existing for $s$ exists for all value less than that value
of $s$.

The key conclusion is the following, see also Theorem 6.17; the proof uses key
differential geometric information about $\Sigma_{u,v}$ such as its Gauss and
mean curvatures.

######  Main Theorem 2.

Suppose a real numerical Chern character $v$ satisfies the Bogomolov
inequality and $v_{0}\neq 0$. Any connected bounded component of a numerical
$\lambda$-wall in $R^{-}_{v}$ for some $s\geq 1/3$ intersects
$\Gamma^{-}_{v,s}$.

Although the same statement is not true of unbounded walls, we can describe
the explicit conditions $u$ must satisfy so that the wall corresponding to $u$
intersects $\Gamma_{v,s}$.

We are then finally in position to prove in Section 7 that strong asymptotic
stability is equivalent to Gieseker stability. Our results can be summarized
as follows.

######  Main Theorem 3.

Let $v$ be a numerical Chern character with $v_{0}\neq 0$ and satisfying the
Bogomolov inequality; fix $s\geq 1/3$.

  1. (1)

If $\gamma$ be an unbounded $\Theta^{-}$-curve, then an object $E\in D^{b}(X)$
is asymptotically $\lambda_{\alpha,\beta,s}$-(semi)stable along $\gamma$ if
and only if $E$ is Gieseker (semi)stable sheaf.

  2. (2)

If $\gamma$ be an unbounded $\Theta^{+}$-curve, then an object $E\in D^{b}(X)$
is asymptotically $\lambda_{\alpha,\beta,s}$-(semi)stable along $\gamma$ if
and only if $E^{\vee}$ is Gieseker (semi)stable sheaf.

Duals of Gieseker semistable sheaves can be described via a technical lemma of
independent interest which characterizes duals of torsion free sheaves on
threefolds (see Proposition 2.15 for the details). We also emphasize that more
is true for the special curve $\Gamma_{v,s}$, which is an example of an
unbounded $\Theta^{-}$-curve: the first part of the previous statement holds
for every $s>0$.

The study of the differential geometry of $\lambda$-walls is also useful to
help locate them. Here again there is a complication. For $\nu$-walls (or
walls for surfaces), a point on an actual $\nu$-wall corresponds to an actual
destabilizing sub-object $F$ of an object $E$ which is $\nu$-stable on one
side of the wall and $\nu$-unstable on the other. This is then always the case
at all points along the numerical $\nu$-wall. In other words, if a portion of
a numerical $\nu$-wall is an actual $\nu$-wall then the whole $\nu$-wall is an
actual $\nu$-wall. It follows that to locate actual $\nu$-walls it suffices to
look along, say, a vertical line from the limiting centre or along
$\Theta_{v}$; we do this in two examples, one in 8.1 and the other in Lemma
8.5.

The situation for $\lambda$-walls is quite different. While we have proved
that numerical $\lambda$-walls to the left of $\Theta_{v}$ must intersect
$\Gamma_{v,s}$, that need not be true of actual $\lambda$-walls. However, the
largest actual $\lambda$-wall (if it exists) must cross $\Gamma_{v,s}$, and
this allows us to find $\lambda$-walls by working along $\Gamma_{v,s}$ from
infinity.

We work out a complete example for the case of the Chern character
$v=(2,0,-1,0)$ on ${\mathbb{P}^{3}}$. This is done for the region to the left
of $\Theta_{v}$ in Proposition 8.10. The key step is to first find what we
call _pseudo-walls_. These are given by destabilizing objects which satisfy
the Bogomolov and generalized Bogomolov inequalities. This approach to finding
$\lambda$-walls complements the method of Schmidt’s [34, Theorem 6.1] which is
to observe that, essentially, the $\lambda$-walls crossing $\Theta_{v}$ must
also cross their associated $\nu$-wall at the same point.

We close the paper with a number of examples for the case of
${\mathbb{P}^{3}}$ in Section 8. First we consider the case of the ideal sheaf
of a line; it illustrates one of two typical situations in which the
$\Gamma_{v,s}$ curve intersects $\Theta_{v}$. In this case, we show that there
must exist a vanishing $\lambda$-wall containing the intersection point (see
Theorem 4.22) and we illustrate this by constructing such a wall for this
example. We also consider the ideal sheaf of a point for which $\Gamma_{v,s}$
and $\Theta_{v}$ do not cross and also provides some additional stability
information for our final example which is the null correlation sheaves on
${\mathbb{P}^{3}}$. In this case, we show there are no $\nu$-walls. This
allows us to use a helpful additional numerical condition for the walls given
in Lemma 8.9 which arises from a general result Proposition 4.3 which gives a
simple numerical condition for when a $\lambda$-wall intersects a $\nu$-wall.
These cases are also considered from a different point of view using
Bridgeland stability in [37].

Throughout the paper we make very full use of the triangulated structure of
the derived category of coherent sheaves and especially the octahedral axiom.
We use a higher dimensional variant of the octahedral axiom which is key to
understanding how objects vary in the heart $\mathcal{A}^{\alpha,\beta}$ along
paths in the upper half plane. We provide a description of this in (2.3) when
describing the objects of $\mathcal{A}^{\alpha,\beta}$ due to Paul Balmer. We
also provide some general technical results about the duals of objects in the
derived category which are hard to find in the literature (see Section 2.4).

### Acknowledgements

MJ is partially supported by the CNPq grant number 302889/2018-3 and the
FAPESP Thematic Project 2018/21391-1; he thanks the University of Edinburgh
for a pleasant and productive visit in 2017, which was supported by the FAPESP
grant number 2016/03759-6. We thank Victor Pretti for his comments on
preliminary versions of this paper, and Arend Bayer for helpful conversations.
We have made extensive use of the software Maxima, Geogebra and Mathematica
for calculations and to produce the various pictures present in this text.

##  2\. Background material and notation

Let $X$ be an irreducible, non-singular projective variety over an
algebraically closed field of characteristic zero of dimension three with
$\operatorname{{Pic}}(X)={\mathbb{Z}}$. Fix an ample generator $L$ of
$\operatorname{{Pic}}(X)$. Our assumptions mean that each object $A\in D^{\rm
b}(X)$ has a well defined numerical Chern character

$\operatorname{ch}(A):=(\operatorname{ch}_{0}(A),\operatorname{ch}_{1}(A)\cdot
L^{2},\operatorname{ch}_{2}(A)\cdot
L,\operatorname{ch}_{3}(A))\in{\mathbb{Z}}\times{\mathbb{Z}}\times\frac{1}{2}{\mathbb{Z}}\times\frac{1}{6}{\mathbb{Z}}.$

Abusing notation we will simply write $\operatorname{ch}_{i}(A)$ for
$\operatorname{ch}_{i}(A)\cdot L^{3-i}$. We will refer to an element of
$\mathbb{R}^{4}=\mathbb{R}\otimes K_{\text{num}}(X)$ as a _real numerical
Chern Character_ and an element $v$ of
${\mathbb{Z}}\times{\mathbb{Z}}\times\frac{1}{2}{\mathbb{Z}}\times\frac{1}{6}{\mathbb{Z}}$
as a Chern character when there is an object $A\in D^{\rm b}(X)$ such that
$v=\operatorname{ch}(A)$. We write the components as
$v=(v_{0},v_{1},v_{2},v_{3})$ corresponding to the Chern characters of objects
so that the underlying real Chern character $v$ which is the numerical Chern
character $\operatorname{ch}(A)$ of an object of $D^{\rm b}(X)$ satisfies
$v_{i}=\operatorname{ch}_{i}(A)$.

Given $\beta\in\mathbb{R}$, recall the definition of the _twisted Chern
character_ $\operatorname{ch}^{\beta}(A):=exp(-\beta
L)\cdot\operatorname{ch}(A)=(\operatorname{ch}_{0}^{\beta}(A),\operatorname{ch}_{1}^{\beta}(A),\operatorname{ch}_{2}^{\beta}(A),\operatorname{ch}_{3}^{\beta}(A))$.
So

$\displaystyle\operatorname{ch}_{0}^{\beta}(A)$
$\displaystyle:=\operatorname{ch}_{0}(A);$
$\displaystyle\operatorname{ch}_{1}^{\beta}(A)$
$\displaystyle:=\operatorname{ch}_{1}(A)-\beta\operatorname{ch}_{0}(A);$
$\displaystyle\operatorname{ch}_{2}^{\beta}(A)$
$\displaystyle:=\operatorname{ch}_{2}(A)-\beta\operatorname{ch}_{1}(A)+\frac{1}{2}\beta^{2}\operatorname{ch}_{0}(A);$
$\displaystyle\operatorname{ch}_{3}^{\beta}(A)$
$\displaystyle:=\operatorname{ch}_{3}(A)-\beta\operatorname{ch}_{2}(A)+\frac{1}{2}\beta^{2}\operatorname{ch}_{1}(A)-\frac{1}{6}\beta^{3}\operatorname{ch}_{0}(A).$

Recall that the $\mu$-slope of a coherent sheaf $E\in\mathcal{C}oh(X)$ is
defined as follows:

$\mu(E):=\begin{cases}\operatorname{ch}_{1}(E)/\operatorname{ch}_{0}(E)&\textrm{if
}E\textrm{ is torsion free},\\\ +\infty,&\textrm{otherwise}.\end{cases}$

In addition, we also define

$\mu^{+}(E):=\max\\{\mu(F)\mid\text{every non-zero subsheaf }F\hookrightarrow
E\\}\quad\text{and}$ $\mu^{-}(E):=\min\\{\mu(G)\mid\text{every non-zero
quotient }E\twoheadrightarrow G\\}.$

As usual, a $E$ is said to be $\mu$-(semi)stable if every subsheaf
$F\hookrightarrow E$ satisfies $\mu(F)<(\leq)\leavevmode\nobreak\ \mu(E)$. So
that $E$ is $\mu$-semistable if and only if $\mu^{+}(E)=\mu(E)$, or
equivalently $\mu^{-}(E)=\mu(E)$.

###  2.1. $\nu$-stability

Given $\beta\in\mathbb{R}$, consider the following torsion pair on
$\mathcal{C}oh(X)$

$\mathcal{T}_{\beta}:=\\{E\in\mathcal{C}oh(X)\leavevmode\nobreak\
|\leavevmode\nobreak\ \textrm{every non-zero quotient }E\twoheadrightarrow
G\textrm{ satisfies }\mu(G)>\beta\\},\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm and}$ $\mathcal{F}_{\beta}:=\\{E\in\mathcal{C}oh(X)\leavevmode\nobreak\
|\leavevmode\nobreak\ \textrm{every non-zero subsheaf }F\hookrightarrow
E\textrm{ satisfies }\mu(F)\leq\beta\\}.$

Tilting on $(\mathcal{F}_{\beta},\mathcal{T}_{\beta})$, one obtains an abelian
subcategory
$\mathcal{B}^{\beta}(X):=\langle\mathcal{F}_{\beta}[1],\mathcal{T}_{\beta}\rangle$
of $D^{\rm b}(X)$, which is the heart of a t-structure on $D^{\rm b}(X)$.

For $B\in D^{\rm b}(X)$, let ${\mathcal{H}}^{p}(B)$ denote cohomology with
respect to $\mathcal{C}oh(X)$. Observe that the objects of
$\mathcal{B}^{\beta}(X)$ are those $B\in D^{\rm b}(X)$ such that:

  * •

${\mathcal{H}}^{p}(B)=0$ for $p\neq-1,0$,

  * •

${\mathcal{H}}^{-1}(B)\in\mathcal{F}_{\beta}$, and

  * •

${\mathcal{H}}^{0}(B)\in\mathcal{T}_{\beta}$.

In particular, from the definition of $\mathcal{F}_{\beta}$,
${\mathcal{H}}^{-1}(B)$ must be a torsion free sheaf.

Introducing a new parameter $\alpha\in\mathbb{R}^{+}$, one considers a group
homomorphism

$Z_{\alpha,\beta}^{\rm tilt}:K_{\rm num}(X)\to\mathbb{C},$

called a _central charge_ , given by

(1) $Z_{\alpha,\beta}^{\rm
tilt}(B):=-\left(\operatorname{ch}_{2}^{\beta}(B)-\frac{1}{2}\alpha^{2}\operatorname{ch}_{0}(B)\right)+\sqrt{-1}\operatorname{ch}_{1}^{\beta}(B),\quad\text{for
}B\in\mathcal{B}^{\beta}(X),$

whose corresponding slope function is

(2)
$\nu_{\alpha,\beta}(B):=\begin{cases}\dfrac{\operatorname{ch}_{2}^{\beta}(B)-\alpha^{2}\operatorname{ch}_{0}(B)/2}{\operatorname{ch}_{1}^{\beta}(B)},&\textrm{
if }\operatorname{ch}_{1}^{\beta}(B)\neq 0,\\\ +\infty,&\text{if
}\operatorname{ch}_{1}^{\beta}(B)=0.\end{cases}$

In addition, we also define

$\nu_{\alpha,\beta}^{+}(B):=\max\\{\nu_{\alpha,\beta}(F)\leavevmode\nobreak\
|\leavevmode\nobreak\ F\hookrightarrow B\textrm{ in }\mathcal{B}^{\beta},\
F\neq 0\\}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}$
$\nu_{\alpha,\beta}^{-}(B):=\min\\{\nu_{\alpha,\beta}(G)\leavevmode\nobreak\
|\leavevmode\nobreak\ B\twoheadrightarrow G\textrm{ in }\mathcal{B}^{\beta},\
G\neq 0\\}.$

An object $B\in D^{\rm b}(X)$ is said to be $\nu_{\alpha,\beta}$-(semi)stable
if $B\in\mathcal{B}^{\beta}(X)$ and every sub-object $F\hookrightarrow B$
within $\mathcal{B}^{\beta}(X)$ satisfies
$\nu_{\alpha,\beta}(F)<(\leq)\leavevmode\nobreak\ \nu_{\alpha,\beta}(B)$. Note
that $E$ is $\nu_{\alpha,\beta}$-semistable if and only if
$\nu_{\alpha,\beta}^{+}(E)=\leavevmode\nobreak\ \nu_{\alpha,\beta}(E)$,
equivalently, $\nu_{\alpha,\beta}^{-}(E)=\leavevmode\nobreak\
\nu_{\alpha,\beta}(E)$.

Every $\mu$-semistable sheaf and every $\nu_{\alpha,\beta}$-semistable object
$B\in\mathcal{B}^{\beta}(X)$ satisfies the usual Bogomolov inequality, which
in our situation is purely numerical, see [7, Corollary 7.3.2]:

(3) $Q^{\rm
tilt}(B):=\operatorname{ch}_{1}(B)^{2}-2\operatorname{ch}_{0}(B)\operatorname{ch}_{2}(B)\geq
0.$

In addition, for certain choices of $X$, every $\nu_{\alpha,\beta}$-semistable
object $B\in\mathcal{B}^{\beta}(X)$ also satisfies the following _generalized
Bogomolov inequality_

(4) $\begin{split}Q_{\alpha,\beta}(B)&=\alpha^{2}Q^{\rm
tilt}(B)+4(\operatorname{ch}^{\beta}_{2}(B))^{2}-6\operatorname{ch}^{\beta}_{1}(B)\operatorname{ch}^{\beta}_{3}(B)\\\
&=Q^{\rm
tilt}(B)(\alpha^{2}+\beta^{2})+\left(6\operatorname{ch}_{0}(B)\operatorname{ch}_{3}(B)-2\operatorname{ch}_{1}(B)\operatorname{ch}_{2}(B)\right)\beta\\\
&\qquad\qquad{}+4\operatorname{ch}_{2}(B)^{2}-6\operatorname{ch}_{1}(B)\operatorname{ch}_{3}(B)\geq
0,\end{split}$

originally proposed in [7, Conjecture 1.3.1]. This inequality was proved to
hold for all Fano and abelian threefolds with Picard rank 1, see [18] and [6,
22], respectively, and for the quintic threefold [19]. We assume from now on
that $X$ is such that the generalized Bogomolov inequality (4) holds for all
$\nu_{\alpha,\beta}$-semistable objects.

Let $\mathbb{H}:=\mathbb{R}^{+}\times\mathbb{R}$, thought as the upper half
plane, with coordinates denoted by $(\alpha,\beta)$. We will want to consider
the slope function as a function of $\alpha$ and $\beta$ and to this end it is
convenient to define the following function on $(\alpha,\beta)\in\mathbb{H}$:

(5) $\rho_{v}(\alpha,\beta)=v_{2}-v_{1}\beta+v_{0}(\beta^{2}-\alpha^{2})/2,$

which coincides with the numerator of $\nu_{\alpha,\beta}(B)$ when
$v=\operatorname{ch}(B)$. To simplify the notation, we define
$\rho_{B}(\alpha,\beta):=\rho_{\operatorname{ch}(B)}(\alpha,\beta)$ for
objects $B\in D^{\rm b}(X)$.

Note that the pair $(\mathcal{B}^{\beta}(X),Z^{\rm tilt}_{\alpha,\beta})$ is a
_weak stability condition_ in $D^{\rm b}(X)$, in the sense of [38, Section 2],
for all pairs $(\alpha,\beta)\in\mathbb{R}^{+}\times\mathbb{R}$. In practical
terms, this gives:

######  Proposition 2.1.

Fix $\beta\in\mathbb{R}$. If $B\in\mathcal{B}^{\beta}$, then
$\operatorname{ch}^{\beta}_{1}(B)\geq 0$, with equality only if
$\rho_{B}(\alpha,\beta)\geq 0$ for all $\alpha>0$.

###  2.2. $\lambda$-stability

The next step is to consider the following torsion pair on
$\mathcal{B}^{\beta}(X)$

$\mathcal{T}_{\alpha,\beta}:=\\{E\in\mathcal{B}^{\beta}(X)\mid\text{every non-
zero quotient }E\twoheadrightarrow G\text{ satisfies
}\nu_{\alpha,\beta}(G)>0\\},\text{ and}$
$\mathcal{F}_{\alpha,\beta}:=\\{E\in\mathcal{B}^{\beta}(X)\mid\text{every non-
zero sub-object }F\hookrightarrow E\text{ satisfies }\nu_{\alpha,\beta}(F)\leq
0\\}.$

Tilting on $(\mathcal{F}_{\alpha,\beta},\mathcal{T}_{\alpha,\beta})$, one
obtains a new abelian subcategory
$\mathcal{A}^{\alpha,\beta}(X):=\langle\mathcal{F}_{\alpha,\beta}[1],\mathcal{T}_{\alpha,\beta}\rangle$
of $D^{\rm b}(X)$, which is also the heart of a t-structure on $D^{\rm b}(X)$.

One then introduces a third parameter $s>0$ in order to define a family of
central charges

$Z_{\alpha,\beta,s}:K_{\rm num}(X)\to\mathbb{C}$

as follows, for $A\in\mathcal{A}^{\alpha,\beta}(X)$:

(6)
$Z_{\alpha,\beta,s}(A):=-\operatorname{ch}_{3}^{\beta}(A)+(s+1/6)\alpha^{2}\operatorname{ch}_{1}^{\beta}(A)+\sqrt{-1}\left(\operatorname{ch}_{2}^{\beta}(A)-\alpha^{2}\operatorname{ch}_{0}(A)/2\right),$

whose corresponding slope function is

(7)
$\lambda_{\alpha,\beta,s}(A):=\begin{cases}\dfrac{\operatorname{ch}_{3}^{\beta}(A)-(s+1/6)\alpha^{2}\operatorname{ch}_{1}^{\beta}(A)}{\operatorname{ch}_{2}^{\beta}(A)-\alpha^{2}\operatorname{ch}_{0}(A)/2},&\textrm{if
}\operatorname{ch}_{2}^{\beta}(A)-\alpha^{2}\operatorname{ch}_{0}(A)/2\neq
0,\\\ +\infty,&\textrm{if
}\operatorname{ch}_{2}^{\beta}(A)-\alpha^{2}\operatorname{ch}_{0}(A)/2=0.\end{cases}$

######  Remark 2.2.

We could also consider a more general central charge, whose real part is

$-\operatorname{ch}^{\beta}_{3}(A)+b\operatorname{ch}^{\beta}_{2}(A)+a\operatorname{ch}^{\beta}_{1}(A)$

for parameters $b\in\mathbb{R}$ and $a\in\mathbb{R}^{+}$, see [6, Lemma 8.3]
and [30]. However, we will only consider the special case where $b=0$, while
$a=\alpha^{2}(s+1/6)$.

An object $A\in D^{\rm b}(X)$ is said to be
$\lambda_{\alpha,\beta,s}$-(semi)stable if $A\in\mathcal{A}^{\alpha,\beta}(X)$
and every sub-object $F\hookrightarrow A$ within
$\mathcal{A}^{\alpha,\beta}(X)$ satisfies
$\lambda_{\alpha,\beta,s}(F)<(\leq)\leavevmode\nobreak\
\lambda_{\alpha,\beta,s}(B)$.

For further reference, we define for each real numerical Chern character
$v\in\mathbb{R}^{4}$, the following the function on
$(\alpha,\beta)\in\mathbb{H}$:

(8)
$\tau_{v,s}(\alpha,\beta)=v_{3}-v_{2}\beta+\frac{1}{2}v_{1}\beta^{2}-\frac{1}{6}v_{0}\beta^{3}-\left(s+\frac{1}{6}\right)(v_{1}-v_{0}\beta)\alpha^{2},$

which coincides with the numerator of $\lambda_{(\alpha,\beta,s)}(A)$ when
$v=\operatorname{ch}(A)$. Again, we define
$\tau_{A,s}(\alpha,\beta):=\tau_{\operatorname{ch}(A),s}(\alpha,\beta)$ for
objects $A\in D^{\rm b}(X)$, and we will also write
$\tau_{v}(\alpha,\beta,s)=\tau_{v,s}(\alpha,\beta)$. Note that, when non-zero,
the denominator of $\lambda_{\alpha,\beta,s}(A)$ is $\rho_{A}(\alpha,\beta)$.

Finally, a direct consequence of the generalized Bogomolov inequality (4) is
that $(\mathcal{A}^{\alpha,\beta}(X),Z_{\alpha,\beta,s})$ is a (numerical)
stability condition, in the sense of [7, Definition 2.1.1], for every triple
$(\alpha,\beta,s)\in\mathbb{R}^{+}\times\mathbb{R}\times\mathbb{R}^{+}$. In
practical terms, this yields:

######  Proposition 2.3.

Fix $(\alpha,\beta)\in\mathbb{H}$. If $A\in\mathcal{A}^{\alpha,\beta}(X)$ is
non-zero, then $\rho_{A}(\alpha,\beta)\geq 0$, with equality only if
$\tau_{A,s}(\alpha,\beta)>0$ for every $s\in\mathbb{R}^{+}$.

The generalized Bogomolov inequality can also be used to prove a form of the
support property for $\lambda_{\alpha,\beta,s}$-semistability. In the case we
are considering where the Picard rank is 1 we can state it as follows:

######  Proposition 2.4.

[6, Thm. 8.7] Suppose $X$ is a smooth threefold with Picard rank 1 such that
the generalized Bogomolov inequality (4) holds for all
$\nu_{\alpha,\beta}$-semistable objects. If $E\in\mathcal{A}^{\alpha,\beta}$
is $\lambda_{\alpha,\beta,s}$-semistable then $Q_{\alpha,\beta}(E)\geq 0$.

######  Remark 2.5.

This implies that $(\mathcal{A}^{\alpha,\beta}(X),Z_{\alpha,\beta,s})$ is a
(full) Bridgeland stability condition on $D^{\rm b}(X)$.

######  Remark 2.6.

Observe that $\lambda_{\alpha,\beta,s}$-semistable objects may not satisfy the
usual Bogomolov inequality:
$A:={\mathcal{O}}_{X}[2]\oplus{\mathcal{O}}_{X}(1)[1]$ is
$\lambda_{1/2,1/2,s}$-semistable for every $s>0$, but $Q^{\rm tilt}(A)=-1$.

When we come to do more detailed computations, it will also be useful to have
a more uniform notation for the various functions of $v\in K_{\rm num}(X)$
introduced above; more precisely, we define the following

(9)
$\displaystyle\operatorname{ch}^{\alpha,\beta}_{0}(v):=v_{0},\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{ch}^{\alpha,\beta}_{2}(v):=\rho_{v}(\alpha,\beta),$
$\displaystyle\operatorname{ch}^{\alpha,\beta}_{1}(v):=\operatorname{ch}^{\beta}_{1}(v),\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{ch}^{\alpha,\beta}_{3}(v):=\tau_{v,1/3}(\alpha,\beta).$

By convention, we set $\operatorname{ch}_{i}^{\alpha,\beta}(v)=0$ for
$i\not\in\\{0,1,2,3\\}$. Alternatively, once can also define

$\operatorname{ch}^{\alpha,\beta}(v)=\operatorname{Re}\bigl{(}\exp(-\beta-\sqrt{-1}\alpha)\cdot
v\bigr{)}.$

The reason for setting $s=1/3$ will become clearer in Section 4.2, but one
technical reason is that the partial derivatives of
$\operatorname{ch}^{\alpha,\beta}_{i}(v)$ with respect to $\alpha$ and $\beta$
behave very well; more precisely:

(10)
$\partial_{\alpha}\operatorname{ch}_{i}^{\alpha,\beta}(v)=-\alpha\operatorname{ch}_{i-2}^{\alpha,\beta}(v)\text{
and
}\partial_{\beta}\operatorname{ch}_{i}^{\alpha,\beta}(v)=-\operatorname{ch}^{\alpha,\beta}_{i-1}(v).$

Note that
$\tau_{v,s}(\alpha,\beta)=\operatorname{ch}^{\alpha,\beta}_{3}(v)-\alpha^{2}(s-1/3)\operatorname{ch}^{\alpha,\beta}_{1}(v)$.

We also introduce

(11)
$\displaystyle\Delta_{ij}(\alpha,\beta):=\operatorname{ch}^{\alpha,\beta}_{i}(u)\operatorname{ch}^{\alpha,\beta}_{j}(v)-\operatorname{ch}^{\alpha,\beta}_{j}(u)\operatorname{ch}^{\alpha,\beta}_{i}(v)\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}$
$\displaystyle\delta_{ij}(u,v):=\operatorname{ch}_{i}(u)\operatorname{ch}_{j}(v)-\operatorname{ch}_{j}(u)\operatorname{ch}_{i}(v)=\Delta_{ij}(0,0).$

In particular, note that $\Delta_{10}(\alpha,\beta)=\delta_{10}(u,v)$.

The following is an easy exercise:

######  Lemma 2.7.

Fix real numerical Chern characters $u$ and $v$.

  1. (1)

$\Delta_{01}\Delta_{23}+\Delta_{02}\Delta_{31}+\Delta_{12}\Delta_{03}=0$.

  2. (2)

The partial derivatives of $\Delta_{ij}$ are given by:

$\displaystyle\partial_{\alpha}\Delta_{ij}(\alpha,\beta)=-\alpha\bigl{(}\Delta_{i-2\,j}(\alpha,\beta)+\Delta_{i\,j-2}(\alpha,\beta)\bigr{)}$
$\displaystyle\partial_{\beta}\Delta_{ij}(\alpha,\beta)=-\bigl{(}\Delta_{i-1\,j}(\alpha,\beta)+\Delta_{i\,j-1}(\alpha,\beta)\bigr{)}.$

  3. (3)

Assume either $\operatorname{ch}^{\alpha,\beta}_{i}(u)\neq 0$ or
$\operatorname{ch}^{\alpha,\beta}_{i}(v)\neq 0$. Then, for any
$i,j,k\in\\{0,1,2,3\\}$, if
$\Delta_{ij}(\alpha,\beta)=0=\Delta_{ik}(\alpha,\beta)$ then
$\Delta_{jk}(\alpha,\beta)=0$.

  4. (4)

The following are equivalent

    1. (a)

there exist $\alpha,\beta$ and $i\in\\{0,1,2,3\\}$ such that
$\operatorname{ch}_{i}^{\alpha,\beta}(v)\neq 0$ (or
$\operatorname{ch}_{i}^{\alpha,\beta}(u)\neq 0$) and for all
$j\in\\{0,1,2,3\\}$, $\Delta_{ij}(\alpha,\beta)=0$.

    2. (b)

for all $\alpha,\beta$ and for all $i,j\in\\{0,1,2,3\\}$,
$\Delta_{ij}(\alpha,\beta)=0$.

    3. (c)

$u\propto v$.

In what follows, we will use ${\mathcal{M}}_{\alpha,\beta,s}(v)$ to denote the
set of $\lambda_{\alpha,\beta,s}$-semistable objects with Chern character $v$;
Piyaratne and Toda proved in [31] that ${\mathcal{M}}_{\alpha,\beta,s}(v)$ has
the structure of an algebraic stack, locally of finite type over $\mathbb{C}$.
Determining whether ${\mathcal{M}}_{\alpha,\beta,s}(v)$ also has the structure
of a projective variety is an important problem.

###  2.3. The second tilt category

We will now collect some useful facts about the objects in the second tilt
category $\mathcal{A}^{\alpha,\beta}(X)$. Much of the following is well known
and is easy to deduce in various ad hoc ways but we give a novel treatment
using higher octahedra which is of independent interest (the idea first
appeared in [3]). We will henceforth drop the $X$ from the notation
$\mathcal{A}^{\alpha,\beta}(X)$. Recall that ${\mathcal{H}}^{i}$ denotes
cohomology in $\mathcal{C}oh(X)$ and ${\mathcal{H}}^{i}_{\mathcal{B}}$ denotes
cohomology in $\mathcal{B}^{\beta}$.

Suppose $A\in\mathcal{A}^{\alpha,\beta}$ for some
$(\alpha,\beta)\in\mathbb{H}$. We write $A_{i}={\mathcal{H}}^{-i}_{\beta}(A)$
and $A_{ij}={\mathcal{H}}^{-j}(A_{i})$. So we have three distinguished
triangles:

(12) $\begin{gathered}A_{11}[1]\to A_{1}\to A_{10};\\\ A_{01}[1]\to A_{0}\to
A_{00};\\\ A_{1}[1]\to A\to A_{0}.\end{gathered}$

Because these triangles intersect we can arrange them into a diagram as
follows:

(13)
$\textstyle{A_{01}}$$\textstyle{A_{0}[-1]}$$\textstyle{A_{1}[1]}$$\textstyle{A_{10}[1]}$$\textstyle{A_{00}[-1]}$$\textstyle{C}$$\textstyle{\widetilde{A}[1]}$$\textstyle{A_{01}[1]}$$\textstyle{A}$$\textstyle{C^{\prime}}$$\textstyle{A_{0}}$$\textstyle{A_{11}[3]}$$\textstyle{A_{1}[2]}$$\textstyle{A_{10}[2]}$

Here the squiggly arrows $X\rightsquigarrow Y$ mean $X\to Y[1]$. The diagram
is meant to repeat infinitely above and to the right by shifting $[-n]$ and
$[n]$. Every square commutes and the triangles along the diagonal are
distinguished. Furthermore, each triple of morphisms formed by composing
horizontally and then vertically and then looping back via the repeated
diagram to the right are distinguished. The additional objects $C$,
$C^{\prime}$ and $\widetilde{A}[1]$ are defined as cones on suitable
composites. Three of the triangles are given in display (12) and the remaining
seven are:

(14) $\begin{gathered}A_{1}[1]\to C\to A_{01}[1]\to A_{1}[2];\\\ A_{10}[1]\to
C^{\prime}\to A_{0}\to A_{10}[2];\\\ A_{10}[1]\to\widetilde{A}[1]\to
A_{01}[1]\to A_{10}[2];\\\ C\to A\to A_{00}\to C[1];\\\ A_{11}[2]\to A\to
C^{\prime}\to A_{11}[3];\\\ A_{11}[2]\to C\to\widetilde{A}[1]\to A_{11}[3];\\\
\widetilde{A}[1]\to C^{\prime}\to A_{00}\to\widetilde{A}[2].\end{gathered}$

The first two tell us what the $\mathcal{B}^{\beta}$-cohomologies of $C$ and
$C^{\prime}$ are. The third tells us that $\widetilde{A}\in\mathcal{C}oh(X)$.
The fourth and fifth tell us what the
$\mathcal{A}^{\alpha,\beta}$-cohomologies of $C$ and $C^{\prime}$ are, and the
final two tell us what the $\mathcal{C}oh(X)$-cohomologies of $C$ and
$C^{\prime}$ are. In particular, we have that

(15) ${\mathcal{H}}^{-2}(A)\simeq A_{11}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{H}}^{-1}(A)=\widetilde{A}\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ {\mathcal{H}}^{0}(A)\simeq
A_{00}.$

There are also five distinguished octahedra which are obtained by removing one
row and column from the diagram in display (13) (for a more concrete example
of an octahedron in this form see the diagrams (22) in the proof of Prop
2.15). The diagram can also be represented as a four dimensional shape given
as a truncated 5-simplex with five octahedral and five tetrahedral faces.
Another way to express the diagram is that $A$ is filtered in $D^{\rm b}(X)$
by

$A_{00}[-3]\to A_{01}[-1]\to A_{10}\to A_{11}[2]\to A$

with factors $A_{0}[-2]$, $\widetilde{A}$ and $A_{1}[1]$ respectively.

Observe that $A_{i1}\in\mathcal{F}_{\beta}$, and these must be torsion free
sheaves, while $A_{i0}\in\mathcal{T}_{\beta}$ and, in particular,
$A_{00}\in\mathcal{A}^{\alpha,\beta}$. In fact, a slightly stronger statement
is true.

######  Lemma 2.8.

If $A\in\mathcal{A}^{\alpha,\beta}$ for some $(\alpha,\beta)\in\mathbb{H}$,
then ${\mathcal{H}}^{-2}(A)$ must be a reflexive sheaf.

###### Proof.

Assume ${\mathcal{H}}^{-2}(A)\simeq A_{11}$ is not reflexive, and
$T:=A_{11}^{**}/A_{11}\neq 0$. Then $T\to A_{11}[1]$ is a monomorphism in
$\mathcal{B}^{\beta}$. Since $A_{11}$ is a $\mathcal{B}^{\beta}$-subobject of
$A_{1}$, it must have $\nu_{\alpha,\beta}^{+}(A_{11})\leq 0$. However,
$\nu_{\alpha,\beta}(T)=+\infty$, providing a contradiction. ∎

The following fact will also be useful later on.

######  Lemma 2.9.

If $A\in\mathcal{A}^{\alpha,\beta}$ for some $(\alpha,\beta)\in\mathbb{H}$
satisfies
${\mathcal{H}}^{-2}(A)={\mathcal{H}}^{-1}({\mathcal{H}}^{0}_{\beta}(A))=0$,
then ${\mathcal{H}}^{p}(A)\simeq{\mathcal{H}}^{p}_{\beta}(A)$ and for
$p=-1,0$.

###### Proof.

Chasing through the seven triangles listed in display (14) with
$A_{11}=A_{01}=0$, one concludes that $\tilde{A}\simeq C\simeq A_{1}$, which
is the same as ${\mathcal{H}}^{-1}(A)\simeq{\mathcal{H}}^{-1}_{\beta}(A)$ by
the isomorphisms in (15). The vanishing of $A_{01}$ also implies that
$A_{0}\simeq A_{00}$ by the sequences in display (12), hence
${\mathcal{H}}^{0}(A)\simeq{\mathcal{H}}^{0}_{\beta}(A)$. ∎

We will need to consider short exact sequences in $\mathcal{A}^{\alpha,\beta}$
and we look at situations where the middle term is in either
$\mathcal{B}^{\beta}$ or $\mathcal{B}^{\beta}[1]$.

######  Lemma 2.10.

If $A\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{B}^{\beta}$ and
$B\hookrightarrow A\twoheadrightarrow C$ is a short exact sequence in
$\mathcal{A}^{\alpha,\beta}$, then there are in $\mathcal{A}^{\alpha,\beta}$ a
short exact sequence $D\hookrightarrow A\twoheadrightarrow C^{\prime}$, a
quotient $B\twoheadrightarrow D$ and an injection $C\hookrightarrow
C^{\prime}$ such that $C^{\prime},D\in\mathcal{B}^{\beta}$.

###### Proof.

Apply $\mathcal{H}_{\mathcal{B}}$ to get a long exact sequence in
$\mathcal{B}^{\beta}$:

$0\to C_{1}\to B\to A\to C_{0}\to 0.$

Split this via $D$. Then $\nu^{-}_{\alpha,\beta}(D)>0$ and so
$D\in\mathcal{A}^{\alpha,\beta}$. Then we have a short exact sequence $D\to
A\to C^{\prime}$ in $\mathcal{A}^{\alpha,\beta}$ together with an injection
$C\hookrightarrow C^{\prime}$ and a surjection $B\twoheadrightarrow D$ also in
$\mathcal{A}^{\alpha,\beta}$ as required. ∎

######  Lemma 2.11.

If $A\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{B}^{\beta}[1]$ and
$B\hookrightarrow A\twoheadrightarrow C$ is a short exact sequence in
$\mathcal{A}^{\alpha,\beta}$ then there are in $\mathcal{A}^{\alpha,\beta}$ a
short exact sequence $B^{\prime}\hookrightarrow A\twoheadrightarrow D$, a
subobject $B^{\prime}\hookrightarrow B$ and a quotient $D\twoheadrightarrow C$
with $B^{\prime},D\in\mathcal{B}^{\beta}[1]$.

###### Proof.

Apply $\mathcal{H}_{\mathcal{B}}$ to get a long exact sequence in
$\mathcal{B}^{\beta}$:

$0\to B_{1}\to A_{1}\to C_{1}\to B_{0}\to 0.$

Split this via $D[-1]$. Then $\nu^{+}_{\alpha,\beta}(D)\leq 0$ and so
$D\in\mathcal{A}^{\alpha,\beta}$. Then we have a short exact sequence
$B_{1}[1]\hookrightarrow A\twoheadrightarrow D$ in
$\mathcal{A}^{\alpha,\beta}$ together with an injection $B[1]\hookrightarrow
B$ and a surjection $D\twoheadrightarrow C_{1}[1]=C$ in
$\mathcal{A}^{\alpha,\beta}$ as required. ∎

When the middle term is in $\mathcal{C}oh(X)$ we can say a lot more:

######  Proposition 2.12.

Suppose $E\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{C}oh(X)$ and
$D\hookrightarrow E$ is a monomorphism in $\mathcal{A}^{\alpha,\beta}$. Then
$E\in\mathcal{B}^{\beta}$. Furthermore, if $D\not\in\mathcal{C}oh(X)$ then
$E/D\in\mathcal{C}oh(X)[2]\cap\mathcal{B}^{\beta}[1]$. Otherwise,
$D\in\mathcal{C}oh(X)\cap\mathcal{B}^{\beta}\cap\mathcal{A}^{\alpha,\beta}$
and there is an
$F\in\mathcal{C}oh(X)\cap\mathcal{B}^{\beta}\cap\mathcal{A}^{\alpha,\beta}$
and a monomorphism $\phi:D\hookrightarrow F$ in $\mathcal{A}^{\alpha,\beta}$
such that one of the following holds for $G:=E/F$ in
$\mathcal{A}^{\alpha,\beta}$:

  1. (1)

$G\in\mathcal{C}oh(X)\cap\mathcal{B}^{\beta}$,

  2. (2)

$G\in\mathcal{C}oh(X)[1]\cap\mathcal{B}^{\beta}[1]$ and $\phi$ is the
identity,

  3. (3)

$G\in\mathcal{C}oh(X)[1]\cap\mathcal{B}^{\beta}$.

###### Proof.

If $A=E\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{C}oh(X)$ then the triangles
(12) and (14) imply $C=0$ and then $A_{1}[1]$ is a sheaf and so $A_{1}=0$.
This establishes $E\in\mathcal{B}^{\beta}$.

Applying cohomology in $\mathcal{B}^{\beta}$ to the triangle $D\to E\to B$,
where $B=E/D$ in $\mathcal{A}^{\alpha,\beta}$ we have that $D_{1}=0$ and a
long exact sequence

(16) $0\to B_{1}\to D\to E\to B_{0}\to 0.$

So $D\in\mathcal{B}^{\beta}\cap\mathcal{A}^{\alpha,\beta}$. If we can split
(16) in the middle via $Q$, say. From $D\to Q\to B_{1}[1]$ we see that
$Q\in\mathcal{B}^{\beta}\cap\mathcal{A}^{\alpha,\beta}$ and $D\to Q$ injects
in $\mathcal{A}^{\alpha,\beta}$. Applying cohomology in $\mathcal{C}oh(X)$ to
the triangle $Q\to E\to B_{0}$ we see that $Q\in\mathcal{C}oh(X)$. Applying it
to $B_{1}\to D\to Q$ we get

$B_{11}\cong D_{01}\text{ and }0\to B_{10}\to D_{00}\to Q\to 0.$

From this it follows that $D_{01}\in\mathcal{A}^{\alpha,\beta}[-2]$.

Suppose first that $D_{01}\neq 0$. Then the composite $D_{01}[1]\to D\to E$
must vanish and so the map $D\to E$ lifts to $D_{00}\to E$. This then fits
into an octahedron

$\textstyle{D_{01}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{01}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E}$

for some object $K$. If $K=0$ we have $E\cong D_{00}$ and then
$E/D=D_{01}[2]\in\mathcal{C}oh(X)[2]\cap\mathcal{B}^{\beta}[1]$. If $K\neq 0$
then applying cohomology in $\mathcal{B}^{\beta}$ to the octahedron we have
$K_{0}=0$, $K_{1}\cong D_{01}[1]$ and $K_{-1}\cong B$ from the vertical
triangle and $K_{0}\to D_{00}\to E\to K_{-1}$ from the horizontal sequence.
But then $0=K_{1}=D_{01}[1]$ which is a contradiction.

We can now assume $D_{01}=0$. Suppose $B_{0}=0$. Then $Q=E$. Now set
$F=D_{00}=D$. Then we have case (2) as
$G:=B_{10}[1]\in\mathcal{B}^{\beta}[1]\cap\mathcal{A}^{\alpha,\beta}$.

Otherwise, $B_{0}\neq 0$. Replacing $D$ with $Q$, we may assume $B_{1}=0$.
Applying cohomology in $\mathcal{C}oh(X)$, we have a long exact sequence:

$0\to B_{01}\to Q\to E\to B_{00}\to 0.$

If $B_{00}=0$ then we take $F=Q$ and we have type (3). Otherwise, we split the
sequence via $F$. From $Q\to F\to B_{01}[1]$ we see that
$F\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{B}^{\beta}$. This is type (1).
Note that the triangles $Q\to F\to B_{01}[1]$ and $B_{01}[1]\to B\to B_{00}$
imply that $B_{01}[1]\in\mathcal{A}^{\alpha,\beta}$ and so $Q\to F$ injects.

The general case can be summarized in a higher octahedron:

(17)
$\textstyle{B_{00}[-1]}$$\textstyle{F}$$\textstyle{\widetilde{B}[1]}$$\textstyle{B_{01}[1]}$$\textstyle{E}$$\textstyle{B}$$\textstyle{B_{0}}$$\textstyle{B_{00}}$$\textstyle{D[1]}$$\textstyle{Q[1]}$$\textstyle{F[1]}$$\textstyle{B_{10}[2]}$$\textstyle{\widetilde{B}[2]}$$\textstyle{B_{01}[2]}$

∎

###  2.4. Duals of semistable sheaves

Given an object $A\in D^{\rm b}(X)$, we denote its derived dual by
$A^{\vee}:=\operatorname{R\mathcal{H}\textit{om}}(A,{\mathcal{O}}_{X})[2]$.
For a sheaf $E$, its derived dual $E^{\vee}$ satisfies
${\mathcal{H}}^{j}(E^{\vee})={\mathcal{E}}{\it xt}^{j+2}(E,{\mathcal{O}}_{X})$
for $j=-2,-1,0,1$, and ${\mathcal{H}}^{j}(E^{\vee})=0$ otherwise. If $E$ is
torsion free, then we have the following short exact sequence in
$\mathcal{C}oh(X)$

(18) $0\to E\to E^{**}\to Q_{E}\to 0\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ Q_{E}:=E^{**}/E,$

where $E^{*}:={\mathcal{H}}{\it om}(E,{\mathcal{O}}_{X})$. Note that $\dim
Q_{E}\leq 1$; letting $Z_{E}$ be the maximal 0-dimensional subsheaf of
$Q_{E}$, define $T_{E}:=Q_{E}/Z_{E}$. Let $E^{\prime}$ be the kernel of the
composed epimorphism $E^{**}\twoheadrightarrow Q_{E}\twoheadrightarrow T_{E}$;
it fits into the following short exact sequence in $\mathcal{C}oh(X)$:

(19) $0\to E\to E^{\prime}\to Z_{E}\to 0.$

We then have that ${\mathcal{H}}^{j}(E^{\vee})=0$ for $j\neq-2,-1,0$, and

${\mathcal{H}}^{-2}(E^{\vee})=E^{*}\simeq E^{\prime*}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{H}}^{-1}(E^{\vee})={\mathcal{E}}{\it
xt}^{1}(E,{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{1}(E^{\prime},{\mathcal{O}}_{X})\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm and}$ ${\mathcal{H}}^{0}(E^{\vee})={\mathcal{E}}{\it
xt}^{2}(E,{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{3}(Z_{E},{\mathcal{O}}_{X}).$

Moreover, ${\mathcal{E}}{\it
xt}^{1}(E,{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{1}(E^{\prime},{\mathcal{O}}_{X})$ fits into the short exact sequence

$0\to{\mathcal{E}}{\it xt}^{1}(E^{**},{\mathcal{O}}_{X})\to{\mathcal{E}}{\it
xt}^{1}(E,{\mathcal{O}}_{X})\to{\mathcal{E}}{\it
xt}^{2}(T_{E},{\mathcal{O}}_{X})\to 0.$

Clearly, $E^{*}\in\mathcal{F}_{\beta}$ for $\beta>\mu^{+}(E^{*})$, while
${\mathcal{E}}{\it xt}^{1}(E,{\mathcal{O}}_{X})\in\mathcal{T}_{\beta}$ for
every $\beta$; it follows that $E^{\prime\vee}[-1]\in\mathcal{B}^{\beta}$ for
every $\beta>\mu^{+}(E^{*})$. In addition, $Z_{E}^{\vee}[1]$ is a
0-dimensional sheaf, so
$Z_{E}^{\vee}[1]\in\mathcal{T}_{\beta}\subset\mathcal{B}^{\beta}$ for every
$\beta$. Comparing with the triangle

(20) $E^{\prime\vee}\to E^{\vee}\to Z_{E}^{\vee}[1],$

obtained from dualizing the sequence in display (19), then we have proved:

######  Proposition 2.13.

If $E$ is a torsion free sheaf and

$0\to E\to E^{\prime}\to Z_{E}\to 0$

is the sequence (19) where $Z_{E}$ is the maximal 0-dimensional subsheaf of
$E^{**}/E$ then

(21) ${\mathcal{H}}_{\beta}^{-1}(E^{\vee})\simeq
E^{\prime\vee}[-1]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\mathcal{H}}_{\beta}^{0}(E^{\vee})\simeq
Z_{E}^{\vee}[1]\simeq{\mathcal{E}}{\it xt}^{2}(E,{\mathcal{O}}_{X}),$

whenever $\beta>\mu^{+}(E^{*})$.

We can also formulate a converse to this construction. Let
$\mathcal{C}oh(X)_{d}$ denote the category coherent sheaves on $X$ of
dimension $\leq d$. We want to be able to characterize when an object $A\in
D^{b}(X)$ which has cohomology in 3 consecutive places $A^{i}$, $A^{i+1}$ and
$A^{i+2}$ such that $A^{i}$ is reflexive and
$A^{i+j}\in\mathcal{C}oh(X)_{2-j}$ for $j=1,2$, is the shift of the dual of a
torsion free sheaf. We will make use of this when we analyze the asymptotics
for $\beta\gg 0$ and we will express them in the form of lifting properties.
First we need a technical lemma as an intermediate step. The equivalent
statement in dimension $2$ is an easy exercise: if $A$ has cohomology in two
places with $A^{i}$ locally-free and $A^{i+1}\in\mathcal{C}oh(X)_{0}$ such
that any subsheaf $S\to A^{i+1}$ does not lift to $A$ then $A$ is the shift of
the dual of a torsion free sheaf.

######  Lemma 2.14.

An object $A\in D^{b}(X)$ satisfies the following conditions

  1. (1)

${\mathcal{H}}^{i}(A)=0$, $i\neq-2,-1,0$,

  2. (2)

$F:={\mathcal{H}}^{-2}(A)$ is a reflexive sheaf,

  3. (3)

$G:={\mathcal{H}}^{-1}(A)\in\mathcal{C}oh(X)_{1}$,

  4. (4)

$S:={\mathcal{H}}^{0}(A)\in\mathcal{C}oh(X)_{0}$,

  5. (5)

the induced map $F^{*}\to{\mathcal{E}}{\it xt}^{2}(G,{\mathcal{O}}_{X})$
surjects with kernel $K$, say,

  6. (6)

the induced map ${\mathcal{E}}{\it
xt}^{1}(F,{\mathcal{O}}_{X})\to{\mathcal{E}}{\it xt}^{3}(G,{\mathcal{O}}_{X})$
is an isomorphism, and

  7. (7)

the induced map $f:K\to{\mathcal{E}}{\it xt}^{3}(S,{\mathcal{O}}_{X})$
surjects,

if and only if $E:=A^{\vee}\cong\ker f$ is a torsion free sheaf.

###### Proof.

To see where these induced maps come from consider the triangles

$\displaystyle F[2]\to A\to\tilde{F}\to F[3]$ $\displaystyle
G[1]\to\tilde{F}\to S\to G[2].$

Then we have maps

${\mathcal{H}}{\it
om}(F,{\mathcal{O}}_{X})\to\operatorname{R\mathcal{H}\textit{om}}(\tilde{F}[-3],{\mathcal{O}}_{X})\to\operatorname{R\mathcal{H}\textit{om}}(G[-2],{\mathcal{O}}_{X})\cong{\mathcal{E}}{\it
xt}^{2}(G,{\mathcal{O}}_{X})$

and ${\mathcal{E}}{\it
xt}^{1}(F,{\mathcal{O}}_{X})\to\operatorname{R\mathcal{H}\textit{om}}(\tilde{F}[-4],{\mathcal{O}}_{X})\to{\mathcal{E}}{\it
xt}^{3}(G,{\mathcal{O}}_{X})$. The final one arises because there is a short
exact sequence of sheaves:

$0\to{\mathcal{E}}{\it xt}^{3}(S,{\mathcal{O}}_{X})\to{\mathcal{E}}{\it
xt}^{3}(\tilde{F},{\mathcal{O}}_{X})\to{\mathcal{E}}{\it
xt}^{2}(G,{\mathcal{O}}_{X})\to 0.$

and so the $K\to{\mathcal{E}}{\it xt}^{3}(\tilde{F},{\mathcal{O}}_{X})$ lifts
to $K\to{\mathcal{E}}{\it xt}^{3}(S,{\mathcal{O}}_{X})$.

The lemma follows immediately from the spectral sequence

${\mathcal{E}}{\it
xt}^{-p}({\mathcal{H}}^{q}(A),{\mathcal{O}}_{X})\Rightarrow{\mathcal{H}}^{q-p+2}(A^{\vee})$

which converges on the third page to a single entry at $(p,q)=(0,-2)$. Note
that $E$ is torsion free because it is the subsheaf of a reflexive sheaf
(namely $F^{*}$). We also have that $E^{\prime}\cong K$,
$Z_{E}\cong{\mathcal{E}}{\it xt}^{3}(S,{\mathcal{O}}_{X})$,
$Q_{E}\cong{\mathcal{E}}{\it xt}^{2}(G,{\mathcal{O}}_{X})$ and
$T_{E}\cong{\mathcal{E}}{\it xt}^{3}(G,{\mathcal{O}}_{X})$. ∎

We can give a more categorical description of the conditions as follows

######  Proposition 2.15.

Suppose $A\in D^{b}(X)$. Then $A$ satisfies the following conditions

  1. (1)

${\mathcal{H}}^{i}(A)=0$, $i\neq-2,-1,0$,

  2. (2)

$F:={\mathcal{H}}^{-2}(A)$ is a reflexive sheaf,

  3. (3)

$G:={\mathcal{H}}^{-1}(A)\in\mathcal{C}oh(X)_{1}$,

  4. (4)

$S:={\mathcal{H}}^{0}(A)\in\mathcal{C}oh(X)_{0}$,

  5. (5)

no monomorphism $S^{\prime}\to S$ in $\mathcal{C}oh(X)$ lifts to $A$, and

  6. (6)

no monomorphism $G^{\prime}\to G$ in $\mathcal{C}oh(X)$ lifts to $A[-1]$.

if and only if $E:=A^{\vee}$ is a torsion free sheaf.

###### Proof.

It is easy to see that the final two conditions are necessary because
otherwise we would have maps $E\to{\mathcal{E}}{\it
xt}^{3}(S^{\prime},{\mathcal{O}}_{X})[-3]$ or $E\to{\mathcal{E}}{\it
xt}^{2}(G^{\prime},{\mathcal{O}}_{X})[-2]$ or $E\to{\mathcal{E}}{\it
xt}^{3}(G^{\prime},{\mathcal{O}}_{X})[-3]$ which are impossible.

For the converse we show that (5) implies Lemma 2.14(7) while (6) implies both
Lemma 2.14(5) and (6) together by considering the contrapositives. For the
proof we consider the octahedron

(22)
$\textstyle{F[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$
or
$\textstyle{S[-1]}$$\textstyle{B[1]}$$\textstyle{G[1]}$$\textstyle{A}$$\textstyle{\widetilde{F}}$$\textstyle{S}$$\textstyle{F[3]}$$\textstyle{B[2]}$$\textstyle{G[2]}$

Observe that the triangle $F^{\vee}[-2]\to G^{\vee}\to B^{\vee}$ shows that
when Lemma 2.14 holds, $B^{\vee}=K[1]$.

Suppose first that $K\to{\mathcal{E}}{\it xt}^{3}(S,{\mathcal{O}}_{X})$ fails
to surject. Let the quotient be denoted by $T$; this is in
$\mathcal{C}oh(X)_{0}$ and let $S^{\prime}$ be ${\mathcal{E}}{\it
xt}(T,{\mathcal{O}}_{X})$. Then $S^{\prime}\to S$ and the composite with $S\to
B[2]$ vanishes and so lifts to $A$ as required.

Now suppose that Lemma 2.14 (5) or (6) fails. Applying cohomology to the left
vertical triangle of our octahedron we have

$0\to B^{-1}\to F^{*}\to{\mathcal{E}}{\it xt}^{2}(G,{\mathcal{O}}_{X})\to
B^{0}\to{\mathcal{E}}{\it xt}^{1}(F,{\mathcal{O}}_{X})\to{\mathcal{E}}{\it
xt}^{3}(G,{\mathcal{O}}_{X})\to B^{1}\to 0$

Then the cone on $B^{-1}[1]\to B$ is the dual of an object $G^{\prime}$
supported in dimension $1$ and there is a map $G^{\prime}\to G$ which lifts to
$G^{\prime}\to B$. But there are no morphisms $G^{\prime}[1]\to T[-1]$ and so
this lifts to a non-zero map $G^{\prime}[1]\to A$ as required. ∎

######  Remark 2.16.

We can rephrase item (5) in Proposition 2.15 to say that if $K\to A$ is a map
from a sheaf in $\mathcal{C}oh(X)_{0}$ then the induced map $K\to S$ must be
zero. Similarly, item (6) becomes the statement that if $K[1]\to A$ is a map
with $K\in\mathcal{C}oh(X)_{1}$ then the induced map $K\to G$ must be zero. In
practice, these are easier to test but we will make more use of the conditions
stated in Proposition 2.15. It is also clear how to extend the statement to
higher dimensions.

###  2.5. Refining notions of stability for sheaves

We complete this section by introducing a stability condition in
$\mathcal{C}oh(X)$ which interpolates between $\mu$-stability and Gieseker
stability; it will play a role in some of the proofs below.

The Hilbert polynomial of a coherent sheaf $E$ on $X$ with respect to the
polarization $L$ is

$P_{E}(t):=\chi(E\otimes L^{\otimes
t})=\operatorname{ch}_{0}(E)\chi(L^{\otimes
t})+\operatorname{ch}_{1}(E)x_{2}(t)+\operatorname{ch}_{2}(E)x_{1}(t)+\operatorname{ch}_{3}(E),$

where

$x_{2}(t):=\left(\dfrac{1}{2}t^{2}L^{2}+t\operatorname{td}_{1}(X)\cdot
L+\operatorname{td}_{2}(X)\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\
x_{1}(t):=\left(tL+\operatorname{td}_{1}(X)\right)\leavevmode\nobreak\ ,$

and $\operatorname{td}_{j}(X)$ denote the Todd classes of $X$. If $F$ is
another coherent free sheaf on $X$, we define, following [32, Section 2]:

$\Lambda(E,F):=\bigl{(}\delta_{10}(E,F),\delta_{20}(E,F),\delta_{30}(E,F),\delta_{21}(E,F),\delta_{31}(E,F),\delta_{32}(E,F)\bigr{)},$

where
$\delta_{ij}(E,F):=\delta_{ij}(\operatorname{ch}(E),\operatorname{ch}(F))$
following the notation introduced in display (11).

We remark that a coherent sheaf $E$ is Gieseker (semi)stable if and only if
every proper, nontrivial subsheaf $F\hookrightarrow E$ satisfies
$\Lambda(E,F)>\leavevmode\nobreak\ (\geq)\leavevmode\nobreak\ 0$ in the
lexicographic order. For instance, assume that $E$ is torsion free, and let
$F\hookrightarrow E$ be a proper subsheaf; letting $p_{E}(t)$ denote the
reduced Hilbert polynomial of the sheaf $E$, we have

$p_{E}(t)-p_{F}(t)=\dfrac{1}{\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\bigl{(}\delta_{10}(E,F)x_{2}(t)+\delta_{20}(E,F)x_{1}(t)+\delta_{30}(E,F)\bigr{)},$

so $E$ is Gieseker (semi)stable if and only if

$\bigl{(}\delta_{10}(E,F),\delta_{20}(E,F),\delta_{30}(E,F)\bigr{)}>\leavevmode\nobreak\
(\geq)\leavevmode\nobreak\ 0$

in the lexicographic order. Similarly, a torsion free sheaf $E$ is Gieseker
(semi)stable if and only if every quotient sheaf $E\twoheadrightarrow G$
satisfies

$\bigl{(}\delta_{10}(E,G),\delta_{20}(E,G),\delta_{30}(E,G)\bigr{)}<\leavevmode\nobreak\
(\leq)\leavevmode\nobreak\ 0$

in the lexicographic order.

In what follows, it will be important to consider another notion of stability
for torsion free sheaves, which is equivalent to the notion of stability in
the category $\mathcal{C}oh_{3,1}(X)$ in the sense of [17, Definition 1.6.3].

######  Definition 2.17.

A torsion free sheaf $E$ on $X$ is said to be _$\mu_{\leq 2}$ -(semi)stable_
if every proper, nontrivial subsheaf $F\hookrightarrow E$ satisfies
$(\delta_{10}(E,F),\delta_{20}(E,F))>\leavevmode\nobreak\
(\geq)\leavevmode\nobreak\ 0$ in the lexicographic order.

For the sheaf $E^{\prime}$ defined as in display (19), we observe that:

  * (i)

$E$ is $\mu$-(semi)stable if and only if $E^{\prime}$ is $\mu$-(semi)stable;

  * (ii)

$E$ is $\mu_{\leq 2}$-(semi)stable if and only if $E^{\prime}$ is $\mu_{\leq
2}$-(semi)stable.

The first claim follows from the fact that $E^{\prime*}=E^{*}$. For the second
claim, note that $\operatorname{ch}_{\leq
2}(E^{\prime})=\operatorname{ch}_{\leq 2}(E)$, since $E^{\prime}/E$ is
0-dimensional; in addition, any subsheaf $F^{\prime}\hookrightarrow
E^{\prime}$ will induce a subsheaf $F\hookrightarrow E$ such that
$\operatorname{ch}_{\leq 2}(F^{\prime})=\operatorname{ch}_{\leq 2}(F)$.

Clearly, one has the following chains of implications:

$\mu\textrm{-stable}\leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \mu_{\leq
2}\textrm{-stable}\leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{Gieseker
stable}\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow$
$\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{Gieseker
semistable}\leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \mu_{\leq
2}\textrm{-semistable}\leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \mu\textrm{-semistable}.$

######  Example 2.18.

It is not hard to find explicit examples that show that the reverse
implications do not hold in general. Indeed, for $X={\mathbb{P}^{3}}$, let $S$
be a rank 2 reflexive sheaf given as an extension of an ideal sheaf of a line
$L\subset{\mathbb{P}^{3}}$ by ${\mathcal{O}}_{\mathbb{P}^{3}}$; note that
$\operatorname{ch}(S)=(2,0,-1,1)$. Let $C\subset{\mathbb{P}^{3}}$ be curve,
and consider an epimorphism $\varphi:S\twoheadrightarrow\mathcal{O}_{C}(k)$;
define $E_{\varphi}:=\ker\varphi$.

  1. (1)

if $C$ is a line not intersecting $L$ and $k>0$, then $E_{\varphi}$ is
$\mu_{\leq 2}$-semistable, but not Gieseker semistable;

  2. (2)

if $C$ is a conic not intersecting $L$ and $k=0$, then $E_{\varphi}$ is
$\mu_{\leq 2}$-stable, but not $\mu$-stable;

  3. (3)

if $C$ is a line intersecting $L$ in a single point, then $E_{\varphi}$ is
Gieseker stable, but not $\mu_{\leq 2}$-stable.

Finally, we also recall the following notion of stability for sheaves of
dimension 2, compare with [17, Definition 1.6.8].

######  Definition 2.19.

A torsion sheaf $T\in\mathcal{C}oh(X)_{2}$ on $X$ is said to be
$\hat{\mu}$-(semi)stable if it is pure, and every subsheaf $U\subset T$
satisfies

$\hat{\mu}(U):=\dfrac{\operatorname{ch}_{2}(U)}{\operatorname{ch}_{1}(U)}<\leavevmode\nobreak\
(\leq)\leavevmode\nobreak\
\dfrac{\operatorname{ch}_{2}(T)}{\operatorname{ch}_{1}(T)}=\hat{\mu}(T).$

Setting
$\Lambda_{2}(E,F):=(\delta_{10}(E,F),\delta_{20}(E,F),\delta_{21}(E,F))$, we
introduce the following version of the Harder–Narasimhan filtration, which
will be useful later on.

######  Lemma 2.20.

Every sheaf $E$ admits a filtration $0=E_{0}\subseteq
E_{1}\subset\cdots\subset E_{n}=E$ satisfying the following conditions:

  1. (1)

$\dim E_{1}\leq 1$, and $\dim E_{k}\geq 2$ for every $k\geq 2$;

  2. (2)

each factor $G_{k}:=E_{k}/E_{k-1}$ for $k\geq 2$ is either
$\hat{\mu}$-semistable
(if $\dim E_{k}/E_{k-1}=2$) or $\mu_{\leq 2}$-semistable (if $E_{k}/E_{k-1}$
is torsion free), and

$\Lambda_{2}(G_{k},G_{k+1})>0\text{ for }k=1,\dots n-1.$

###### Proof.

Let $E_{1}$ be the maximal subsheaf of $E$ of dimension at most 1, so that
$F:=E/T$ has dimension at least 2. Note that this $E_{1}$ might be the zero
sheaf.

Next, let $F=F_{0}\twoheadrightarrow F_{1}\twoheadrightarrow\cdots\subset
F_{n}=0$ be the Harder–Narasimhan cofiltration of $F$ as an object in the
category $\mathcal{C}oh_{3,1}(X)$, in the sense of [17, Theorem 1.6.7]; each
$F_{k}$ is a sheaf of dimension at least 2 either $\hat{\mu}$\- or $\mu_{\leq
2}$-semistable. Composing with $E\twoheadrightarrow F$ provides our required
cofiltration and hence filtration in the usual way. ∎

##  3\. Regions of $\mathbb{H}$

Let $v$ be a real numerical Chern character with $v_{0}\neq 0$ and satisfying
$Q^{\rm tilt}(v)\geq 0$. It follows that the curve

$\Theta_{v}:=\\{\rho_{v}(\alpha,\beta)=0\\}$

is a hyperbola in $\mathbb{H}$, centered around the vertical line
$M_{v}:=\\{\beta=\mu(v)\\}$; explicitly:

(23) $\rho_{v}(\alpha,\beta)=0\leavevmode\nobreak\ \leavevmode\nobreak\
\Longleftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\left(\beta-\mu(v)\right)^{2}-\alpha^{2}=\frac{Q^{\rm tilt}(v)}{v_{0}^{2}}.$

When $v$ fails the Bogomolov inequality we still occasionally still consider
$\Theta_{v}$ but now it is a single branch hyperbola cutting the
$\alpha$-axis.

The hyperbola $\Theta_{v}$ divides $\mathbb{H}$ up into three regions
$R^{-}\sqcup R^{0}\sqcup R^{+}$ defined as follows:

$\displaystyle R^{-}_{v}$
$\displaystyle=\\{(\alpha,\beta)\mid\text{$\rho_{v}(\alpha,\beta)>0$ and
$\beta<\mu(v)$}\\},$ $\displaystyle R^{0}_{v}$
$\displaystyle=\\{(\alpha,\beta)\mid\text{$\rho_{v}(\alpha,\beta)<0$ or
$\rho_{v}(\alpha,\beta)=0$ and $\beta<\mu(v)$}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}$ $\displaystyle R^{+}_{v}$
$\displaystyle=\\{(\alpha,\beta)\mid\text{$\rho_{v}(\alpha,\beta)\geq 0$ and
$\beta>\mu(v)$}\\}.$

Notice that $R^{-}_{v}$ does not include any of $\Theta_{v}$ whereas $R^{0}$
and $R^{+}_{v}$ include the branch of $\Theta_{v}$ to their left. In addition,
$R^{0}_{v}$ is split in half by the vertical line $\\{\beta=v_{1}/v_{0}\\}$;
we shall denote these regions by $R^{0+}_{v}$ and $R^{0-}_{v}$ (to include the
vertical line) to the left and right, respectively. These regions are
illustrated in Figure 1.

######  Remark 3.1.

The various cases which arise in Proposition 2.12 can be rewritten usefully in
terms of these regions. In the notation of the statement we have $F\in
R^{-}_{\operatorname{ch}(F)}$ and then the cases are

  1. (1)

$G\in R^{-}_{\operatorname{ch}(G)}$,

  2. (2)

$G\in R^{0+}_{\operatorname{ch}(G)}$,

  3. (3)

$G\in R^{0-}_{\operatorname{ch}(G)}$.

The categories $\mathcal{A}^{\alpha,\beta}$ along the curve $\Theta_{v}$
satisfy some strong conditions. These form the basis for Theorem 4.21 (see
also [34, Lemma 6.2 and 6.4]).

######  Proposition 3.2.

If $A\in\mathcal{A}^{\alpha,\beta}(X)$ with $\operatorname{ch}(A)=v$ and
$(\alpha,\beta)\in\Theta_{v}$, then, for every $s>0$, $A$ is
$\lambda_{\alpha,\beta,s}$-semistable,
${\mathcal{H}}^{0}_{\beta}(A)\in\mathcal{C}oh(X)_{0}$, and
${\mathcal{H}}^{-1}_{\beta}(A)$ is $\nu_{\alpha,\beta}$-semistable.

Conversely, if $B\in\mathcal{B}^{\beta}(X)$ with $\operatorname{ch}_{\leq
2}(B)=\operatorname{ch}_{\leq 2}(v)$ is $\nu_{\alpha,\beta}$-semistable for
some $(\alpha,\beta)\in\Theta_{v}$ and $P\in\mathcal{C}oh(X)_{0}$, then every
non-trivial extension $A\in\operatorname{Ext}^{1}(P,B[1])$ with
$\operatorname{ch}(A)=v$ belongs to $\mathcal{A}^{\alpha,\beta}(X)$ and is
$\lambda_{\alpha,\beta,s}$-semistable for every $s>0$.

###### Proof.

Suppose $F\hookrightarrow A$ is a subobject in
$\mathcal{A}^{\alpha,\beta}(X)$. Since $\rho_{A}(\alpha,\beta)=0$, we have
that $\rho_{F}(\alpha,\beta)=0$ as well, thus
$\lambda_{\alpha,\beta,s}(F)=\infty$. Similarly for quotient objects. Hence,
$E$ is $\lambda_{\alpha,\beta,s}$-semistable for every $s>0$.

Now write $A$ as

$0\to A_{1}[1]\to A\to A_{0}\to 0$

in $\mathcal{A}^{\alpha,\beta}(X)$. As in the argument above, we also conclude
that $\rho_{A_{0}}(\alpha,\beta)=0$. However, this contradicts
$\nu_{\alpha,\beta}^{-}(A_{0})>0$ unless $A_{0}\in\mathcal{C}oh(X)_{0}$. Since
$A_{1}\in\mathcal{F}_{\alpha,\beta}$, we have
$\nu_{\alpha,\beta}^{+}(A_{1})\leq 0=\nu_{\alpha,\beta}(A_{1})$ and so $A_{1}$
is $\nu_{\alpha,\beta}$-semistable.

For the second statement, note that every sub-object $F\hookrightarrow B$
within $\mathcal{B}^{\beta}(X)$ satisfies
$\nu_{\alpha,\beta}(F)\leq\nu_{\alpha,\beta}(B)=0$, since $B$ is
$\nu_{\alpha,\beta}$-semistable and $(\alpha,\beta)\in\Theta_{v}$. It follows
that $B\in\mathcal{F}_{\alpha,\beta}$, thus
$B[1]\in\mathcal{A}^{\alpha,\beta}(X)$. Since $P\in\mathcal{C}oh(X)_{0}$, then
$P\in\mathcal{T}_{\alpha,\beta}$ for every $(\alpha,\beta)\in\mathbb{H}$, thus
any extension $A\in\operatorname{Ext}^{1}(P,B[1])$ does belong to
$\mathcal{A}^{\alpha,\beta}(X)$. For every $s>0$,
$\lambda_{\alpha,\beta,s}(A)=+\infty$ because $(\alpha,\beta)\in\Theta_{v}$,
thus $A$ must be $\lambda_{\alpha,\beta,s}$-semistable. ∎

###  3.1. Variation of $\mathcal{A}^{\alpha,\beta}$ along paths

Fix a numerical Chern character $v$ and a point $(\alpha,\beta)\in R^{-}_{v}$
and some numerical Chern character $v$ corresponding to an object $A$ of
$\mathcal{A}^{\alpha,\beta}$. Suppose $\gamma:[0,b)\to\mathbb{H}$ is a path in
the upper half plane with $\gamma(0)=(\alpha,\beta)$. Assume first that
$\beta(t):=\pi_{2}(\gamma(t))$ is constant and $\alpha(t):=\pi_{1}(\gamma(t))$
is monotonic increasing. As before we write $A_{0}$ and $A_{1}$ for the
$\mathcal{B}^{\beta}$ cohomology of $A$. Observe that the Harder–Narasimhan
factors of $A_{0}$ and $A_{1}$ with respect to $\nu_{\alpha,\beta}$ are
locally constant (they change only on $\lambda$-walls corresponding to the
factors). Let $A^{-}_{0}$ be the Harder–Narasimhan factor of $A_{0}$ with
$\nu_{\alpha,\beta}(A^{-}_{0})=\nu_{\alpha,\beta}^{-}(A_{0})$. Then as we move
along $\gamma$ (upwards), $A$ remains in $\mathcal{A}^{\alpha,\beta}$ until we
cross $\Theta^{-}_{\operatorname{ch}(A^{-}_{0})}$ (in a Harder–Narasimhan
chamber) at which point $\mathcal{H}^{1}_{\mathcal{A}}(A)=A^{-}_{0}$. But
since $\nu^{+}_{\alpha,\beta}(A_{1})<0$ and we are traversing the path
upwards, for any Harder–Narasimhan factor $A^{\prime}_{1}$ of $A_{1}$ with
$\operatorname{ch}(A)^{\prime}_{1})=v$, we start in the region $R^{0}_{v}$.
Hence we can only cross $\Theta^{+}_{\operatorname{ch}(A_{1}^{\prime})}$ but
this does not take $A$ out of $\mathcal{A}^{\gamma(t)}$. The reverse happens
if $\alpha(t)$ is monotonic decreasing. In this case, if $A^{+}_{1}$ is the
Harder–Narasimhan factor corresponding to $\nu_{\gamma(T)}^{+}(A_{1})$ then as
we cross $\Theta^{+}_{\operatorname{ch}(A^{+}_{1})}$, $A$ moves out of
$\mathcal{A}^{\gamma(t)}$ with $\mathcal{H}_{\mathcal{A}}^{-1}(A)=A^{+}_{1}$.
Otherwise, $A$ remains in $\mathcal{A}^{\gamma(t)}$.

This also applies to horizontal paths but now we need to consider what happens
when we cross $M_{u}$ corresponding to $u$ equal to Chern characters of
$\mu$-Harder-Harasimhan factors of $A_{00}$, $A_{10}$, $A_{01}$ and $A_{11}$.
In this case, we see that crossing these lines does not affect $A$. To see
this consider one of the cases where $\mu_{-}(A_{10})=\mu(B)$ and
$\widetilde{A}_{10}=\ker(A_{10}\to B)$ in $\mathcal{C}oh(X)$. Then we have an
octahedron

$\textstyle{A_{1}}$$\textstyle{A_{10}}$$\textstyle{B}$$\textstyle{A_{11}[2]}$$\textstyle{Q[1]}$$\textstyle{A_{1}[1]}$$\textstyle{\widetilde{A}_{10}[1]}$$\textstyle{A_{10}[1]}$$\textstyle{B[1]}$

for some new object $Q$. From this it follows that $Q\in\mathcal{B}^{\beta}$
and that $\nu^{*}_{\alpha,\beta}(Q)\leq 0$ as $Q$ is a
$\mathcal{B}^{\beta}$-subobject of $A_{1}$. So
$Q[1]\in\mathcal{A}^{\alpha,\beta}$. Now we add the triangle $A_{1}[1]\to A\to
A_{0}$ to the octahedra by inserting a second column and second row to form a
higher octahedron (see Section 2.3).

(24)
$\textstyle{A_{0}[-2]}$$\textstyle{A_{1}}$$\textstyle{A_{10}}$$\textstyle{B}$$\textstyle{A[-1]}$$\textstyle{C^{\prime}[-1]}$$\textstyle{D[-1]}$$\textstyle{A_{0}[-1]}$$\textstyle{A_{11}[2]}$$\textstyle{Q[1]}$$\textstyle{A_{1}[1]}$$\textstyle{\widetilde{A}_{10}[1]}$$\textstyle{A_{10}[1]}$$\textstyle{B[1]}$

with a new object $D$ and $C^{\prime}$ is from (14). The triangle $B[1]\to
D\to A_{0}$ implies that $D\in\mathcal{A}^{\alpha,\beta}$. We also see that
$Q[1]\to A\to D$ is a triangle and so $A\in\mathcal{A}^{\alpha,\beta}$. Hence,
$A$ remains in $\mathcal{A}^{\alpha,\beta}$ through $M_{u}$. The other cases
follow similarly.

In conclusion we have proved

######  Theorem 3.3.

Suppose $(\alpha_{0},\beta_{0})\in\mathbb{H}$ and
$A\in\mathcal{A}^{\alpha_{0},\beta_{0}}$. Let $A^{-}_{0}$ be the
Harder–Narasimhan factor of $A_{0}\in\mathcal{B}^{\beta_{0}}$ with least
$\nu_{\alpha_{0},\beta_{0}}$ and $A^{+}_{1}$ the Harder–Narasimhan factor of
$A_{1}$ with the greatest $\nu_{\alpha_{0},\beta_{0}}$. Suppose also that
$\Theta^{+}_{\operatorname{ch}(A^{+}_{1})}$ do not contain
$(\alpha_{0},\beta_{0})$. Then there is an open neighbourhood $U$ of
$(\alpha_{0},\beta_{0})$ such that $A\in\mathcal{A}^{\alpha,\beta}$ for all
$(\alpha,\beta)\in U$. Furthermore, any boundary of $U$ consists of
$\Theta^{+}_{w}$ curves associated to subobjects in $\mathcal{B}^{\beta}$ or
$\Theta^{-}_{w}$ curves of quotient objects.

###  3.2. The $\Gamma$ Curve

The next step is to understand the vanishing locus of $\lambda$-slope for a
given numerical Chern vector $v$, and to study its relative position with
respect to $\Theta_{v}$.

######  Definition 3.4.

For any real numerical Chern character $v$ and $s\geq 0$, we define the curve

$\Gamma_{v,s}:=\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
\mid\leavevmode\nobreak\ \tau_{v,s}(\alpha,\beta)=0\\}.$

In addition, we set $\Gamma_{v,s}^{\bullet}:=\Gamma_{v,s}\cap
R_{v}^{\bullet}$, with $\bullet=-,0,+$.

The curve $\Gamma_{v,s}$ is given explicitly by

(25)
$v_{0}\beta^{3}-3v_{1}\beta^{2}+\left(6v_{2}-(6s+1)v_{0}\alpha^{2}\right)\beta+\left((6s+1)v_{1}\alpha^{2}-6v_{3}\right)=0.$

If $v_{0}\neq 0$, equation (25) is cubic on $\beta$, hence $\Gamma_{v,s}$ has
at most three connected components. In fact, there exists $\alpha_{0}>0$ such
that

$\Gamma_{v,s}\cap\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
|\leavevmode\nobreak\ \alpha>\alpha_{0}\\}$

has exactly three connected components. Two of these components are asymptotic
to lines $\alpha\sqrt{6s+1}=\pm(\beta-v_{1}/v_{0})$, and therefore lie in the
regions $R^{\pm}_{v}$; these are the curves $\Gamma_{v,s}^{\pm}$ in Definition
3.4 above. The third component, namely $\Gamma_{v,s}^{0}$, either coincides
with or is asymptotic to the vertical line $\beta=v_{1}/v_{0}$.

######  Example 3.5.

When $v=(m,0,-n,0)$ for some $m,n>0$, then equation (25) simplifies to

$\left(m\beta^{2}-(6s+1)m\alpha^{2}-6n\right)\beta=0,$

so $\Gamma_{v,s}$ consists of exactly three components: $\Gamma^{\pm}_{v,s}$
are the two branches of the hyperbola $\beta^{2}-(6s+1)\alpha^{2}=6n/m$, and
$\Gamma^{0}_{v,s}$ coincides with the vertical axis $\\{\beta=0\\}$.

In the limit $s\to 0$, assuming also that the Bogomolov inequality
$v_{1}^{2}\geq 2v_{0}v_{2}$ holds for $v$, the curves $\Gamma^{\pm}_{v,s=0}$
are co-asymptotic with the two branches of the the hyperbola $\Theta_{v}$.

It turns out that the position of the different branches of cubic curve
$\Gamma_{v,s}$ relative to the two branches of the hyperbola $\Theta_{v}$ has
an important geometric meaning, which is unraveled in the following two
statements.

######  Proposition 3.6.

Let $v$ be a real numerical Chern character with $v_{0}\neq 0$ and satisfying
the Bogomolov inequality (3). If, for each $(\alpha,\beta)\in\mathbb{H}$,
there is an object $E\in\mathcal{A}^{\alpha,\beta}(X)$ with
$\operatorname{ch}(E)=v$, then $\Gamma_{v,s}$ consists of exactly three
connected components:

$\Gamma^{-}_{v,s}\sqcup\Gamma^{0}_{v,s}\sqcup\Gamma^{+}_{v,s}$

and,

(26)
$q(E):=3v_{1}^{2}v_{2}^{2}-6v_{1}^{3}v_{3}+18v_{0}v_{1}v_{2}v_{3}-8v_{0}v_{2}^{3}-9v_{0}^{2}v_{3}^{2}\geq
0.$

###### Proof.

The hypotheses imply that the three asymptotic branches
$\Gamma^{\bullet}_{v,s}$ remain in their respective regions $R^{\bullet}_{v}$
for $\bullet=+,-,0$ all the way down to the horizontal axis $\\{\alpha=0\\}$,
and so we have three distinct irreducible components as given.

Recall that a cubic equation admits three distinct real roots exactly when its
discriminant is positive. Regarding equation (25) as a cubic in $\beta$, an
easy computation gives its discriminant as ($108$ times)

(27)
$\begin{split}q_{\alpha}(E):=\alpha^{6}v_{0}^{4}(6s+1)^{3}+9\alpha^{4}v_{0}^{2}(6s+1)^{2}\left(v_{1}^{2}-2v_{0}v_{1}\right)+\\\
27\alpha^{2}(6s+1)\left(v_{1}^{2}-2v_{0}v_{1}\right)^{2}+\\\
27\left(3v_{1}^{2}v_{2}^{2}-6v_{1}^{3}v_{3}+18v_{0}v_{1}v_{2}v_{3}-8v_{0}v_{2}^{3}-9v_{0}^{2}v_{3}^{2}\right).\end{split}$

Setting $\alpha=0$ yields (26). Finally we need to know that $q_{\alpha}$ can
only change sign once as $\alpha$ varies. To see that observe that

$\frac{dq_{\alpha}(E)}{d\alpha^{2}}=3(6s+1)\bigl{(}v_{0}^{2}(6s+1)\alpha^{2}+3(v_{1}^{2}-2v_{0}v_{2})\bigr{)}^{2}>0.$

So as a cubic in $\alpha^{2}$, $q_{\alpha}$ has a single point of inflection
and so can change sign at most once. Note that, since $v_{1}^{2}\geq
2v_{0}v_{1}$, the point of inflection does not occur for any $\alpha>0$. ∎

We also make use of the following:

######  Definition 3.7.

Suppose $v_{0}\neq 0$ and $Q^{\rm tilt}(v)\geq 0$. We define the region
$R^{L}_{v,s}$ to be

$\\{(\alpha,\beta)\in\mathbb{H}\mid\alpha<\alpha_{0},\text{ whenever
}(\alpha_{0},\beta)\in\Gamma^{-}_{v,s}\\}\cap R^{-}_{v}.$

Alternatively,

$R^{L}_{v,s}=\\{(\alpha,\beta)\in\mathbb{H}\mid
v_{0}\tau_{v,s}(\alpha,\beta)\geq 0\\}\cap R^{-}_{v}.$

Note that, if $q(v)<0$, then $\Gamma_{v,s}$ intersects $\Theta_{v}$. This is
illustrated in Figure 2. On the other hand, away from $\Theta_{v}$ and
$M_{v}$, $\Gamma^{-}_{v,1/3}$ is strictly monotonic. This is because
$\partial_{\beta}\operatorname{ch}^{\alpha,\beta}_{3}(v)=-\operatorname{ch}^{\alpha,\beta}_{2}(v)$.

######  Proposition 3.8.

Let $v$ be a real numerical Chern character with $v_{0}\neq 0$ and satisfying
the Bogomolov inequality (3). If $q(v)<0$ then $\Gamma_{v,s}$ intersects the
left branch of $\Theta_{v}$ if and only if

$v_{0}^{2}v_{3}>v_{0}v_{1}v_{2}-\frac{1}{3}v_{1}^{3}$

###### Proof.

This is easiest to see by considering the sign of
$\lambda_{\alpha,\beta,s}(v)$ for very small $\alpha$ (we can just take
$\alpha=0$). Observe that for $\beta\to-\infty$ the sign is positive and it
changes sign when we cross either $\Gamma_{v,s}$ or $\Theta_{v}$. If $q(v)<0$
then $\Gamma_{v,s}$ crosses the $\beta$ axis exactly once. Observe that
$\lambda_{0,\beta,s}(v)=0$ for $\beta>\mu(v)$ if and only if $\Gamma_{v,s}$
intersects the left branch of $\Theta_{v}$. We can easily read off the
numerical condition from this by writing $\tau_{v,s}(m,c,d,e)$ at $\alpha=0$
as

$6v_{3}-\frac{6v_{1}v_{2}}{v_{0}}+\frac{2v_{1}^{3}}{v_{0}^{2}}+\left(\beta-\frac{v_{1}}{v_{0}}\right)\left(\frac{3v_{1}^{2}}{v_{0}}-6v_{2}\right)-v_{0}\left(\beta-\frac{v_{1}}{v_{0}}\right)^{3}$

∎

Figure 2. This graph contains an example of the situation described in
Proposition 3.8 for $v=(3,4,2,2/3)$, which is the Chern character of the
tangent bundle of ${\mathbb{P}^{3}}$. We set $s=1/3$. The curve $\Gamma_{v,s}$
(in red) intersects the left branch of the hyperbola $\Theta_{v}$ (in blue) at
the point $(\alpha,\beta)\approx(0.27,0.62)$. In addition, $\Gamma_{v,s}^{0}$
is asymptotic to the vertical line $M_{v}$, while the component
$\Gamma_{v,s}^{+}$ does not intersect $\Theta_{v}^{+}$. We also illustrate
Theorem 4.22 here, with the vanishing $\nu$\- and $\lambda$-walls represented
by the curves $\Xi_{u,v}$ (in magenta) and $\Upsilon_{u,v,s}$ (in black),
respectively, where $u=\operatorname{ch}({\mathcal{O}}_{\mathbb{P}^{3}}(1))$.
Finally, Lemma 4.5 is also represented, since both of these walls cross
$\Theta^{-}_{v}$ at the same point.

The following is an easy exercise.

######  Proposition 3.9.

The quartic form $q$ in equation (26) is invariant under $E\mapsto
E\otimes{\mathcal{O}}_{X}(k)$ and $E\mapsto E^{\vee}$.

##  4\. Walls

###  4.1. $\nu$-walls

Given a real numerical Chern character $v$, a curve
$\Xi_{u,v}\subset\mathbb{R}^{+}\times\mathbb{R}$ is called a _numerical
$\nu$-wall_ for $v$ if there is a real numerical Chern character $u$ such that

$\Xi_{u,v}:=\\{t_{u,v}(\alpha,\beta)=0\\},\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm where}$ (28)
$t_{u,v}(\alpha,\beta):=\rho_{u}(\alpha,\beta)\operatorname{ch}_{1}^{\beta}(v)-\rho_{v}(\alpha,\beta)\operatorname{ch}_{1}^{\beta}(u)=\Delta_{21}(\alpha,\beta),$

compare with the notation introduced in display (11). Note that this coincides
with the numerator of the difference
$\nu_{\alpha,\beta}(u)-\nu_{\alpha,\beta}(v)$, so its vanishing locus is
precisely where it changes sign. Observe that numerical $\nu$-walls are
invariant as subsets of the upper half plane under the changes of coordinates
$u\mapsto\kappa v+u$, $u\mapsto\zeta u$ and $(u,v)\mapsto(v,u)$ for any real
constants $\kappa$ and $\zeta\neq 0$.

######  Lemma 4.1.

Suppose that $v_{0}>0$, $v$ satisfies the Bogomolov inequality and
$u=(0,1,x,y)$ for real $x$ and $y$. Then the following are equivalent

  1. (1)

$\Xi_{u,v}$ is empty.

  2. (2)

for all real $\kappa$, $\kappa v+u$ satisfies the Bogomolov inequality.

  3. (3)

$(x-\mu(v))^{2}\leq\mu(v)^{2}-2v_{2}/v_{0}$.

  4. (4)

$\delta_{02}(u,v)^{2}\leq 4\delta_{01}(u,v)\delta_{12}(u,v)$.

###### Proof.

The condition in (3) is the discriminant condition for (2) which is a
quadratic condition on $\lambda$ and it happens to be the same as the
quadratic condition for $t_{u,v}(\alpha,\beta)=0$ to have solutions. ∎

The structure of numerical $\nu$-walls was described in [20] for the case of
projective surfaces, but the same results also hold for projective threefolds,
see for instance [34, Theorem 3.3] and [24, Remark 9.2]. Precisely, numerical
$\nu$-walls are non-intersecting semicircles centered along the horizontal
axis, and cross the hyperbola $\Theta_{v}$ at their maximum point. The
semicircles tend to a fixed point $(0,C_{0})$ on the $\beta$-axis as the
radius tends to zero.

######  Definition 4.2.

A numerical $\nu$-wall $\Xi_{u,v}$ is called an _actual $\nu$-wall_ if for
some $(\alpha_{0},\beta_{0})\in\Xi_{u,v}$ there is an object
$B\in\mathcal{B}^{\beta_{0}}$ with $\operatorname{ch}(B)=v$ and a
$\nu_{\alpha_{0},\beta_{0}}$-semistable sub-object $F\hookrightarrow B$ with
$\operatorname{ch}(F)=u$ such that the quotient $E/F$ is also
$\nu_{\alpha_{0},\beta_{0}}$-semistable.

It then follows that the same property holds for every point of $\Xi_{u,v}$,
see [34, Theorem 3.3]. Actual $\nu$-walls are _locally finite_ (that is, any
compact subset of the upper half plane intersects only finitely many numerical
$\nu$-walls), see [24, Lemma 6.23], and there are $C_{\rm max}\in\mathbb{R}$
and $R_{\rm max}>0$ such that every numerical $\nu$-wall is contained in the
semicircle centered at $(0,C_{\rm max})$ with radius $R_{\rm max}$. In
particular, there is $\overline{\alpha}>0$ for which there are only finitely
many actual $\nu$-walls above the horizontal line
$\\{\alpha=\overline{\alpha}\\}$. However, actual $\nu$-walls may accumulate
towards the point $\Theta_{v}\cap\\{\alpha=0\\}$, see [27] and [39] for
examples on abelian surfaces.

Another observation is that, since $\nu$-walls are nested, there can be at
most two vanishing $\nu$-walls for a given Chern character $v$, one on each
side of the vertical line $M_{v}$.

Combining Lemma 4.1 with Proposition 2.12 we can deduce a the following
technical result that will be useful later on. For a Chern character $v$, we
let

$Y_{v}=\\{\gamma\mid\gamma=\delta_{02}(u,v)/\delta_{01}(u,v)\text{ for some
actual $\nu$-wall given by $u$}\\}$

Then we have:

######  Proposition 4.3.

Suppose $\operatorname{ch}(E)=v=(a,b,c/2,d/6)$ and
$\operatorname{ch}(F)=u=(r,x,y/2,z/6)$ are two numerical Chern characters
which satisfy the Bogomolov inequality, $a>0$ and $\delta_{01}(u,v)\neq 0$. If
$F\hookrightarrow E\twoheadrightarrow G$ is a short exact sequence in
$\mathcal{B}^{\beta}$ for $\beta<\mu(E)$ then if
$\delta_{02}(u,v)/\delta_{01}(u,v)\not\in Y_{v}$,

$\delta_{01}(u,v)\delta_{12}(u,v)\geq 0\text{ and }|\delta_{02}(u,v)|\leq
2\sqrt{\delta_{01}(u,v)\delta_{12}(u,v)}.$

###  4.2. $\lambda$-walls

Given in a real numerical Chern character $v$, a curve
$\Upsilon_{u,v,s}\subset\mathbb{R}^{+}\times\mathbb{R}$ is called a _numerical
$\lambda$-wall_ for $v$ if there is a real numerical Chern character $u$ such
that

$\Upsilon_{u,v,s}:=\\{f_{u,v,s}(\alpha,\beta)=0\\},\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm where}$ (29)
$f_{u,v,s}(\alpha,\beta):=\tau_{u,s}(\alpha,\beta)\rho_{v}(\alpha,\beta)-\tau_{v,s}(\alpha,\beta)\rho_{u}(\alpha,\beta).$

Note that this coincides with the numerator of the difference
$\lambda_{\alpha,\beta,s}(u)-\lambda_{\alpha,\beta,s}(v)$, so its vanishing
locus is precisely where it changes sign. Comparing with the notation
introduced in display (11), we remark that

$f_{u,v,s}(\alpha,\beta)=\Delta_{32}-\alpha^{2}(s-1/3)\Delta_{21}$

thus, in particular, $f_{u,v,1/3}(\alpha,\beta)=\Delta_{32}(\alpha,\beta)$.

More explicitly, using the notation from display (11)

$\delta_{ij}=\delta_{ij}(u,v)=\operatorname{ch}_{i}(u)\operatorname{ch}_{j}(v)-\operatorname{ch}_{j}(u)\operatorname{ch}_{i}(v),$

we have:

(30)
$\begin{split}f_{u,v,s}(\alpha,\beta)=&\dfrac{6s+1}{12}\delta_{10}\alpha^{4}+\\\
&+\left(\dfrac{3s-1}{6}\delta_{10}\beta^{2}-\dfrac{3s-1}{3}\delta_{20}\beta+\dfrac{6s+1}{6}\delta_{21}-\dfrac{1}{2}\delta_{30}\right)\alpha^{2}+\\\
&+\left(\dfrac{1}{12}\delta_{10}\beta^{4}-\dfrac{1}{3}\delta_{20}\beta^{3}+\dfrac{1}{2}(\delta_{30}+\delta_{21})\beta^{2}-\delta_{31}\beta+\delta_{32}\right).\end{split}$

Numerical $\lambda$-walls that are bounded as subsets of $\mathbb{H}$ have a
simple numerical characterization.

######  Proposition 4.4.

Let $v$ be a real numerical Chern character with $v_{0}\neq 0$ and satisfying
the Bogomolov inequality. A numerical $\lambda$-wall $\Upsilon_{u,v,s}$ for
$v$ is bounded if and only if $\delta_{10}(u,v)\neq 0$.

Notice that if $\Upsilon_{u,v,s}$ is bounded for some value of the parameter
$s$, then it is bounded for every value of $s$.

###### Proof.

Assuming $\delta_{10}(u,v)\neq 0$, turn $f_{u,v,s}(\alpha,\beta)$ into a
homogeneous polynomial of degree 4 by adding a new variable $\gamma$; let us
denote this new function by $\overline{f_{u,v,s}}(\alpha,\beta,\gamma)$. The
set $\\{\overline{f_{u,v,s}}(\alpha,\beta,\gamma)=0\\}$ can be regarded as a
hypersurface in $\mathbb{R}{\mathbb{P}^{2}}$, and $\Upsilon_{u,v,s}$ is
bounded if and only if $\\{\overline{f_{u,v,s}}(\alpha,\beta,\gamma)=0\\}$
does not intersect the line at infinity $\\{\gamma=0\\}$.

Comparing with equation (30), note that

$\overline{f_{u,v,s}}(\alpha,\beta,0)=\dfrac{\delta_{10}}{12}\left(6s\alpha^{2}(\alpha^{2}+\beta^{2})+(\alpha^{2}-\beta^{2})^{2}\right).$

Since the right hand side vanishes if and only if $\alpha=\beta=0$, we
conclude that $\\{\overline{f_{u,v,s}}(\alpha,\beta,\gamma)=0\\}$ does not
intersect the line at infinity $\\{\gamma=0\\}$, hence $\Upsilon_{u,v,s}$ is
bounded.

Next, if $\delta_{10}(u,v)=0$ and $\delta_{20}(u,v)\neq 0$, then the
homogenisation of $f_{u,v,s}(\alpha,\beta)$ yields a polynomial
$\overline{f_{u,v,s}}(\alpha,\beta,\gamma)$ of degree 3. Comparing with
equation (30), note that

(31)
$\overline{f_{u,v,s}}(\alpha,\beta,0)=-\dfrac{\delta_{20}}{3}\left((3s-1)\alpha^{2}+\beta^{2}\right)\beta,$

so $\overline{f_{u,v,s}}(1,0,0)=0$, thus $\Upsilon_{u,v,s}$ is not bounded.

Finally, if $\delta_{10}(u,v)=\delta_{20}(u,v)=0$ also, then
$f_{u,v,s}(\alpha,\beta)$ becomes a constant multiple of
$\rho_{u}(\alpha,\beta)=\rho_{v}(\alpha,\beta)$, hence the curve
$\Upsilon_{u,v,s}$ coincides with the hyperbola $\Theta_{V}$ which is not
bounded. ∎

Let us now analyse unbounded numerical $\lambda$-walls. Note that if
$\delta_{10}(u,v)=0$, then the expression in equation (30) reduces to a cubic
polynomial in $\beta$, with coefficients depending on $\alpha^{2}$, so it
always has a real root; in other words, unbounded numerical $\lambda$-walls
intersect every horizontal line. Furthermore, unbounded numerical
$\lambda$-walls with different values of the parameter $s$ can look very
different, see Figure 3; in particular an unbounded numerical $\lambda$-wall
might not be connected and one of its connected components may be bounded, see
Figure 4.

Figure 3. The numerical $\lambda$-wall $\Upsilon_{u,v,s}$ for $v=(2,0,-3,0)$
and $u=(0,0,-1,1)$ is plotted for $s=0.01$ (red curve), which has two separate
connected components, and for $s=2.5$ (purple curve), which has a single
connected component. Both curves are asymptotic to the vertical line
$M_{v}=\\{\beta=\mu(v)=0\\}$, while the red curve is also asymptotic both
branches of $\Theta_{v}$ (in blue).

Next, we observe an interesting relation between a numerical $\lambda$-wall
$\Upsilon_{u,v,s}$ and its the associated numerical $\nu$-wall $\Xi_{u,v}$: if
one intersects the hyperbola $\Theta_{v}$, then so does the other.

######  Lemma 4.5.

Let $v$ be a real numerical Chern character satisfying the Bogomolov
inequality and $v_{0}\neq 0$. If a numerical $\lambda$-wall
$\Upsilon_{u,v_{s}}$ intersects $\Theta_{v}$ at a point
$(\alpha_{0},\beta_{0})$ for which an object $A$ with $\operatorname{ch}(A)=v$
exists in $\mathcal{A}^{\alpha_{0},\beta_{0}}$, then the associated numerical
$\nu$-wall $\Xi_{u,v}$ also passes through $(\alpha_{0},\beta_{0})$.

Conversely, if a numerical $\nu$-wall $\Xi_{u,v}$ intersects $\Theta_{v}$ at a
point $(\alpha_{0},\beta_{0})$, then the associated numerical $\lambda$-wall
$\Upsilon_{u,v,s}$ also passes through $(\alpha_{0},\beta_{0})$.

###### Proof.

Since $\rho_{v}(\alpha_{0},\beta_{0})=0$, we have
$\rho_{u}(\alpha_{0},\beta_{0})\tau_{v,s}(\alpha_{0},\beta_{0})=0$. However,
Lemma 2.3 guarantees that $\tau_{v,s}(\alpha_{0},\beta_{0})>0$ for every $s$,
thus $\rho_{u}(\alpha_{0},\beta_{0})=0$. It follows that
$t_{u,v}(\alpha_{0},\beta_{0})=0$, as desired.

For the second statement, if
$t_{u,v}(\alpha_{0},\beta_{0})=\rho_{v}(\alpha_{0},\beta_{0})=0$, then also
$\rho_{u}(\alpha_{0},\beta_{0})=0$ because
$\operatorname{ch}_{1}^{\beta_{0}}(v)\neq 0$, thus
$f_{u,v}(\alpha_{0},\beta_{0})=0$. ∎

######  Lemma 4.6.

If the curves $\Theta_{v}$ and $\Gamma_{v,s}$ intersect, then every numerical
$\lambda$-wall for $v$ passes through the point of intersection.

###### Proof.

Let $(\alpha_{0},\beta_{0})$ be the point of intersection, so that
$\rho_{v}(\alpha_{0},\beta_{0})=\tau_{v,s}(\alpha_{0},\beta_{0})=0$. By the
expression in display (29), it follows that
$f_{u,v,s}(\alpha_{0},\beta_{0})=0$, for every real numerical Chern character
$u$ and every $s>0$. ∎

Unbounded numerical $\lambda$-walls are trickier to describe, as their
qualitative behaviour is highly dependent on the value of the parameter $s$,
as illustrated in Figure 3 above. Their main properties are described in the
following series of lemmas. First, we consider the case $s>1/3$.

######  Lemma 4.7.

Let $s>1/3$. If $u$ and $v$ are real numerical Chern characters satisfying
$\delta_{10}(u,v)=0$ and $\delta_{20}(u,v)\neq 0$, then there exists
$\beta_{\rm max}>0$ (depending on $u$, $v$ and $s$) such that
$\Upsilon_{u,v,s}\subset\mathbb{R}^{+}\times[-\beta_{\rm max},\beta_{\rm
max}]$.

###### Proof.

If $\delta_{10}=0$, then equation (30) reduces, after dividing by
$\delta_{20}$, to

(32)
$\underbrace{\left((s-1/3)\beta+\cdots\right)}_{a_{2}}\alpha^{2}+\underbrace{\left(\beta^{3}+\cdots\right)}_{a_{0}}=0$

If $s>1/3$ and either $\beta\ll 0$ or $\beta\gg 0$, then both coefficients
$a_{2}$ and $a_{0}$ in equation (32) have the same sign, hence equation (32)
does not admit any solutions. ∎

######  Lemma 4.8.

Let $s\neq 1/3$. If $u$ and $v$ are real numerical Chern characters satisfying
$\delta_{10}(u,v)=0$ and $\delta_{20}(u,v)\neq 0$, then one of the connected
components of $\Upsilon_{u,v,s}$ is asymptotic to the vertical line
$\\{\beta=\beta^{\prime}\\}$, where

$\beta^{\prime}:=\dfrac{3}{(6s-2)\delta_{20}(u,v)}\left(\dfrac{6s+1}{3}\delta_{21}(u,v)-\delta_{30}(u,v)\right).$

###### Proof.

If $s\neq 1/3$, then the equation $f_{u,v}(\alpha,\beta)=0$ can be rewritten
in the following way:

$\alpha^{2}=\dfrac{\dfrac{1}{3}\delta_{20}\beta^{3}-\dfrac{1}{2}(\delta_{30}+\delta_{21})\beta^{2}+\delta_{31}\beta-\delta_{32}}{-\dfrac{3s-1}{3}\delta_{20}\beta+\dfrac{6s+1}{6}\delta_{21}-\dfrac{1}{2}\delta_{30}}.$

Note that the denominator vanishes at $\beta=\beta^{\prime}$ given above;
therefore $\alpha$ goes to infinity as $\beta$ approaches $\beta^{\prime}$. ∎

######  Lemma 4.9.

Let $s\leq 1/3$. If $u$ and $v$ are numerical Chern characters satisfying
$\delta_{10}(u,v)=0$ and $\delta_{20}(u,v)\neq 0$, then there exist
$\alpha_{\rm max}>0$ (depending on $u$, $v$ and $s$) such that

$\Upsilon_{u,v,s}\cap\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
|\leavevmode\nobreak\ \alpha>\alpha_{\rm
max}\\}\subset\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
|\leavevmode\nobreak\ \rho_{v}(\alpha,\beta)<0\\}.$

###### Proof.

If $s<1/3$, then equation (32) does admit solutions, which are asymptotically
of the form

$\alpha\sqrt{1/3-s}=\pm(\beta-\beta^{\prime}),$

for some $\beta^{\prime}\in\mathbb{R}$ given explicitly in Lemma 4.8 above. In
other words, there is $\alpha_{\rm max}>0$ such that
$\upsilon_{u,v,s}\cap\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
|\leavevmode\nobreak\ \alpha>\overline{\alpha}\\}\subset R^{0}_{v}$.

If $s=1/3$ and $\delta_{21}-\delta_{30}\neq 0$, then equation
$f_{u,v,s}(\alpha,\beta)=0$ reduces to

$\alpha^{2}+\left(\dfrac{2}{3}\dfrac{\delta_{20}}{\delta_{21}-\delta_{30}}\beta^{3}+\cdots\right)=0.$

Depending on the sign of $\delta_{20}/(\delta_{21}-\delta_{30})$, such
equation does admit solutions either for $\beta\ll 0$ or for $\beta\gg 0$; in
both cases, $\alpha$ grows like $|\beta|^{3/2}$, so $\Upsilon_{u,v,s}$ will
lie in the region $R^{0}_{v}$ once $\alpha$ is sufficiently large. The same
conclusion holds when $\delta_{30}=\delta_{21}$, since then
$f_{u,v,s}(\alpha,\beta)$ becomes just a cubic polynomial on $\beta$, not
depending on $\alpha$. ∎

######  Remark 4.10.

Note that there are no $\nu$-walls corresponding to unbounded $\lambda$-walls
and so by continuity the unbounded component remains in $R^{0}_{v}$ as it
cannot intersect $\Theta_{v}$.

However, an unbounded $\lambda$-wall for $v$ may have a bounded connected
component contained in $R^{-}_{v}$, as pictured in Figure 4 below.

Figure 4. The numerical $\lambda$-wall $\Upsilon_{u,v,s}$ for $v=(2,0,-1,0)$
and $u=(1,0,-1,1)$, so that $\delta_{01}(u,v)=0$, is plotted for $s=1/3$
(black curve). It has two connected components: one is bounded, and fully
contained in $R_{v}^{-}$; the other is unbounded, and fully contained in
$R_{v}^{0+}$. The bounded component crosses $\Gamma_{v,s}^{-}$. The curve
$\Theta_{v}$ and the other connected components of $\Gamma_{v,s}$, namely
$\Gamma_{v,s}^{0}$ and $\Gamma_{v,s}^{+}$, are also shown, with the former
coinciding with the $\alpha$-axis.

We now show that intersection between numerical $\lambda$-walls and the curve
$\Gamma_{v,s}$ is independent of $s$.

######  Lemma 4.11.

If a numerical $\lambda$-wall $\Upsilon_{u,v,\overline{s}}$ crosses
$\Gamma_{v,\overline{s}}$ for some $\overline{s}\geq 0$ away from
$\Theta_{v}$, then $\Upsilon_{u,v,s}$ crosses $\Gamma_{v,s}$ for every $s\geq
0$ at points whose $\beta$-coordinate does not depend on $s$.

###### Proof.

Assume that $(\alpha_{0},\beta_{0})\in\mathbb{H}$ satisfies
$f_{u,v,\overline{s}}(\alpha_{0},\beta_{0})=\tau_{v,\overline{s}}(\alpha_{0},\beta_{0})=0$
and we also assume that $\rho_{v}(\alpha_{0},\beta_{0})\neq 0$; in other
words,
$(\alpha_{0},\beta_{0})\in\upsilon_{u,v,\overline{s}}\cap\Gamma_{v,\overline{s}}$
lies away from the hyperbola $\Theta_{v}$. It follows that
$\tau_{u,\overline{s}}(\alpha_{0},\beta_{0})=0$, i.e. the curve $\Gamma_{u,s}$
also passes through $(\alpha_{0},\beta_{0})$, so that

$\operatorname{ch}_{3}^{\beta_{0}}(u)=\left(\overline{s}+\dfrac{1}{6}\right)\operatorname{ch}_{1}^{\beta_{0}}(u)\alpha_{0}^{2}.$

Since $\tau_{v,\overline{s}}(\alpha_{0},\beta_{0})=0$ as well, we conclude
that

(33)
$\dfrac{\operatorname{ch}_{3}^{\beta_{0}}(u)}{\operatorname{ch}_{1}^{\beta_{0}}(u)}=\dfrac{\operatorname{ch}_{3}^{\beta_{0}}(v)}{\operatorname{ch}_{1}^{\beta_{0}}(v)}=\left(\overline{s}+\dfrac{1}{6}\right)\alpha_{0}^{2}.$

For any $s\geq 0$, there is an $\alpha_{s}$ such that
$(s+1/6)\alpha_{s}^{2}=(s+1/6)\alpha_{0}^{2}$; substituting back into equation
(33), we conclude that
$\tau_{u,s}(\alpha_{s},\beta_{0})=\tau_{v,s}(\alpha_{s},\beta_{0})=0$, thus
$f_{u,v,s}(\alpha_{s},\beta_{0})=0$, meaning that
$(\alpha_{s},\beta_{0})\in\Upsilon_{u,v,s}\cap\Gamma_{v,s}\cap\Gamma_{u,s}$.
which is independent of the parameter $s\geq 0$. ∎

Given $v\in K_{\rm num}(X)$, we define an equivalence relation in $K_{\rm
num}(X)$ as follows:

(34) $u\sim_{v}u^{\prime}\text{ if there exists real numbers }\psi\neq 0\text{
and }\phi\text{ such that }u^{\prime}=\phi v+\psi u.$

Note that $u\sim_{v}u^{\prime}$ implies
$\delta_{ij}(u^{\prime},v)=\psi\delta_{ij}(u,v)$ and thus
$\Upsilon_{u,v,s}=\Upsilon_{u^{\prime},v,s}$. In other words, the set of
numerical $\lambda$-walls for a fixed $v\in K_{\rm num}(X)$ only depend on the
equivalence class under $\sim_{v}$.

######  Lemma 4.12.

Suppose $u$ and $v$ are real numerical Chern characters and $v_{0}\neq 0$.
Then:

  1. (1)

$u\sim_{v}(0,1,\delta_{02}(u,v)/\delta_{01}(u,v),\delta_{03}(u,v)/\delta_{01}(u,v))$
if $\delta_{01}(u,v)\neq 0$ and

  2. (2)

$u\sim_{v}(0,0,1,\delta_{03}(u,v)/\delta_{02}(u,v))$ if $\delta_{01}(u,v)=0$
and $\delta_{02}(u,v)\neq 0$.

###### Proof.

If $\delta_{01}(u,v)\neq 0$ take $\phi=u_{0}/\delta_{01}(u,v)$ and
$\psi=-v_{0}/\delta_{01}(u,v)$. If $\delta_{01}(u,v)=0$, take
$\phi=u_{0}/\delta_{02}(u,v)$ and $\psi=-v_{0}/\delta_{02}(u,v)$. ∎

It follows that equivalence classes for $\sim_{v}$ come in three types when
$v_{0}\neq 0$, according to Table 1 below ($x$ and $y$ are arbitrary rational
numbers). In the degenerate case, the numerical $\lambda$-wall
$\Upsilon_{u,v,s}$ coincides with $\Theta_{v}$.

Type | Characterization | Canonical element
---|---|---
Bounded | $\delta_{01}\neq 0$ | (0,1,x,y)
Unbounded |  $\delta_{01}=0$, $\delta_{02}\neq 0$ | (0,0,1,x)
Degenerate |  $\delta_{01}=\delta_{02}=0$, $\delta_{03}\neq 0$ | (0,0,0,1)
Table 1. Equivalence classes $\sim_{v}$ when $v_{0}\neq 0$.

This gives us a quick way to see that various families of numerical
$\lambda$-walls do not intersect each other, and it provides a form of
Bertram’s Nested Wall Theorem.

######  Theorem 4.13.

Suppose $v_{0}\neq 0$ and $u\not\sim_{v}u^{\prime}$.

  1. (1)

The numerical $\nu$-walls corresponding to $u$ and $u^{\prime}$ do not
intersect away from $M_{v}$.

  2. (2)

If $\delta_{01}(u,v)=0=\delta_{01}(u^{\prime},v)$ then the numerical
$\lambda$-walls corresponding to $u$ and $u^{\prime}$ do not intersect.

  3. (3)

If $\delta_{01}(u,v)\neq 0$ and $\operatorname{ch}_{\leq
2}(u)=\operatorname{ch}_{\leq 2}(u^{\prime})$, then the numerical
$\lambda$-walls corresponding for $v$ to $u$ and $u^{\prime}$ only intersect
on $\Theta_{v}$.

###### Proof.

We do (3) and leave (1) and (2) as similar exercises. We simply observe that
the equation for the numerical $\lambda$-wall $\Upsilon_{u,v,s}$ can be
written in the following manner

$u_{3}\rho_{v}(\alpha,\beta)=g(\alpha,\beta,v,\operatorname{ch}_{\leq 2}(u)),$

for some function $g$. Hence, if $(\alpha,\beta)$ is not on $\Theta_{v}$ but
lies on $\Upsilon_{u,v,s}\cap\Upsilon_{u^{\prime},v,s}$, then we must have
$u_{3}=\operatorname{ch}_{3}(u^{\prime})$. ∎

We conclude that distinct unbounded numerical $\lambda$-walls never intersect
one another. On the other hand, it is interesting to observe that, in contrast
with numerical $\nu$-walls, two distinct bounded numerical $\lambda$-walls for
$\Upsilon_{u,v,s}$ and $\Upsilon_{u^{\prime},v,s}$ can intersect both along
$\Theta_{v}$ (if $\operatorname{ch}_{\leq 2}(u)=\operatorname{ch}_{\leq
2}(u^{\prime})$, as illustrated in item (3) of Theorem 4.13) and away from it,
see Figure 5.

Figure 5. The numerical $\lambda$-walls $\Upsilon_{u,v,s}$ and
$\Upsilon_{u^{\prime},v,s}$ for $v=(2,0,-3,0)$ with $u=(1,-1,1/2,-1/6)$ and
$u^{\prime}=(1,-2,2,-4/3)$ are plotted for $s=1/3$; they cross at the point
$(\alpha=0.5,\beta=-1.48)$. The curves $\Gamma_{v,1/3}$ (in red) and
$\Theta_{v}$ (in blue) are also shown.

Our normal form for $u$ can also be used to give sufficient conditions for
when a numerical $\lambda$-wall for $v$ intersects $\Gamma_{v,s}$:

######  Proposition 4.14.

Let $u$ and $v$ be real numerical Chern characters such that $v$ satisfies the
Bogomolov inequality, $ch_{0}(v)\neq 0$ and $\delta_{01}(u,v)\neq 0$. If
$\delta_{01}(u,v)\delta_{03}(u,v)\geq\delta_{02}(u,v)^{2}/2$ then the
numerical $\lambda$-wall $\Upsilon_{u,v,s}$ intersects $\Gamma_{v,s}^{0}$. If
$\delta_{01}(u,v)\delta_{03}(u,v)\leq\delta^{2}_{02}/2$ then,
$\Upsilon_{u,v,s}$ intersects both $\Gamma_{v,s}^{+}$ and $\Gamma_{v,s}^{-}$.

###### Proof.

Writing $u$ in the canonical form $(0,1,x,y)$ observe that $\Gamma_{u,s}$ is
given by the hyperbola:

$(\beta-x)^{2}/2-\alpha^{2}(s+1/6)=x^{2}/2-y.$

The conditions in the statement about $\delta_{ij}$ are equivalent to the
hyperbola crossing the $\beta$-axis or not. The hyperbolae are asymptotic to
$\beta=x\pm\sqrt{2(s+1/6)}\alpha$ while $\Gamma_{v,s}$ is either asymptotic to
$M_{v}$ or to $\beta=\mu\pm\sqrt{6(s+1/6)}\alpha$. Then the hypotheses
guarantee that $\Gamma_{u,s}$ and $\Gamma_{v,s}$ intersect as stated. ∎

######  Remark 4.15.

Observe that an unbounded numerical $\lambda$-wall $\Upsilon_{u,v,s}$
associated to $u\sim_{v}(0,0,1,x)$ intersects $\Gamma_{v,s}$ (necessarily away
from $\Theta_{v}$, since unbounded $\lambda$-walls for $v$ never cross
$\Theta_{v}$) at a point with coordinates $(\alpha,x)$ for some uniquely
determined $\alpha$ so long as $x$ is the $\beta$ coordinate of some point on
$\Gamma_{v,s}$. Since the wall $\Upsilon_{u,v,s}$ can only cross $\Theta_{v}$
at its intersection with $\Gamma_{v,s}$ and its asymptotes are in $R^{0}_{v}$,
it follows that, for $|x|$ sufficiently large, there is a component of the
wall which is bounded and which intersects $\Gamma^{\pm}_{v,s}$.

###  4.3. Actual, pseudo and vanishing $\lambda$-walls

######  Definition 4.16.

An _actual $\lambda$-wall_ $W_{u,v,s}$ is the subset of points
$(\alpha,\beta)$ in $\Upsilon_{u,v,s}$ for which there is an object
$E\in\mathcal{A}^{\alpha,\beta}$ with $\operatorname{ch}(E)=v$ and a path
$\gamma:(-1,1)\to\mathbb{R}_{>0}\times\mathbb{R}$ such that
$\gamma(0)=(\alpha,\beta)$ and $E\in{\mathcal{A}}^{\gamma(t)}$ is
$\lambda$-stable for $t<0$ and $\lambda$-unstable for $t>0$.

This is a rather stronger condition than the usual one, which asks simply that
$A$ is properly $\lambda_{\alpha,\beta,s}$-semistable at the point
$(\alpha,\beta)$. Our definition ensures that actual $\lambda$-walls are one-
dimensional and avoids degenerate situations like the one described in Remark
2.6. In addition, it allows us to state:

######  Lemma 4.17.

An actual $\lambda$-wall $W_{u,v,s}$ is a union of segments of arc within the
underlying numerical $\lambda$-wall $\Upsilon_{u,v,s}$ whose endpoints lie
either on another actual $\lambda$-wall for $u$ or $v$,
$\Theta_{u}\cap\Theta_{v}$, or on the $\beta$-axis.

###### Proof.

Assume first that $p\in W_{u,v,s}$ is an isolated point, that is, there exists
an open neighbourhood $B$ of $p$ on which there are no other actual
$\lambda$-walls. In contrast with the path $\gamma$ going through $p$ along
which the moduli space ${\mathcal{M}}_{\gamma(t),s}(v)$ changes as explained
above, one can find a path $\gamma^{\prime}:[0,1]\to B$ such that
$\gamma^{\prime}(0)=\gamma(0)$ and $\gamma^{\prime}(1)=\gamma(1)$ which does
not cross any actual $\lambda$-wall, implying that
${\mathcal{M}}_{\gamma(0),s}(v)={\mathcal{M}}_{\gamma(1),s}(v)$, and thus
providing a contradiction.

It follows that $W_{u,v,s}$ is a union of segments of arc within
$\Upsilon_{u,v,s}$. The same argument shows that if an endpoint
$(\alpha_{0},\beta_{0})$ of such an arc does not lie on another actual
$\lambda$-wall for a short exact sequence $0\to A\to E\to B\to 0$ in
$\mathcal{A}^{\alpha,\beta}$ corresponding to the wall, then $E$, $A$ or $B$
must go out of the category. But if $A$ or $B$ go out of the category it is
because they have a sub or quotient object which has a $\Theta$ curve going
through that point. But then the $\lambda$-slope goes to infinity and this
would contradict the existence of the wall in a small neighbourhood of the
point. Consequently, the wall can only end on $\Theta_{\operatorname{ch}(E)}$
and so also $\Theta_{\operatorname{ch}(A)}$. ∎

In another direction, if an object is strictly destabilized along a curve then
it can only become stable if either the curve crosses an actual $\lambda$-wall
or a $\Theta$ curve. Note that if an actual $\lambda$-wall $W_{u,v}$ crosses
$\Theta_{u}$ then it must also cross $\Theta_{v}$.

######  Proposition 4.18.

Fix $s>0$. Suppose $\gamma:(-a,b)\to\mathbb{H}$ is a curve for some $a,b>0$
and $A\in\mathcal{A}^{\gamma(t)}$ for all $-a<t<b$. Suppose $B\hookrightarrow
A\twoheadrightarrow C$ is a destabilizing sequence for $-a<t<0$.

  1. (1)

If $A$ is not $\lambda_{\gamma(t),s}$-semistable for $0<t<b$ then there exists
$\epsilon,\epsilon^{\prime}>0$ and some destabilizing sequence
$K\hookrightarrow A\twoheadrightarrow Q$ defined for all
$-\epsilon^{\prime}<t<\epsilon$ with
$\lambda_{\gamma(t),s}(K)\geq\lambda_{\gamma(t),s}(B)$, respectively
$\lambda_{\gamma(t),s}(Q)\leq\lambda_{\gamma(t),s}(C)$ with equalities if and
only if $C=Q$, respectively $B=K$.

  2. (2)

If $A$ is $\lambda_{\gamma(t),s}$-stable for $0<t<b$ then there is an actual
$\lambda$-wall corresponding to $B$ containing $\gamma(0)$.

###### Proof.

In either case, we have a long exact sequence for $0<t<\epsilon$ in
$\mathcal{A}^{\gamma(t)}$:

$0\to C^{-1}\to B^{0}\to A\to C^{0}\to B^{1}\to 0.$

This is because as we cross $t=0$ the phase changes continuously and the
$\mathcal{A}^{\gamma(t)}$-cohomology of any object in
$\mathcal{A}^{\gamma(t)}$ for $-\epsilon^{\prime}<t<0$ in nearby $\gamma(t)$
is concentrated in positions $-1$, $0$ and/or $1$. Split the sequence via
$B^{0}\to K\to A\to Q\to C^{0}$ and note that, for $-1\ll t<0$,
$C^{-1}[1],B^{1}[-1]\in\mathcal{A}^{\gamma(t)}$ and $\lim_{t\nearrow
0}\lambda_{\gamma(t),s}(C^{-1}[1])=+\infty$ and $\lim_{t\nearrow
0}\lambda_{\gamma(t),s}(B^{1}[-1])=-\infty$. We also have short exact
sequences for $-1\ll t<0$:

$C^{-1}[1]\to C\to C^{0}\text{ and }B^{0}\to B\to B^{1}[-1].$

It follows that $K\to A\to Q$ is short exact in $\mathcal{A}^{\gamma(t)}$ for
$-\epsilon<t<\epsilon$ and that, for $-1\ll t<0$,
$\lambda_{\gamma(t),s}(K)\geq\lambda_{\gamma(t),s}(B)$ with equality only if
$C^{-1}=0$. Similarly, $\lambda_{\gamma(t),s}(Q)\leq\lambda_{\gamma(t),s}(C)$
with equality only if $B^{1}=0$.

If $A$ is $\lambda_{\gamma(t),s}$\- stable beyond $t=0$ then we must have
equality in one of these. If it is both then the $0\to B\to A\to C\to 0$
provides an actual $\lambda$-wall at $\gamma(0)$. If only one is equality then
$\gamma(0)\in\Theta_{u}$ where $u=\operatorname{ch}(B)$ or
$u=\operatorname{ch}(C)$. But then, $\lambda_{\gamma(0),s}(A)=+\infty$ and so
$\gamma(0)\in\Theta_{\operatorname{ch}(A)}$ as well and so, again, this
sequence provides an actual $\lambda$-wall. ∎

If our aim is to find all actual $\lambda$-walls then it is easier to first
consider a list of necessary numerical conditions in order to reduce the
possibilities to a small list of examples (this is exactly what we will do in
the example presented in Proposition 8.10 below).

######  Definition 4.19.

By a _pseudo $\lambda$-wall_ $W_{u,v,s}$ we mean the subset of points
$(\alpha,\beta)$ of a numerical $\lambda$-wall $\Upsilon_{u,v,s}$ for which

  1. (1)

there are objects $E,F,G\in\mathcal{A}^{\alpha,\beta}$ satisfying
$\operatorname{ch}(E)=v$, $\operatorname{ch}(F)=u$,
$\operatorname{ch}(G)=v-u$;

  2. (2)

$Q_{\alpha,\beta}(u)\geq 0$, $Q_{\alpha,\beta}(v-u)\geq 0$,
$Q_{\alpha,\beta}(u)+Q_{\alpha,\beta}(v-u)\leq Q_{\alpha,\beta}(v)$.

The support property, see Proposition 2.4, implies that an actual
$\lambda$-wall is also a pseudo $\lambda$-wall. We shall see an example in
Proposition 8.10(65) of a pseudo wall which is not an actual wall but for
which there is a destabilizing sequence.

######  Remark 4.20.

In practice, we would start by replacing (1) with the necessary conditions
that
$u\in{\mathbb{Z}}\times{\mathbb{Z}}\times{\mathbb{Z}}/2\times{\mathbb{Z}}/6$
and $\chi(u)\in{\mathbb{Z}}$. The third condition of (2) is an observation of
Schmidt ([34, Lemma 2.7]) and follows from the fact that the bilinear form
$Q_{\alpha,\beta}(v,v)=Q_{\alpha,\beta}(v)\geq 0$.

Unlike actual $\nu$-walls, distinct actual $\lambda$-walls for a given Chern
character $v$ do intersect. This was first noticed by Schmidt in [34], where
he establishes a refinement of Lemma 4.5 providing a relationship between
actual $\nu$\- and $\lambda$-walls for $v$ along $\Theta_{v}^{-}$; more
precisely, he proves the following statement,

######  Theorem 4.21.

[34, Thm. 6.1] Let $v$ be a Chern character satisfying the Bogomolov
inequality and $v_{0}\neq 0$, and let $(\alpha_{0},\beta_{0})$ be a point
$\Theta_{v}^{-}$.

  * •

If there is an actual $\lambda$-wall for $v$ containing
$(\alpha_{0},\beta_{0})$ given by the exact sequence $0\to F\to A\to G\to 0$
in $\mathcal{A}^{\alpha_{0},\beta_{0}}$, then there is an actual $\nu$-wall
containing $(\alpha_{0},\beta_{0})$ given by the exact sequence $0\to F[-1]\to
A[-1]\to G[-1]\to 0$ in $\mathcal{B}^{\beta_{0}}$.

  * •

Conversely, if there is an actual $\nu$-wall for $v$ containing
$(\alpha_{0},\beta_{0})$ given by the sequence $0\to F\to A\to G\to 0$ in
$\mathcal{B}^{\beta_{0}}$, then there is an actual $\lambda$-wall for $v$
which is defined by the same sequence in $\mathcal{A}^{\alpha_{0},\beta_{0}}$.

We also provide a concrete example of two distinct actual $\lambda$-walls for
the Chern character $v:=(2,0,-1,0)$ which intersect away from $\Theta_{v}$ in
Section 8.4 below.

Furthermore, an actual $\lambda$-wall $W_{u,v,s}$ is called a _vanishing
$\lambda$-wall_ for $v$ if for each $p\in W_{u,v,s}$ there exists a path
$\gamma:(-1,1)\to\mathbb{H}$ crossing $W_{u,v,s}$ transversely at
$\gamma(0)=p$ for some $w\in(0,1)$ such that
${\mathcal{M}}_{\gamma(t),s}(v)\neq\emptyset$ for $t<0$, while
${\mathcal{M}}_{\gamma(t),s}(v)=\emptyset$ and $t>0$.

The meaning of the quartic function $q(v)$ on $K_{\rm num}(X)$ defined in
equation (26) can now be expressed in the following theorem.

######  Theorem 4.22.

Let $v$ be a Chern character for which there exists a Gieseker semistable
sheaf $E$ with $\operatorname{ch}(E)=v$. If $q(v)<0$ then there must exist
vanishing $\nu$\- and $\lambda$-walls for $v$.

###### Proof.

Since $E$ is Gieseker semistable, $v$ satisfies the Bogomolov inequality and
so the curve $\Theta_{v}$ divides the plane into three regions, as explained
in Section 3 above.

If $q(E)<0$, then there is $\alpha_{0}>0$ such that
$\Gamma_{v,s}\cap\\{\alpha=\alpha_{0}\\}$ consists of a single point, thus
either the asymptotic components $\Gamma^{-}_{v,s}$ and $\Gamma^{0}_{v,s}$ or
$\Gamma^{+}_{v,s}$ and $\Gamma^{0}_{v,s}$ must belong to the same connected
component of $\Gamma_{v,s}$, which must then cross the hyperbola $\Theta_{v}$.

Since Proposition 2.3 fails for $(\alpha,\beta)\in\Theta_{v}$ below the point
of intersection $(\tilde{\alpha},\tilde{\beta}):=\Gamma_{v,s}\cap\Theta_{v}$,
there can not exist any objects with Chern character equal to $v$ in
$\mathcal{A}^{\alpha,\beta}(X)$ for $(\alpha,\beta)\in\Theta_{v}$ with
$\alpha\leq\tilde{\alpha}$.

However, if $\beta$ is sufficiently large as to lie above any $\nu$-wall for
$v$, then $E$ is $\nu_{\alpha,\beta}$-semistable and Proposition 3.2
guarantees the existence of $\lambda_{\alpha,\beta,s}$-semistable objects in
$\mathcal{A}^{\alpha,\beta}$ for $(\alpha,\beta)\in\mathbb{H}$. It follows
that there must exist vanishing $\nu$\- and $\lambda$-walls for $v$ crossing
$\Theta_{v}$ above the point of intersection
$(\tilde{\alpha},\tilde{\beta}):=\Gamma_{v,s}\cap\Theta_{v}$. ∎

We will see an example of the situation described in Theorem 4.22 in Section
8.1 below, where we study in detail the ideal sheaf of a line in
${\mathbb{P}^{3}}$.

##  5\. Asymptotic $\nu_{\alpha,\beta}$-stability

One of the main goals of this paper is to characterize which objects in
$D^{\rm b}(X)$ are $\nu_{\alpha,\beta}$\- and
$\lambda_{\alpha,\beta,s}$-semistable _at infinity_ , that is, for large
values of the parameters $\alpha$, $\beta$. More formally, let
$\gamma:[0,\infty)\to\mathbb{H}$ be an unbounded path; we consider the
following definition. The dual situation is also considered in [30, 3.2] but
from a different perspective.

######  Definition 5.1.

An object $A\in D^{\rm b}(X)$ is _asymptotically
$\nu_{\alpha,\beta}$-(semi)stable along $\gamma$_ if the following two
conditions hold:

  * (i)

there is $t_{0}>0$ such that $A\in\mathcal{B}^{\gamma(t)}$ for every
$t>t_{0}$;

  * (ii)

for every sub-object $F\hookrightarrow A$ within $\mathcal{B}^{\gamma(t)}$
with $t>t_{0}$, there is $t_{1}>t_{0}$ such that
$\nu_{\gamma(t)}(F)<\leavevmode\nobreak\ (\leq)\leavevmode\nobreak\
\nu_{\gamma(t)}(A)$ for $t>t_{1}$.

In this section we characterize asymptotically $\nu_{\alpha,\beta}$-semistable
objects. More precisely, we establish the following results.

######  Theorem 5.2.

Let $B\in D^{\rm b}(X)$ be an object with $\operatorname{ch}_{0}(B)\neq 0$.

  1. (1)

$B$ is asymptotically $\nu_{\alpha,\beta}$-(semi)stable along a path
$\gamma(t)=(\alpha(t),\beta(t))$ such that $\beta(t)<\mu(B)$ for $t\gg 0$ if
and only if $B$ is a $\mu_{\leq 2}$-(semi)stable sheaf.

  2. (2)

$B$ is asymptotically $\nu_{\alpha,\beta}$-(semi)stable along along a path
$\gamma(t)=(\alpha(t),\beta(t))$ such that $\beta(t)>\mu(B)$ for $t\gg 0$ if
and only if $S:=B^{\vee}[-1]$ is a $\mu_{\leq 2}$-(semi)stable sheaf such that
$S^{**}/S$ either is empty or has pure dimension 1.

Theorem 5.2 is not new, though its statement and the second part especially,
is different from previous versions. The proof of item (1) is essentially the
same as [10, Proposition 14.2], which covers the case of $K3$ surfaces,
compare also with [36, Proposition 2.5]. Part (2) is to be compared with [1,
Proposition 2.3] in the case of surfaces, and [7, Proposition 5.1.3]. Since we
will use Theorem 5.2 and the arguments in its proof in the subsequent
sections, we include a full proof here. Our proof is longer then previous ones
for two reasons. Firstly, we have a stronger form of asymptotic stability and
secondly we do not assume that there are only finitely many $\nu$-walls above
a certain horizontal line. Although the latter fact is true, we want to
illustrate how it can be avoided, since it is not known for
$\lambda$-stability. If we do assume that the $\nu$-walls are bounded above,
then it follows that the $t_{1}$ of the definition of asymptotic
$\nu$-stability can be chosen uniformly in $E$, that is, $t_{1}$ only depends
on $\operatorname{ch}_{\leq 2}(E)$.

The study of asymptotic $\nu_{\alpha,\beta}$-stability is considerably
simplified by the following observation:

######  Lemma 5.3.

  1. (1)

If $B\in\mathcal{B}^{\beta}$ for all $\beta\ll 0$, then
$B\in\mathcal{C}oh(X)$.

  2. (2)

Suppose $E$ is a $\mu$-semistable sheaf and suppose there is some
$(\alpha,\beta)$ with $\beta<\mu(E)$ lies on an actual $\nu$-wall so that
there is a $\nu_{\alpha,\beta}$-semistable object $F$ and a monomorphism
$F\hookrightarrow E$ in $\mathcal{B}^{\beta}$. Then this is the only
$\nu$-wall for $\beta<\mu(E)$ at which $E$ is destabilized.

###### Proof.

If ${\mathcal{H}}^{-1}(B)\neq 0$, then

$\operatorname{ch}_{1}^{\beta}({\mathcal{H}}^{-1}(B)[1])=-\operatorname{ch}_{1}({\mathcal{H}}^{-1}(B))+\operatorname{ch}_{0}({\mathcal{H}}^{-1}(B))\beta<0\text{
for }\beta\ll 0,$

so $B$ cannot lie in $\mathcal{B}^{\beta}$ for every $\beta\ll 0$.

For the second item, the idea is that $E$ must be $\nu_{\alpha,\beta}$-stable
above the $\nu$-wall defined by the exact sequence. The local finiteness of
the actual $\nu$-walls ensures that $E$ is $\nu_{\alpha,\beta}$-stable either
immediately below or immediately above this wall. But we can show that the
latter holds as follows. First observe that $\delta_{01}(E,F)\leq 0$. This is
because if we split the $\mathcal{C}oh(X)$ sequence

$0\to{\mathcal{H}}^{-1}(G)\to F\to E\to{\mathcal{H}}^{0}(G)\to 0,$

where $G=E/F$ in $\mathcal{B}^{\beta}$, via $K\to E$ then $\mu(K)\leq\mu(E)$
by the $\mu$-semistability of $E$ while
$\mu({\mathcal{H}}^{-1}(G))\leq\beta<\mu(F)$ and so $\mu(F)<\mu(K)\leq\mu(E)$.
We use Lemma 2.7(2) to compute the partial derivatives

$\partial_{\alpha}\bigl{(}\nu_{\alpha,\beta}(E)-\nu_{\alpha,\beta}(F)\bigr{)}=\partial_{\alpha}\frac{\Delta_{21}(E,F)}{\operatorname{ch}^{\beta}_{1}(E)\operatorname{ch}^{\beta}_{1}(F)}=-\frac{\alpha\delta_{01}(E,F)}{\operatorname{ch}_{1}^{\beta}(E)\operatorname{ch}_{1}^{\beta}(F)}\geq
0$

as $E$ is $\mu$-semistable. Consequently, $E$ cannot be destabilized on
another wall outside of the this $\nu$-wall because it would need to be both
stable and unstable immediately below which is impossible.

To complete the proof, observe that $F\hookrightarrow E\twoheadrightarrow G$
remains a short exact sequence in $\mathcal{B}^{\beta}$ at all points inside
the $\nu$-wall corresponding to $F$. ∎

We will see that a similar statement holds for $\beta>\mu(E)$. So, for a given
$E$, there is at most one actual $\nu$-wall destabilizing it one on each side
of the vertical line $M_{v}$, and then we can reduce the proof of Theorem 5.2
to the study of asymptotic $\nu_{\alpha,\beta}$-stability along horizontal
lines. To this end we define:

(35) $\displaystyle\Lambda_{\overline{\alpha}}^{-}$
$\displaystyle:=\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
|\leavevmode\nobreak\ \alpha=\overline{\alpha},\beta<0\\}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\Lambda_{\overline{\alpha}}^{+}$
$\displaystyle:=\\{(\alpha,\beta)\in\mathbb{H}\leavevmode\nobreak\
|\leavevmode\nobreak\ \alpha=\overline{\alpha},\beta>0\\}.$

###  5.1. Asymptotics along $\Lambda_{\overline{\alpha}}^{-}$

The first part of Theorem 5.2 is proved in two separate lemmas.

######  Lemma 5.4.

If $B\in D^{\rm b}(X)$ is asymptotically $\nu_{\alpha,\beta}$-semistable along
the horizontal half-line $\Lambda_{\overline{\alpha}}^{-}$, then $B$ is a
$\mu_{\leq 2}$-semistable sheaf.

###### Proof.

By Lemma 5.3 we may assume $B\in\mathcal{C}oh(X)\cap\mathcal{B}^{\beta}$ for
all $\beta\ll 0$.

Let $T\hookrightarrow B$ be a torsion subsheaf of $B$; if
$\operatorname{ch}_{1}(T)\neq 0$, then

$\lim_{\beta\to-\infty}\dfrac{-1}{\beta}(\nu_{\overline{\alpha},\beta}(T)-\nu_{\overline{\alpha},\beta}(B))=\dfrac{1}{2}$

contradicting asymptotic $\nu_{\alpha,\beta}$-semistability. If
$\operatorname{ch}_{1}(T)=0$, then

$\nu_{\alpha,\beta}(B)<\nu_{\alpha,\beta}(T)=+\infty$

for every $(\alpha,\beta)\in\mathbb{H}$, again contradicting asymptotic
$\nu_{\alpha,\beta}$-semistability. So $B$ must be a torsion free sheaf.

Let $F$ be a subsheaf of $B$; since $F\in\mathcal{B}^{\beta}(X)$ for
$\beta<\mu^{-}(F)$, $F$ is also a sub-object of $B$ within
$\mathcal{B}^{\beta}(X)$ for $\beta<\min\\{\mu^{-}(F),\mu^{-}(B)\\}$. Note
that

$\lim_{\beta\to-\infty}\left(\nu_{\overline{\alpha},\beta}(F)-\nu_{\overline{\alpha},\beta}(B)\right)=-\dfrac{1}{2}\dfrac{\delta_{10}(B,F)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(B)}\leq
0$

by asymptotic $\nu_{\alpha,\beta}$-semistability. It follows that
$\delta_{10}(B,F)\geq 0$, thus $B$ is $\mu$-semistable.

If $\delta_{10}(B,F)=0$, then

$\lim_{\beta\to-\infty}(-\beta)(\nu_{\overline{\alpha},\beta}(F)-\nu_{\overline{\alpha},\beta}(B))=-\dfrac{\delta_{20}(B,F)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(B)}\leq
0$

by asymptotic $\nu_{\alpha,\beta}$-semistability, thus $\delta_{20}(B,F)\geq
0$, as desired. ∎

Before going on to look at the converse we can make an interesting deduction
from this:

######  Proposition 5.5.

If $E$ is a $\mu$-semistable sheaf which is not $\mu_{\leq 2}$-semistable,
then $E$ is not $\nu_{\alpha,\beta}$-semistable for any
$(\alpha,\beta)\in\mathbb{H}$.

###### Proof.

Suppose otherwise. Then by item (2) in Lemma 5.3 it follows that $E$ is
asymptotically $\nu_{\alpha,\beta}$-semistable and then by Lemma 5.4 it must
be $\mu_{\leq 2}$-semistable, a contradiction. ∎

Now we consider the converse to Lemma 5.4:

######  Lemma 5.6.

Fix $\overline{\alpha}>0$. If $E$ is $\mu_{\leq 2}$-semistable, then there is
$\beta_{0}<0$ (depending only on $\operatorname{ch}_{\leq 2}(E)$ and
$\overline{\alpha}$) such that $E$ is
$\nu_{\overline{\alpha},\beta}$-semistable for every $\beta<\beta_{0}$.

###### Proof.

First, note that $E$ is $\mu$-semistable, thus
$E\in\mathcal{T}_{\beta}\subset\mathcal{B}^{\beta}$ whenever $\beta<\mu(E)$.
Then Lemma 5.3(2) implies that there is at most one actual $\nu$-wall for $E$
containing the point $(\overline{\alpha},\beta_{0})$, say (set
$\beta_{0}=\mu(E)$ if the wall does not exist). It follows that $E$ is
$\nu_{\overline{\alpha},\beta}$-stable for $\beta<\beta_{0}$. ∎

This completes the proof of the first part of Theorem 5.2.

######  Remark 5.7.

We will also need the following version of Lemma 5.6 for torsion sheaves:
given any fixed $\overline{\alpha}>0$, if $T\in\mathcal{C}oh(X)_{2}$ is
$\hat{\mu}$-semistable, then there is $\beta_{0}<0$ such that $T$ is
$\nu_{\overline{\alpha},\beta}$-semistable for every $\beta<\beta_{0}$. The
proof of this claim is similar to the proof of Lemma 5.6.

We can now describe an _asymptotic Harder–Narasimhan filtration_ for unstable
objects as well.

######  Proposition 5.8.

Let $\gamma(t)=(\alpha(t),\beta(t))$ be a path satisfying
$\lim_{t\to\infty}\beta(t)=-\infty$. If $B\in D^{\rm b}(X)$ is an object for
which there is $t_{0}>0$ such that $B\in{\mathcal{B}}^{\beta(t)}$ for all
$t>t_{0}$, then $B$ admits a filtration in ${\mathcal{B}}^{\beta(t)}$

$0=B_{0}\subseteq B_{1}\subset B_{2}\subset\cdots\subset B_{n}=B,$

whose factors $G_{k}:=B_{k}/B_{k-1}$ for are asymptotically
$\nu_{\alpha,\beta}$-semistable and satisfy

  1. (1)

$\nu_{\gamma(t)}(B_{1})=+\infty$ for every $t>0$ if $B_{1}\neq 0$;

  2. (2)

for each $k=2,\dots,n$, there is $t_{k}>t_{0}$ such that
$\nu_{\gamma(t)}(G_{k})-\nu_{\gamma(t)}(G_{k+1})>0$ for every $t>t_{k}$.

###### Proof.

The first item of Lemma 5.3 implies that $B$ must be a sheaf, so it admits a
filtration as described in Lemma 2.20; this is the filtration we are looking
for.

Indeed, each factor in the filtration of Lemma 2.20 belongs to
$\mathcal{B}^{\beta}$ for $\beta\ll 0$, according to Lemma 5.6 and Remark 5.7.
The property in item (1) is clear, since $B_{1}\in\mathcal{C}oh_{1}(X)$; item
(2) is a consequence of the following claim: given $E,F\in\mathcal{C}oh(X)$
with $\operatorname{ch}_{\leq 1}(E),\operatorname{ch}_{\leq 1}(F)\neq 0$, then
$\nu_{\gamma(t)}(E)>\nu_{\gamma(t)}(F)$ for $t\gg 0$ if and only if
$\Lambda_{2}(E,F)>0$. This can be explicitly checked for the path
$\gamma(t)=(\overline{\alpha},-t)$ using the limits calculated in the proof of
Lemma 5.4; the verification for more general paths is similar. ∎

###  5.2. Unbounded $\Theta^{*}$-curves

It is tempting to think that if $B\in\mathcal{B}^{\beta}$ (respectively,
$A\in\mathcal{A}^{\alpha,\beta}$) for some $\alpha,\beta$ then
$B^{\vee}\in{\mathcal{B}}^{-\beta}$ (respectively,
$A^{\vee}\in\mathcal{A}^{\alpha,-\beta}$). The problem is that the duals of
objects $B$ in $\mathcal{F}_{\beta}$ are not necessarily in
$\mathcal{T}_{-\beta}$ because it might be that $\mu^{+}(B)=\beta$ and then
$\mu^{-}(B^{\vee})=-\beta$. Similarly for objects $A$ in
$\mathcal{F}_{\alpha,\beta}$. However, an _asymptotic_ version of this
statement does hold.

First we prove a technical lemma which allows us to turn Proposition 5.8 about
the existence of asymptotic Harder–Narasimhan filtrations into a more precise
bound on $\nu_{\alpha,\beta}^{-}$ so long as we constrain the unbounded curve
we move along. To this end, we introduce the following definition.

######  Definition 5.9.

A path $\gamma:(0,\infty)\to\mathbb{H}$ with $\gamma(t)=(\alpha(t),\beta(t))$
is called _an unbounded $\Theta^{-}$-curve_ if
$\lim_{t\to\infty}\beta(t)=-\infty$ and for all $v$ such that $v_{\leq
1}\neq(0,0)$, $\nu_{\gamma(t)}(v)>0$ for all $t$ sufficiently large.
Equivalently, the curve is asymptotically bounded by $\Theta^{-}_{v}$, ie

$\lim_{t\to\infty}\frac{\dot{\alpha}(t)}{\dot{\beta}(t)}>-1.$

We define the _dual curve_ $\gamma^{*}$ is defined to be
$\gamma^{*}(t)=(\alpha(t),-\beta(t))$. Similarly, we define $\gamma$ to be _an
unbounded $\Theta^{+}$-curve_ if $\lim_{t\to\infty}\beta(t)=+\infty$ and for
all $v$ such that $v_{\leq 1}\neq(0,0)$, $\nu_{\gamma(t)}(v)<0$ for all $t$
sufficiently large.

In particular, $\gamma$ is an unbounded $\Theta^{-}$-curve if and only if
$\gamma^{*}$ is an unbounded $\Theta^{+}$-curve.

######  Remark 5.10.

If $v_{0}=0$ while $v_{1}\neq 0$ then the first condition implies the second;
indeed, we have $\nu_{\gamma(t)}(v)=v_{2}/v_{1}-\beta(t)$ and the first
condition implies that this is positive for all $t\gg 0$.

When $v_{0}\neq 0$, the condition on $\nu_{\gamma(t)}$ is satisfied, for
example, if there is an $\epsilon>0$ such that for all $t\gg 0$,
$\alpha(t)^{2}<\beta(t)^{2}(1-\epsilon)$: we have

$\lim_{t\to\infty}\dfrac{\nu_{\gamma(t)}(v)}{-\beta(t)}>\lim_{t\to\infty}\frac{v_{2}-\beta(t)v_{1}+\beta(t)^{2}\epsilon
v_{0}/2}{-v_{1}\beta(t)+v_{0}\beta(t)^{2}}=\dfrac{\epsilon}{2}>0.$

So again this is positive for all $t\gg 0$.

######  Example 5.11.

Note that, since $s>0$, $\alpha^{2}<\beta^{2}(1-\epsilon)$ sufficiently far
along $\Gamma^{-}_{v,s}$ and so $\Gamma^{-}_{v,s}$ is an unbounded
$\Theta^{-}$-curve.

######  Lemma 5.12.

Let $\gamma$ be an unbounded $\Theta^{-}$-curve. If
$B\in\mathcal{B}^{\beta(t)}$ for all $t\gg 0$ and $\operatorname{ch}_{\leq
1}(B)\neq(0,0)$, then there exists $t_{0}>0$ such that
$\nu^{-}_{\gamma(t)}(B)>0$ for all $t>t_{0}$.

###### Proof.

Using Remark 5.8, there is a $t_{0}$ large enough so that there is a fixed
Harder–Narasimhan factor of $B$, say $B_{0}\in\mathcal{B}^{\beta(t)}$, which
satisfies $\nu_{\gamma(t)}(B_{0})=\nu^{-}_{\gamma(t)}(B)$ for all $t>t_{0}$.
If $\operatorname{ch}_{0}(B_{0})=\operatorname{ch}_{1}(B_{0})=0$ then
$\nu_{\gamma(t)}(B_{0})=\infty$ for all $t\gg 0$ and we are done. Otherwise,
$\operatorname{ch}_{\leq 1}(B_{0})\neq(0,0)$. Then $\nu_{\gamma(t)}(B_{0})$ is
asymptotically positive by the assumption on $\gamma(t)$, thus
$\nu_{\gamma(t)}^{-}(B)>0$ for $t\gg 0$, as desired. ∎

Combining this with Proposition 2.13 we deduce:

######  Proposition 5.13.

Consider an unbounded $\Theta^{-}$-curve $\gamma$ as above. Then the following
are equivalent for an object $E\in D^{b}(X)$.

  1. (1)

$E\in\mathcal{A}^{\gamma(t)}$ for all $t\gg 0$,

  2. (2)

$E\in\mathcal{B}^{\beta(t)}$ for all $t\gg 0$,

  3. (3)

$E\in\mathcal{C}oh(X)$.

If $E\in\mathcal{C}oh(X)$ contains no subsheaf of dimension $0$ then there is
a $t_{0}>0$ such that for all $t>t_{0}$,
$E^{\vee}\in\mathcal{A}^{\gamma^{*}(t)}$.

###### Proof.

When $E$ is torsion free, the claim follows immediately from Proposition 2.13
and the observation that
$\nu^{+}_{\gamma^{*}(t)}(E^{\prime\vee})=-\nu^{-}_{\gamma(t)}(E^{\prime})\to-\infty$
as $t\to\infty$. Otherwise, let $T\subset E$ be its maximal torsion subsheaf.
Then the hypothesis implies that $T^{\vee}\in{\mathcal{A}}^{\gamma^{*}(t)}$
for all $t\gg 0$ and so dualizing $T\to E\to E/T$ we deduce the last part. ∎

######  Corollary 5.14.

Suppose $\gamma$ is an unbounded $\Theta^{-}$-curve $\gamma$ and $E$ is a
sheaf in $\mathcal{A}^{\gamma(t)}$ for all $t\geq 0$. If $0\to F\to E\to G\to
0$ is a short exact sequence in $\mathcal{A}^{\gamma(0)}\cap\mathcal{C}oh(X)$
then this remains a short exact sequence in $\mathcal{A}^{\gamma(t)}$ for all
$t\geq t_{0}$ for some $t_{0}\geq 0$.

###### Proof.

Observe that $0\to F\to E\to G\to 0$ is also a short exact sequence in
$\mathcal{B}^{\beta(0)}$ and so also in $\mathcal{B}^{\gamma(t)}$ for all
$t>0$. But $\nu^{-}_{\gamma(t)}(G)\geq\nu^{-}_{\gamma(t)}(E)$. So
$G\in\mathcal{A}^{\gamma(t)}$ for all $t>0$. The same is not necessarily true
for $F$. But $\nu^{-}_{\gamma(t)}(F)>0$ for all $t\gg 0$. ∎

###  5.3. Asymptotics along $\Lambda_{\overline{\alpha}}^{+}$

We now move to the proof of the second part of Theorem 5.2, starting with a
characterization of objects lying in $\mathcal{B}^{\beta}$ for $\beta\gg 0$.

######  Lemma 5.15.

Let $B\in D^{\rm b}(X)$. There is $\beta_{0}>0$ such that
$B\in\mathcal{B}^{\beta}$ for every $\beta>\beta_{0}$ if and only if the
following conditions hold:

  1. (1)

${\mathcal{H}}^{p}(B)=0$ for $p\neq-1,0$;

  2. (2)

${\mathcal{H}}^{-1}(B)$ is a torsion free sheaf;

  3. (3)

${\mathcal{H}}^{0}(B)$ is a torsion sheaf.

###### Proof.

Set $E:={\mathcal{H}}^{-1}(B)$ and $P:={\mathcal{H}}^{0}(B)$. If
$B\in\mathcal{B}^{\beta}$ for every $\beta\gg 0$, then the first two items
follow immediately. For the third one, just note that

$\lim_{\beta\to\infty}\dfrac{\operatorname{ch}_{1}^{\beta}(P)}{\beta}=-\operatorname{ch}_{0}(P)\geq
0,$

thus $\operatorname{ch}_{0}(P)=0$.

Conversely, we have that $E\in\mathcal{F}_{\beta}$ for $\beta\geq\mu^{+}(E)$
and $P\in\mathcal{T}_{\beta}$ for every $\beta$; it immediately follows that
$B\in\mathcal{B}^{\beta}$ for $\beta\geq\mu^{+}(E)$. ∎

######  Lemma 5.16.

If $B\in D^{\rm b}(X)$ is asymptotically $\nu_{\alpha,\beta}$-semistable along
$\Lambda_{\overline{\alpha}}^{+}$, then $S:=B^{\vee}[-1]$ is a $\mu_{\leq
2}$-semistable sheaf such that $S^{**}/S$ either is empty or has pure
dimension 1.

###### Proof.

For simplicity, set $E:={\mathcal{H}}^{-1}(B)$ and $P:={\mathcal{H}}^{0}(B)$.
We start by cheking that $E$ is reflexive and $\dim P\leq 1$.

Indeed, if $E$ is not reflexive, then
$Q_{E}:=E^{**}/E\in\mathcal{T}_{\beta}\subset\mathcal{B}^{\beta}(X)$ for every
$\beta$, thus $Q_{E}$ is a sub-object of $B$ within $\mathcal{B}^{\beta}(X)$;
but $\nu_{\alpha,\beta}(Q_{E})=+\infty$ for every
$(\alpha,\beta)\in\mathbb{H}$, so we have a contradiction with the asymptotic