# Toric sheaves and flips

Andrew Clarke  Andrew Clarke, Instituto de Matemática, Universidade Federal do
Rio de Janeiro, Av. Athos da Silveira Ramos 149, Rio de Janeiro, RJ,
21941-909, Brazil<EMAIL_ADDRESS>,  Achim NAPAME  Achim Napame, Univ
Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique,
France<EMAIL_ADDRESS>and  Carl Tipler  Carl Tipler, Univ Brest,
UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, France
<EMAIL_ADDRESS>
###### Abstract.

Any toric flip naturally induces an equivalence between the associated
categories of equivariant reflexive sheaves, and we investigate how slope
stability behaves through this functor. On one hand, for a fixed toric sheaf,
and natural polarisations that make the exceptional loci small, we provide a
simple numerical criterion that characterizes when slope stability is
preserved through the flip. On the other hand, for a given flip, we introduce
full subcategories of logarithmic toric sheaves and characterize when
polystability is preserved for all toric sheaves in those subcategories at
once.

###### 2010 Mathematics Subject Classification:

Primary: 53C07, Secondary: 53C55, 14J60

##  1\. Introduction

Introduced by Mumford [16] and generalized by Takemoto [23], slope stability
of vector bundles, and more generally of torsion-free coherent sheaves, can be
used as a device to produce moduli spaces. While slope stability is not a GIT
notion in higher dimension, it behaves well with respect to tensor products
and restrictions, and has a differential geometric interpretation in gauge
theory through the Hitchin–Kobayashi correspondence (see e.g. [14] and
reference therein). In particular, stable bundles, and more generally stable
reflexive sheaves ([1]), are of particular interest in gauge theory and
mathematical physics (see e.g. [13] for a survey on stable sheaves on toric
varieties addressed to the mathematical physics community).

Despite its usefulness, checking stability in practice remains a difficult
problem. Our goal is to add on to the list of known methods to produce stable
sheaves via transformations of the underlying polarised manifold. In the
equivariant context of toric geometry, the behaviour of slope stability
through descent under GIT quotients was studied in [4], while the problem of
pulling back stable sheaves on fibrations was considered in [18]. In this
paper, we study how slope stability is affected through a toric flip between
polarised (simplicial) toric varieties. Those transformations are of
particular interest in several aspects. From the complex geometry point of
view, they form building blocks for toric Minimal Model Program (see [6,
Chapter 15] and references therein), and together with fibrations and blow-ups
addressed in [18], our results complete the study of slope stability through
any type of extremal contraction arising in toric MMP. From the mathematical
physics point of view, toric flips can be seen as singular transitions between
toric varieties. Those transitions are of particular importance given the
construction of Calabi–Yau hypersurfaces in Fano reflexive toric varieties
([2]) and the connections between various Calabi–Yau vacua through conifold
transitions ([3, 21]). Our results then provide a toy model in the study of
stable sheaves through singular transitions between toric varieties (see [5]
for a differential geometric approach to stability of the tangent bundle
through conifold transitions).

Consider a toric flip (see Section 2 for precise definitions):

${X}$${X^{\prime}}$${X_{0}}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$

between two simplicial toric varieties $X$ and $X^{\prime}$. There is a
$\mathbb{Q}$-Cartier divisor $D_{+}\subset X$ naturally attached to the flip,
such that $-D_{+}$ is $\phi$-ample and restricts to the anticanonical divisor
of the $\phi$-contracted fibers (see Section 2.2). By abuse of notation, we
will still denote $D_{+}$ the divisor $\psi_{*}(D_{+})\subset X^{\prime}$.
Then, for some ample Cartier divisor $L_{0}$ on $X_{0}$, there exists
$\varepsilon_{0}>0$ such that the divisors

$L_{-\varepsilon}:=\phi^{*}L_{0}-\varepsilon D_{+}\subset X$

and

$L_{\varepsilon}:=(\phi^{\prime})^{*}L_{0}+\varepsilon D_{+}\subset
X^{\prime}$

define ample $\mathbb{Q}$-Cartier divisors for
$\varepsilon\in(0,\varepsilon_{0})$. Then, our first result (Theorem 4.4)
relates slope stability of toric sheaves on $(X_{0},L_{0})$,
$(X,L_{-\varepsilon})$ and $(X^{\prime},L_{\varepsilon})$ :

######  Theorem 1.1.

Let $\mathcal{E}$ be a torus equivariant reflexive sheaf on $X$. Then, up to
shrinking $\varepsilon_{0}$, we have for all
$\varepsilon\in(0,\varepsilon_{0})$ :

  1. (i)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-stable, then $\mathcal{E}$ (resp.
$\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-stable on $X$ (resp.
$L_{\varepsilon}$-stable on $X^{\prime}$).

  2. (ii)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-unstable, then $\mathcal{E}$ (resp.
$\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-unstable on $X$ (resp.
$L_{\varepsilon}$-unstable on $X^{\prime}$).

  3. (iii)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-semistable, let $\mathfrak{E}$ be the
finite set of equivariant and saturated reflexive subsheaves
$\mathcal{F}\subseteq\phi_{*}\mathcal{E}$ arising in a Jordan–Hölder
filtration of $\phi_{*}\mathcal{E}$. If for any $\mathcal{F}\in\mathfrak{E}$,

(1.1) $\dfrac{c_{1}(\mathcal{E})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{E})}<\dfrac{c_{1}((\phi^{*}\mathcal{F})^{\vee\vee})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{F})}$

then $\mathcal{E}$ is $L_{-\varepsilon}$-stable on $X$.

A similar statement as $(iii)$ holds for $\psi_{*}\mathcal{E}$ with the
reverse inequalities in (1.1) (see Theorem 4.4). The intersection number that
appears in (1.1) is the first order term in the $\varepsilon$-expansion of the
$L_{-\varepsilon}$-slope. As Theorem 1.1 shows, if $\phi_{*}\mathcal{E}$ is
strictly semistable, this term will never allow for both $\mathcal{E}$ and
$\psi_{*}\mathcal{E}$ to be stable at the same time, for the considered
polarisations. As $\mathfrak{E}$ is finite, the numerical criterion in $(iii)$
can be used in practice to produce examples of toric sheaves that go from
being unstable to stable through a toric flip (see Section 4.3).

######  Remark 1.2.

Theorem 1.1 actually holds uniformly for flat families of toric sheaves with
fixed characteristic function (see the discussion at the end of Section 4.2).

While our first result focuses on specific flat families of sheaves for a
given flip, our second result describes toric flips that preserve slope
polystability for all equivariant reflexive sheaves at once, in some adapted
full subcategories. Let’s denote by $\mathfrak{Ref}^{T}(X)$ the category of
torus equivariant reflexive sheaves on $X$. For any given torus invariant
divisor $D\subset X$, we introduce in Section 3.3 a full subcategory
$\mathfrak{Ref}^{T}(X,D)$ of logarithmic toric sheaves. Our terminology is
inspired by the fact that the logarithmic tangent sheaf $\mathcal{T}_{X}(-\log
D)$ belongs to $\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D))$. Setting
$D^{\prime}=\psi_{*}D$, the birational transformation $\psi:X\dashrightarrow
X^{\prime}$ induces an equivalence of categories (still denoted $\psi_{*}$)
between $\mathfrak{Ref}^{T}(X,D)$ and
$\mathfrak{Ref}^{T}(X^{\prime},D^{\prime})$. We will say that the functor
$\psi_{*}$ preserves polystability for a pair of ample classes
$(\alpha,\alpha^{\prime})\in\mathrm{Pic}(X)_{\mathbb{Q}}\otimes\mathrm{Pic}(X^{\prime})_{\mathbb{Q}}$
if for any $\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D))$,
$\mathcal{E}$ is polystable on $(X,\alpha)$ if and only if
$\psi_{*}\mathcal{E}$ is polystable on $(X^{\prime},\alpha^{\prime})$. Then,
our result is as follows (see Theorem 5.1) :

######  Theorem 1.3.

Let $\Sigma$ be the fan of $X$, and let

$D:=\sum_{\rho\in\Delta}D_{\rho}\subset X$

be a torus invariant divisor, for $\Delta\subset\Sigma(1)$. Then, the
following assertions are equivalent, for a pair of ample classes
$(\alpha,\alpha^{\prime})\in\mathrm{Pic}(X)_{\mathbb{Q}}\otimes\mathrm{Pic}(X^{\prime})_{\mathbb{Q}}$.

  1. (i)

The functor
$\psi_{*}:\mathfrak{Ref}^{T}(X,D)\to\mathfrak{Ref}^{T}(X^{\prime},D)$
preserves polystability for $(\alpha,\alpha^{\prime})$.

  2. (ii)

There is $c\in\mathbb{Q}_{>0}$ such that for all
$\mathcal{E}\in\mathfrak{Ref}^{T}(X,D)$,

$\mu_{\alpha}(\mathcal{E})=c\;\mu_{\alpha^{\prime}}(\psi_{*}\mathcal{E}).$

  3. (iii)

There is $c\in\mathbb{Q}_{>0}$ such that for all $\rho\notin\Delta$,

$\deg_{\alpha}D_{\rho}=c\;\deg_{\alpha^{\prime}}D_{\rho}^{\prime}.$

In the above statement, $D_{\rho}$ stands for the torus invariant divisor
associated to a ray $\rho\in\Sigma(1)$, and $\mu_{\alpha}$ stands for the
slope on $(X,\alpha)$. We should point out that condition $(iii)$ becomes very
restrictive when $\Delta$ is small, while $\mathfrak{Ref}^{T}(X,D)$ becomes
smaller for larger $\Delta$’s. Nevertheless, Theorem 1.3 provides a simple
numerical criterion for pairs of classes on $X$ and $X^{\prime}$ to preserve
polystability of specific toric sheaves through the toric flip.

### Acknowledgments

The authors are partially supported by the grant BRIDGES ANR–FAPESP
ANR-21-CE40-0017. AN and CT also benefited from the grant MARGE
ANR-21-CE40-0011.

### Notations and conventions

All varieties we will consider will be normal toric varieties over the complex
numbers. For such a variety $X$, we denote by $T$ its torus, $N$ its lattice
of one parameter subgroups, $M$ its lattice of characters. For $\mathbb{K}$ a
field of characteristic zero, and a lattice $W$, we denote
$W_{\mathbb{K}}:=W\otimes_{\mathbb{Z}}\mathbb{K}$. The fan of $X$ will be
denoted $\Sigma_{X}$, or simply $\Sigma$ when the situation is clear enough,
and we will also use the notation $X_{\Sigma}$ for the variety associated to
the fan $\Sigma$. For a given cone $\sigma\in\Sigma$, we let
$U_{\sigma}=\mathrm{Specm}(\mathbb{C}[\sigma^{\vee}\cap M])$ the affine chart
in $X$, $O(\sigma)\subset X$ the orbit associated to $\sigma$ by the orbit-
cone correspondence, and $V(\sigma)$ the closure in $X$ of $O(\sigma)$.
Finally, for a fan $\Sigma$ and a subset $S\subset N_{\mathbb{R}}$, we denote
by $\Sigma_{S}=\\{\sigma\in\Sigma\>|\>\sigma\subset S\\}$.

##  2\. Background on toric flips

###  2.1. Toric flips

We recall in this section the basics on toric flips (and refer the reader to
[6, Section 3.3] for the definition of toric morphisms and their fan
description). While our presentation differs slightly from [6, Chapter 15], we
will keep most of the notations from this book, and the properties that we
list here can be recovered from [6, Lemma 15.3.7, Lemma 15.3.11 and Theorem
15.3.13]. Let $N$ be a rank $n$ lattice, with $n\geq 3$.

######  Definition 2.1.

A full dimensional strictly convex cone $\sigma_{0}\subset N_{\mathbb{R}}$
will be called a flipping cone if there exist primitive elements
$\\{\nu_{1},\ldots,\nu_{n+1}\\}\in N^{n+1}$ such that

  1. (1)

The cone $\sigma_{0}$ is spanned by the $\nu_{i}$’s :

$\sigma_{0}=\mathrm{Cone}(\nu_{1},\ldots,\nu_{n+1}),$

  2. (2)

there exists $(b_{1},\ldots,b_{n+1})\in\mathbb{Z}^{n+1}$ such that

$\sum_{i=0}^{n+1}b_{i}\nu_{i}=0,$

  3. (3)

the sets $J_{-}=\\{i\>|\>b_{i}<0\\}$ and $J_{+}=\\{i\>|\>b_{i}>0\\}$ both
contain at least $2$ elements.

For a given flipping cone $\sigma_{0}\subset N_{\mathbb{R}}$ as in Definition
2.1, we will denote

$J_{0}=\\{i\>|\>b_{i}=0\\}.$

We also introduce the notation, for any $J\subset\\{1,\ldots,n+1\\}$,

$\sigma_{J}=\mathrm{Cone}(\nu_{i}\>|\>i\in J)$

together with the fans

$\Sigma_{-}=\\{\sigma_{J}\>|\>J_{+}\nsubseteq J\\}$

and

$\Sigma_{+}=\\{\sigma_{J}\>|\>J_{-}\nsubseteq J\\}.$

Identifying $\sigma_{0}$ with the fan of its faces, we see that $\Sigma_{-}$
and $\Sigma_{+}$ provide refinements of $\sigma_{0}$. Those refinements induce
toric morphisms $\phi_{\pm}:X_{\Sigma_{\pm}}\to U_{\sigma_{0}}$, whose
properties are listed below (see [6, Lemma 15.3.11.(c)]) :

######  Lemma 2.2.

The morphisms $\phi_{\pm}:X_{\Sigma_{\pm}}\to U_{\sigma_{0}}$ are surjective
and birational. Their exceptional loci are given by
$V(\sigma_{J_{\pm}})\subset X_{\Sigma_{\pm}}$ and satisfy
$\phi_{\pm}(V(\sigma_{J_{\pm}}))=V(\sigma_{J_{-}\cup J_{+}})$ and $\dim
V(\sigma_{J_{\pm}})=n-|J_{\pm}|$, while $\dim V(\sigma_{J_{-}\cup
J_{+}})=|J_{0}|$.

We now introduce the notion of toric flips that we will use in this paper.

######  Definition 2.3.

Let $X$ and $X^{\prime}$ be two $n$-dimensional simplicial toric varieties
with fans $\Sigma$ and $\Sigma^{\prime}$ and common lattice of one-parameter
subgroups $N$. We will say that they are related by a toric flip if there
exists a normal toric variety $X_{0}$ with fan $\Sigma_{0}$ containing a
flipping cone $\sigma_{0}\in\Sigma_{0}$ such that

$\Sigma_{|\sigma_{0}}=\Sigma_{+},\>\Sigma_{|\sigma_{0}}^{\prime}=\Sigma_{-}\>\textrm{
and
}\Sigma_{|N_{\mathbb{R}}\setminus\sigma_{0}}=\Sigma_{|N_{\mathbb{R}}\setminus\sigma_{0}}^{\prime}=(\Sigma_{0})_{|N_{\mathbb{R}}\setminus\sigma_{0}}.$

In that situation, the refinements $\Sigma$ and $\Sigma^{\prime}$ of
$\Sigma_{0}$ induce toric morphisms $\phi:X\to X_{0}$ and
$\phi^{\prime}:X^{\prime}\to X_{0}$, the latter being called the flip of the
former.

In Definition 2.3, the fans $\Sigma_{\pm}$ are the one associated to the
flipping cone $\sigma_{0}$ as described above. Note that the definition,
together with Lemma 2.2, imply that $X$ and $X^{\prime}$ are birational and
isomorphic in codimension $2$, and that $X_{0}\setminus(\sigma_{J_{-}\cup
J_{+}})$ is simplicial. The situation of Definition 2.3 can be better
summarized in the following commutative diagram :

${X}$${X^{\prime}}$${V(\sigma_{J_{+}})}$${X_{0}}$${V(\sigma_{J_{-}})}$${V(\sigma_{J_{+}\cup
J_{-}})}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\psi}$$\scriptstyle{\iota_{+}}$$\scriptstyle{\phi}$$\scriptstyle{\phi_{-}}$$\scriptstyle{\iota_{-}}$$\scriptstyle{\phi_{+}}$$\scriptstyle{\iota_{0}}$

The maps $\phi$ and $\phi^{\prime}$ are the toric morphisms induced by the
refinements $\Sigma$ and $\Sigma^{\prime}$ of $\Sigma_{0}$, while the maps
$\iota_{+}$, $\iota_{-}$ and $\iota_{0}$ denote inclusions. Finally,

$\psi=(\phi^{\prime})^{-1}\circ\phi:X\dashrightarrow X^{\prime}$

is the birational morphism that is an isomorphism away from
$V(\sigma_{J_{\pm}})$. From now on, we fix $X$ and $X^{\prime}$ two simplicial
toric varieties related by a toric flip $\psi$, and keep the notations from
the previous diagram.

###  2.2. The exceptional loci and relatively ample divisors

We will be interested in stability later on, whose definition requires
polarisations. We hence turn to the description of the fibers of $\phi_{\pm}$,
and describe some $\phi$-ample (and $\phi^{\prime}$-ample) divisors. For a
cone $\sigma\subset N_{\mathbb{R}}$, we denote $N_{\sigma}\subset N$ the
sublattice

$N_{\sigma}=\mathrm{Span}(\sigma)\cap N,$

and the quotient map

$\pi_{\sigma}:N\to N(\sigma):=N/N_{\sigma}.$

Recall that from the orbit-cone correspondence (see [6, Section 3.2,
Proposition 3.2.7]), the toric variety $V(\sigma)$ can be obtained as the
toric variety associated to the fan of cones in $(N/N_{\sigma})_{\mathbb{R}}$
:

$\mathrm{Star}(\sigma)=\\{\pi_{\sigma}(\tau)\>|\>\sigma\preceq\tau\\}.$

In particular, the lattices of one parameter subgroups of $V(\sigma_{J_{-}})$
and $V(\sigma_{J_{-}\cup J_{+}})$ are respectively $N/N_{\sigma_{J_{-}}}$ and
$N/N_{\sigma_{J_{-}\cup J_{+}}}$. One can show that the projection map

$N/N_{\sigma_{J_{-}}}\to N/N_{\sigma_{J_{-}\cup J_{+}}}$

is compatible with the fans $\mathrm{Star}(\sigma_{J_{-}})$ and
$\mathrm{Star}(\sigma_{J_{-}\cup J_{+}})$ and induces the toric morphism

$\phi_{-}:V(\sigma_{J_{-}})\to V(\sigma_{J_{-}\cup J_{+}}).$

Those lattices fit naturally in the sequence

$0\to N_{\sigma_{J_{-}\cup J_{+}}}/N_{\sigma_{J_{-}}}\to
N/N_{\sigma_{J_{-}}}\to N/N_{\sigma_{J_{-}\cup J_{+}}}\to 0.$

As we are interested in the fibers of $\phi_{-}$, we introduce the quotient
lattice

$N_{\mathcal{R}}:=N_{\sigma_{J_{-}\cup J_{+}}}/N_{\sigma_{J_{-}}},$

and denote the projection $N_{\sigma_{J_{-}\cup J_{+}}}\to N_{\mathcal{R}}$,
and its $\mathbb{R}$-linear extension, by $u\mapsto\overline{u}$. We finally
introduce the fan

$\Sigma_{\mathcal{R}}:=\\{\overline{\sigma}_{J}\>|\>J\subsetneq J_{+}\\},$

and the associated toric variety $X_{\mathcal{R}}$.

######  Remark 2.4.

We keep the notation $X_{\mathcal{R}}$ to be consistent with [6], where the
$\mathcal{R}$ stands for an extremal ray being responsible for the flip in the
context of toric MMP.

For the following, see [6, Lemma 15.4.2 and Proposition 15.4.5.(c)].

######  Proposition 2.5.

The fibers of $\phi_{-}$ are isomorphic to the $\mathbb{Q}$-Fano toric variety
$X_{\mathcal{R}}$. Moreover, $X_{\mathcal{R}}$ has dimension $|J_{+}|-1$ and
Picard rank one.

From the above, we deduce that the anticanonical divisor of $X_{\mathcal{R}}$

$-K_{\mathcal{R}}=\sum_{\rho\in\Sigma_{\mathcal{R}}(1)}D_{\rho}$

is $\mathbb{Q}$-Cartier and ample. Note that by construction it can be written

$-K_{\mathcal{R}}=\sum_{i\in
J_{+}}D_{\overline{\mathbb{R}_{+}\cdot\nu_{i}}}=\sum_{i\in
J_{+}}D_{\overline{\rho}_{i}}$

where we set

$\rho_{i}=\mathbb{R}_{+}\cdot\nu_{i},$

and $D_{\rho}$ stands for the torus-invariant divisor associated to the ray
$\rho$. An easy exercise, using the orbit-cone correspondence, together with
[6, Proposition 15.5.1], shows that

######  Proposition 2.6.

The $\mathbb{Q}$-Cartier torus invariant divisor

(2.1) $-D_{J_{+}}:=-\sum_{i\in J_{+}}D_{\rho_{i}}\in\mathrm{Div}(X)$

is $\phi$-ample, while the $\mathbb{Q}$-Cartier invariant divisor

(2.2) $D_{J_{+}}^{\prime}:=\psi_{*}(D_{J_{+}})=\sum_{i\in
J_{+}}D_{\rho_{i}}^{\prime}\in\mathrm{Div}(X^{\prime})$

is $\phi^{\prime}$-ample.

######  Remark 2.7.

In the above proposition, we let a ′ to $D^{\prime}$ to remember that the
torus invariant divisor $D^{\prime}$ is taken as the orbit closure of some ray
in $X^{\prime}$. As $\psi:X\dashrightarrow X^{\prime}$ is an isomorphism in
codimension $2$, it induces an isomorphism $\psi_{*}$ between the groups of
torus invariant $\mathbb{Q}$-Cartier divisors on $X$ and $X^{\prime}$, which
can be written on a basis $D_{\rho}\mapsto D_{\rho}^{\prime}$ for any
$\rho\in\Sigma(1)=\Sigma^{\prime}(1)$. Then, to ease notation later on, we
will omit the ′ to $D_{\rho}^{\prime}$, the situation being clear enough
whether $D_{\rho}$ is considered as a divisor on $X$ or $X^{\prime}$. We will
also simply denote $D_{J_{+}}$ and $D_{J_{+}}^{\prime}$ by

$D_{+}=\sum_{i\in J_{+}}D_{\rho_{i}},$

so that the conclusion of Proposition 2.6 is that $-D_{+}$ is $\phi$-ample on
$X$ and $D_{+}$ is $\phi^{\prime}$-ample on $X^{\prime}$.

Finally, we note that we could have considered the divisor

$D_{-}=\sum_{i\in J_{-}}D_{\rho_{i}},$

which is $\phi$-ample (while $-D_{-}$ is $\phi^{\prime}$-ample). However, the
wall relation

$\sum_{i\in J_{-}}b_{i}\nu_{i}+\sum_{i\in J_{+}}b_{i}\nu_{i}=0$

from Definition 2.1 implies that from the intersection theory point of view,
computing slopes with $D_{+}$ or $D_{-}$ will produce the same results
regarding stability notions (see e.g. [6, Section 6.4, Proposition 6.4.4]).

##  3\. The flip functor

###  3.1. Equivariant sheaves and Klyachko’s equivalence

We turn now to the description of the flip functor, which requires first
introducing the categories of torus equivariant reflexive sheaves. For a given
toric variety $X$ with fan $\Sigma$, a torus equivariant reflexive sheaf is a
reflexive sheaf $\mathcal{E}$ on $X$ together with an isomorphism

$\varphi:\alpha^{*}\mathcal{E}\to\pi_{2}^{*}\mathcal{E}$

satisfying some cocyle conditions, where $\alpha:T\times X\to X$ and
$\pi_{2}:T_{N}\times X\to X$ stand for the torus action and the projection on
$X$ respectively (see for example [20, Section 5]).

######  Definition 3.1.

A toric sheaf is a torus equivariant reflexive sheaf.

Klyachko has shown (see [12] for locally free sheaves and [20] in general)
that any toric sheaf is uniquely described by a family of filtrations, denoted

$(E,E^{\rho}(i))_{\rho\in\Sigma(1),i\in\mathbb{Z}}.$

Here, $E$ stands for a finite dimensional complex vector space of dimension
$\mathrm{rank}(\mathcal{E})$, and for each ray $\rho\in\Sigma(1)$,
$(E^{\rho}(i))_{i\in\mathbb{Z}}$ is a bounded increasing filtration of $E$ (we
will use increasing filtrations as in [20], rather than decreasing ones as in
[12]). Then, the equivariant reflexive sheaf $\mathcal{E}$ is recovered from
the formula, for $\sigma\in\Sigma$ :

$\Gamma(U_{\sigma},\mathcal{E}):=\bigoplus_{m\in
M}\bigcap_{\rho\in\sigma(1)}E^{\rho}(\langle
m,u_{\rho}\rangle)\otimes\chi^{m}$

where $u_{\rho}\in N$ is the primitive generator of $\rho$,
$\langle\cdot,\cdot\rangle$ the duality pairing and $\chi^{m}$ the weight $m$
character function.

A morphism between two families of filtrations

$b:(E_{1},E_{1}^{\rho}(i))_{\rho\in\Sigma(1),i\in\mathbb{Z}}\to(E_{2},E_{2}^{\rho}(i))_{\rho\in\Sigma(1),i\in\mathbb{Z}}$

is a linear map $b:E_{1}\to E_{2}$ that satisfies

$b(E_{1}^{\rho}(i))\subset E_{2}^{\rho}(i)$

for any $\rho\in\Sigma(1)$ and $i\in\mathbb{Z}$. Any such morphism corresponds
uniquely to a morphism between the associated reflexive sheaves, and a
fundamental result of Klyachko and Perling ([12, 20]) asserts that the
categories of families of filtrations and of toric sheaves are equivalent. For
our purposes, it seems more natural to restrict a bit the definition of
morphisms. We will consider morphisms between toric sheaves $\mathcal{E}_{i}$
to be equivariant morphisms of coherent sheaves
$\beta:\mathcal{E}_{1}\to\mathcal{E}_{2}$ that satisfy that
$\mathrm{Im}(\beta)$ is a saturated reflexive subsheaf of $\mathcal{E}_{2}$.
Those morphisms correspond through Klyachko’s equivalence to linear maps
$b:E_{1}\to E_{2}$ such that

$b(E_{1}^{\rho}(i))=b(E)\cap E_{2}^{\rho}(i)$

for any $(\rho,i)$, as can be seen via [18, Lemma 2.15] for example. We denote
by $\mathfrak{Ref}^{T}(X)$ on one hand and by $\mathfrak{Filt}(X)$ on the
other hand, the categories of toric sheaves and of families of filtrations,
endowed with those classes of morphisms. We will denote by

$\mathfrak{Kl}:\mathfrak{Filt}(X)\to\mathfrak{Ref}^{T}(X)$

Klyachko’s functor as described above. Then, Klyachko and Perling’s work
readily implies the following :

######  Theorem 3.2 ([12, 20]).

The functor $\mathfrak{Kl}$ is an equivalence of categories.

######  Remark 3.3.

A nice feature of the categories $\mathfrak{Filt}(X)$ and
$\mathfrak{Ref}^{T}(X)$ is that they are abelian. This is no longer true when
we consider all morphisms of reflexive sheaves, as for example the quotient of
$\mathcal{O}_{\mathbb{P}^{1}}$ by the subsheaf
$\mathcal{O}_{\mathbb{P}^{1}}(1)$ is torsion, and hence not reflexive.

###  3.2. Flip functor

Assume now that $\phi^{\prime}:X^{\prime}\to X_{0}$ is the flip of $\phi:X\to
X_{0}$ as in the previous section. As $\phi$ and $\phi^{\prime}$ are
isomorphisms in codimension $2$, we have

$\Sigma(1)=\Sigma_{0}(1)=\Sigma^{\prime}(1).$

We deduce that there are equivariant injections $i$ (resp. $i_{0}$ and
$i^{\prime}$) of the $T$-invariant Zariski open set

$U:=\bigcup_{\tau\in\Sigma(0)\cup\Sigma(1)}U_{\tau}$

into $X$ (resp. $X_{0}$ and $X^{\prime}$). Then, from [10, Proposition 1.6],
we deduce that for any toric sheaf
$\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X))$ (resp.
$\mathcal{F}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X_{0}))$ and
$\mathcal{G}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X^{\prime}))$), we have

$i_{*}(\mathcal{E}_{|U})\simeq\mathcal{E}$

(resp. $(i_{0})_{*}(\mathcal{F}_{|U})\simeq\mathcal{F}$ and
$i^{\prime}_{*}(\mathcal{G}_{|U})\simeq\mathcal{G}$). As reflexive sheaves are
normal, meaning their sections extend over codimenion $2$ Zariski closed sets,
we have :

######  Proposition 3.4.

The pushforward $i_{*}$ (resp. $i^{\prime}_{*}$ and $(i_{0})_{*}$) induces an
equivalence of categories

$i_{*}:\mathfrak{Ref}^{T}(U)\to\mathfrak{Ref}^{T}(X)$

(resp. $\mathfrak{Ref}^{T}(U)\simeq\mathfrak{Ref}^{T}(X_{0})$ and
$\mathfrak{Ref}^{T}(U)\simeq\mathfrak{Ref}^{T}(X^{\prime})$). Hence, we have
equivalences

$\mathfrak{Ref}^{T}(X)\simeq\mathfrak{Ref}^{T}(X_{0})\simeq\mathfrak{Ref}^{T}(X^{\prime}).$

It is straightforward to check that the equivalence

$\mathfrak{Ref}^{T}(X)\simeq\mathfrak{Ref}^{T}(X_{0})$

is induced by the pushforward $\phi_{*}$ while the equivalence

$\mathfrak{Ref}^{T}(X^{\prime})\simeq\mathfrak{Ref}^{T}(X_{0})$

is induced by $\phi_{*}^{\prime}$. Moreover, the categories of families of
filtrations on $U$, $X$, $X^{\prime}$ and $X_{0}$ are readily the same, and
the above equivalence of categories simply correspond to the self-equivalence
of $\mathfrak{Filt}(X)$ induced by the identity on objects and morphisms.

######  Definition 3.5.

We define the flip functor

$\psi_{*}:\mathfrak{Ref}^{T}(X)\to\mathfrak{Ref}^{T}(X^{\prime})$

to be the composition of functors induced by $((\phi^{\prime})^{-1})_{*}$ and
$\phi_{*}$.

We conclude this section by noting that the flip functor sends the tangent
sheaf $\mathcal{T}_{X}$ of $X$ to the tangent sheaf $\mathcal{T}_{X^{\prime}}$
of $X^{\prime}$:

######  Lemma 3.6.

We have

$\phi_{*}\mathcal{T}_{X}=\mathcal{T}_{X_{0}}=\phi^{\prime}_{*}\mathcal{T}_{X^{\prime}}.$

###### Proof.

It follows directly from the description of the family of filtrations
associated to the tangent sheaf, that only depends on $N$, $\Sigma(1)$ and the
primitive generators of elements $\rho\in\Sigma(1)$ (see [12, Example 2.3(5)
on page 350]). As this data is the same for $X$, $X^{\prime}$ and $X_{0}$, and
as the functors induced by $\phi_{*}$ and $\phi^{\prime}_{*}$ correspond to
the identity functor on

$\mathfrak{Filt}(X)=\mathfrak{Filt}(X_{0})=\mathfrak{Filt}(X^{\prime}),$

the result follows. ∎

###  3.3. The logarithmic subcategories

In Section 5, we will be interested in specific subcategories of
$\mathfrak{Ref}^{T}(X)$ and $\mathfrak{Ref}^{T}(X^{\prime})$. For any
$\Delta\subset\Sigma(1)$, we introduce the torus invariant divisor

$D_{\Delta}:=\sum_{\rho\in\Delta}D_{\rho}$

and the full subcategory $\mathfrak{Ref}^{T}(X,D_{\Delta})$ of
$\mathfrak{Ref}^{T}(X)$ whose objects are the toric sheaves on $X$ whose
associated families of filtrations
$(E,E^{\rho}(i))_{\rho\in\Sigma(1),i\in\mathbb{Z}}$ satisfy for all
$\rho\in\Delta$:

(3.1) $E^{\rho}(i)=\left\\{\begin{array}[]{ccc}0&\mathrm{if}&i<0\\\
E&\mathrm{if}&i\geq 0.\end{array}\right.$

######  Remark 3.7.

From [17, Theorem 1.1], the logarithmic tangent sheaf $\mathcal{T}_{X}(-\log
D_{\Delta})$ belongs to $\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D_{\Delta}))$,
which justify our choice of terminology.

It is then straightforward to see that the flip functor $\psi_{*}$ induces an
equivalence between $\mathfrak{Ref}^{T}(X,D_{\Delta})$ and
$\mathfrak{Ref}^{T}(X^{\prime},D_{\Delta}^{\prime})$, where we use
$D_{\Delta}^{\prime}$ to denote $\sum_{\rho\in\Delta}D_{\rho}^{\prime}$. Note
also that $\psi_{*}$ sends $\mathcal{T}_{X}(-\log D_{\Delta})$ to
$\mathcal{T}_{X^{\prime}}(-\log D_{\Delta}^{\prime})$.

##  4\. Flips and stability for a given sheaf

###  4.1. Slope stability of toric sheaves

Let $(X,L)$ be a polarised complex variety. Recall that a reflexive sheaf
$\mathcal{E}$ on $(X,L)$ is said to be slope stable (resp. slope semistable),
or simply stable (resp. semistable) for short, if for any coherent and
saturated subsheaf $\mathcal{F}\subset\mathcal{E}$ of strictly smaller rank,
one has

$\mu_{L}(\mathcal{F})<\mu_{L}(\mathcal{E}),$

(resp. $\mu_{L}(\mathcal{F})\leq\mu_{L}(\mathcal{E})$), where for any coherent
torsion-free sheaf $\mathcal{F}$, the slope $\mu_{L}(\mathcal{F})$ is defined
by

$\mu_{L}(\mathcal{F})=\frac{c_{1}(\mathcal{F})\cdot
L^{n-1}}{\mathrm{rank}(\mathcal{F})}\in\mathbb{Q}.$

A polystable sheaf is a direct sum of stable ones with the same slope. A sheaf
will be called unstable if it is not semistable.

######  Remark 4.1.

When referring to a specific polarisation $L$ used to define stability
notions, we will use the terminology $L$-stable (resp. $L$-unstable,
$L$-semistable, etc).

A remarkable fact, proved by Kool ([15, Proposition 4.13]), is that if we
assume $X$ and $\mathcal{E}$ to be toric, to check stability for
$\mathcal{E}$, it is enough to compare slopes with equivariant and saturated
reflexive subsheaves, that is sub objects of $\mathcal{E}$ in
$\mathfrak{Ref}^{T}(X)$ (note that this was proved in the smooth case by Kool,
but it was noticed in [4] that the proof extends in the normal case). If
$(E,E^{\rho}(\bullet))_{\rho\in\Sigma(1)}$ stands for the family of
filtrations of $\mathcal{E}$, any saturated equivariant reflexive subsheaf of
$\mathcal{E}$ is associated to a family of filtrations of the form $(F,F\cap
E^{\rho}(i))_{\rho\in\Sigma(1),i\in\mathbb{Z}}$ for some vector subspace
$F\subsetneq E$ (see [18, Lemma 2.15]). Moreover, Klyachko’s formula for the
first Chern class ([4, Corollary 2.18]) is

(4.1)
$\mu_{L}(\mathcal{E})=-\frac{1}{\mathrm{rank}(\mathcal{E})}\sum_{\rho\in\Sigma(1)}\iota_{\rho}(\mathcal{E})\,\deg_{L}(D_{\rho}),$

where $\deg_{L}(D_{\rho})$ is the degree with respect to $L$, and

$\iota_{\rho}(\mathcal{E}):=\sum_{i\in\mathbb{Z}}i\left(\dim(E^{\rho}(i))-\dim(E^{\rho}(i-1))\right).$

To sum up, we have

######  Proposition 4.2.

The toric sheaf associated to
$(E,E^{\rho}(i))_{\rho\in\Sigma(1),i\in\mathbb{Z}}$ is stable if and only if
for any subspace $F\subsetneq E$, we have

$\frac{1}{\dim(F)}\sum_{\rho\in\Sigma(1)}\iota_{\rho}(F)\,\deg_{L}(D_{\rho})>\frac{1}{\dim(E)}\sum_{\rho\in\Sigma(1)}\iota_{\rho}(\mathcal{E})\,\deg_{L}(D_{\rho}),$

where

$\iota_{\rho}(F):=\displaystyle\sum_{i\in\mathbb{Z}}i\left(\dim(F\cap
E^{\rho}(i))-\dim(F\cap E^{\rho}(i-1))\right).$

The similar statement holds for semistability by replacing the strict
inequality by a large inequality.

######  Remark 4.3.

As observed in Section 3.2, for a given flip as in Section 2.1, the families
of filtrations for $\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X))$,
$\psi_{*}\mathcal{E}$ and $\phi_{*}\mathcal{E}$ are the same. We thus have the
equalities
$\iota_{\rho}(\mathcal{E})=\iota_{\rho}(\phi_{*}\mathcal{E})=\iota_{\rho}(\psi_{*}\mathcal{E})$.
Then, to compare slopes on $X$, $X_{0}$ and $X^{\prime}$, only the terms
coming from the degrees of the invariant divisors $D_{\rho}$’s will vary
according to the polarisations on each variety.

###  4.2. Main result and its proof

Consider now $\phi:X\to X_{0}$ and its toric flip $\phi^{\prime}:X^{\prime}\to
X_{0}$ as defined in Section 2. From Proposition 2.6 (recall also Remark 2.7),
for any ample Cartier divisor $L_{0}$ on $X_{0}$, there exists
$\varepsilon_{0}>0$ such that the divisors

$L_{-\varepsilon}:=\phi^{*}L_{0}-\varepsilon D_{+}$

on $X$ and

$L_{\varepsilon}:=(\phi^{\prime})^{*}L_{0}+\varepsilon D_{+}$

on $X^{\prime}$ define ample $\mathbb{Q}$-Cartier divisors for
$\varepsilon\in(0,\varepsilon_{0})$. We will then be interested in the
behaviour of stability for toric sheaves related by the flip functor on
$(X,L_{-\varepsilon})$ and $(X^{\prime},L_{\varepsilon})$, for
$0<\varepsilon<\varepsilon_{0}$. Note that a necessary condition for stability
of an element $\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X))$ under those
polarisations is $L_{0}$-semistability of $\phi_{*}\mathcal{E}$.

Conversely, if the sheaf $\phi_{*}\mathcal{E}$ is $L_{0}$-semistable, it then
admits a Jordan-Hölder filtration

$0=\mathcal{E}_{1}\subseteq\mathcal{E}_{2}\subseteq\ldots\subseteq\mathcal{E}_{\ell}=\phi_{*}\mathcal{E}$

by slope semistable coherent and saturated subsheaves with stable quotients of
same slope as $\phi_{*}\mathcal{E}$ (see e.g. [11]). We denote by

$\mathrm{Gr}(\phi_{*}\mathcal{E}):=\bigoplus_{i=1}^{\ell-1}\mathcal{E}_{i+1}/\mathcal{E}_{i}$

the graded object of $\phi_{*}\mathcal{E}$ and by $\mathfrak{E}$ the finite
set of equivariant and saturated reflexive subsheaves
$\mathcal{F}\subseteq\phi_{*}\mathcal{E}$ arising in a Jordan-Hölder
filtration of $\phi_{*}\mathcal{E}$. Note that by Proposition 3.4, for any
$\mathcal{F}\in\mathfrak{E}$, $(\phi^{*}\mathcal{F})^{\vee\vee}$ (resp.
$((\phi^{\prime})^{*}\mathcal{F})^{\vee\vee}$) is saturated in $\mathcal{E}$
(resp. in $\psi_{*}\mathcal{E}$).

######  Theorem 4.4.

Let $\mathcal{E}$ be a toric sheaf on $X$. Then, up to shrinking
$\varepsilon_{0}$, we have for all $\varepsilon\in(0,\varepsilon_{0})$ :

  1. (i)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-stable, then $\mathcal{E}$ (resp.
$\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-stable on $X$ (resp.
$L_{\varepsilon}$-stable on $X^{\prime}$).

  2. (ii)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-unstable, then $\mathcal{E}$ (resp.
$\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-unstable on $X$ (resp.
$L_{\varepsilon}$-unstable on $X^{\prime}$).

  3. (iii)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-semistable, and if for any
$\mathcal{F}\in\mathfrak{E}$,

$\dfrac{c_{1}(\mathcal{E})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{E})}<\dfrac{c_{1}((\phi^{*}\mathcal{F})^{\vee\vee})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{F})}$

then $\mathcal{E}$ (resp. $\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-stable
on $X$ (resp. $L_{\varepsilon}$-unstable on $X^{\prime}$).

  4. (iv)

If $\phi_{*}\mathcal{E}$ is $L_{0}$-semistable, and if for any
$\mathcal{F}\in\mathfrak{E}$,

$\dfrac{c_{1}(\mathcal{E})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{E})}>\dfrac{c_{1}((\phi^{*}\mathcal{F})^{\vee\vee})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{F})}$

then $\mathcal{E}$ (resp. $\psi_{*}\mathcal{E}$) is
$L_{-\varepsilon}$-unstable on $X$ (resp. $L_{\varepsilon}$-stable on
$X^{\prime}$).

######  Remark 4.5.

Note that, given semistability of $\phi_{*}\mathcal{E}$, the numerical
criterion in points $(iii)$ and $(iv)$ only requires to test a finite number
of inequalities, as $\mathfrak{E}$ is finite. This makes this criterion useful
in practice.

Before proving this theorem, we first recall some facts on intersection
products in toric varieties that will be used. Let $\\{u_{1},\ldots,u_{k}\\}$
be a set of primitive elements of $N$ such that
$\sigma=\mathrm{Cone}(u_{1},\ldots,u_{k})$ is simplicial. We define
$\mathrm{mult}(\sigma)$ as the index of the sublattice
$\mathbb{Z}u_{1}+\ldots+\mathbb{Z}u_{k}$ in
$N_{\sigma}=\mathrm{Span}(\sigma)\cap N$. If $\Sigma$ is simplicial, according
to [9, Section 5.1], one can consider intersections of cycles or cycle classes
with rational coefficients. The Chow group

$A^{\bullet}(X)_{\mathbb{Q}}=\bigoplus_{p=0}^{n}A^{p}(X)\otimes\mathbb{Q}=\bigoplus_{p=0}^{n}A_{n-p}(X)\otimes\mathbb{Q}$

has the structure of a graded $\mathbb{Q}$-algebra and by [6, Lemma 12.5.2],
if $\rho_{1},\ldots,\rho_{d}\in\Sigma(1)$ are distinct, then in
$A^{\bullet}(X)_{\mathbb{Q}}$, we have

(4.2)
$[D_{\rho_{1}}]\cdot[D_{\rho_{2}}]\cdots[D_{\rho_{d}}]=\left\\{\begin{array}[]{ll}\dfrac{1}{\mathrm{mult}(\sigma)}[V(\sigma)]&\text{if
$\sigma=\rho_{1}+\ldots+\rho_{d}\in\Sigma$}\\\
0&\text{otherwise.}\end{array}\right..$

If $\chi^{m}$ is the weighted $m$ character function on $X_{\Sigma}$, then by
[6, Proposition 4.1.2 and (12.5.4)], the divisor of $\chi^{m}$ is given by

(4.3) ${\rm div}(\chi^{m})=\sum_{\rho\in\Sigma(1)}\langle m,u_{\rho}\rangle
D_{\rho}$

and ${\rm div}(\chi^{m})\sim_{\rm lin}0$ in $A^{1}(X_{\Sigma})$.

###### Proof of Theorem 4.4.

We first prove that for any $\rho\in\Sigma(1)$,

(4.4) $\psi_{\ast}(D_{\rho}\cdot D_{+})=D^{\prime}_{\rho}\cdot
D^{\prime}_{+}.$

We use the notations of Section 2.1 for the toric flip, and set

$\Delta=\\{\mathrm{Cone}(\nu_{i}):i\in J_{+}\\}.$

Recall from Definition 2.1 that $J_{+}$ and $J_{-}$ have at least two
elements. It then follows from the definition of $\Sigma_{\pm}$ that for any
$\rho\in\sigma_{0}(1)\setminus\Delta$, and any $j\in J_{+}$,
$\rho+\mathrm{Cone}(\nu_{j})$ is a two-dimensional cone of $\Sigma_{+}$ and
$\Sigma_{-}$. Together with the definition of $\psi$, we deduce that in the
Chow ring $A^{\bullet}(X^{\prime})_{\mathbb{Q}}$,

$\psi_{\ast}(D_{\rho}\cdot D_{\rho_{j}})=D^{\prime}_{\rho}\cdot
D^{\prime}_{\rho_{j}}.$

If $\rho\in\Sigma(1)\setminus\sigma_{0}(1)$, then for any $j\in J_{+}$,

$\rho+\mathrm{Cone}(\nu_{j})\notin\\{\tau:\tau\preceq\sigma_{0}\\}.$

As by Definition 2.3,

$\Sigma_{|N_{\mathbb{R}}\setminus\sigma_{0}}=\Sigma_{|N_{\mathbb{R}}\setminus\sigma_{0}}^{\prime}=(\Sigma_{0})_{|N_{\mathbb{R}}\setminus\sigma_{0}},$

we deduce that:

  * •

either
$\rho+\mathrm{Cone}(\nu_{j})\in\Sigma_{0}\setminus\\{\tau:\tau\preceq\sigma_{0}\\}$,
and then $\psi_{\ast}(D_{\rho}\cdot D_{\rho_{j}})=D^{\prime}_{\rho}\cdot
D^{\prime}_{\rho_{j}}$ in $A^{\bullet}(X^{\prime})_{\mathbb{Q}}$;

  * •

or
$\rho+\mathrm{Cone}(\nu_{j})\notin\Sigma_{0}\setminus\\{\tau:\tau\preceq\sigma_{0}\\}$,
in which case $D_{\rho}\cdot D_{\rho_{j}}=0$ in $A^{\bullet}(X)_{\mathbb{Q}}$
and $D^{\prime}_{\rho}\cdot D^{\prime}_{\rho_{j}}=0$ in
$A^{\bullet}(X^{\prime})_{\mathbb{Q}}$.

This proves that for any $\rho\in\Sigma(1)\setminus\Delta$ and any $j\in
J_{+}$,

$\psi_{*}(D_{\rho}\cdot D_{\rho_{j}})=D^{\prime}_{\rho}\cdot
D^{\prime}_{\rho_{j}}$

and then by linearity and definition of $D_{+}$ :

$\psi_{*}(D_{\rho}\cdot D_{+})=D^{\prime}_{\rho}\cdot D^{\prime}_{+}.$

We now assume that $\rho\in\Delta$. By Lemma 2.2, one has $\dim
V(\sigma_{J_{+}})=n-|J_{+}|$; therefore $\sigma_{J_{+}}$ is a simplicial cone
and then $\\{\nu_{j}:j\in J_{+}\\}$ form a part of a $\mathbb{Q}$-basis of
$N\otimes_{\mathbb{Z}}\mathbb{Q}$. Let $\\{\nu_{j}^{*}:j\in J_{+}\\}$ be a
part of a $\mathbb{Q}$-basis of $M\otimes_{\mathbb{Z}}\mathbb{Q}$ such that
for any $i,j\in J_{+}$,

$\langle\nu_{j}^{*},\nu_{i}\rangle=\left\\{\begin{array}[]{ll}0&\text{if
$i\neq j$}\\\ 1&\text{if $i=j$}\end{array}\right..$

For any $j\in J_{+}$, there is $a_{j}\in\mathbb{N}^{*}$ such that
$a_{j}\nu_{j}^{*}\in M$. By using (4.3) with $m=a_{j}\nu_{j}^{*}$, we get

$a_{j}D_{\rho_{j}}\sim_{\rm lin}-\sum_{\rho\in\Sigma(1)\setminus\Delta}\langle
a_{j}\nu_{j}^{*},u_{\rho}\rangle D_{\rho}.$

on $X$ and $X^{\prime}$. By the first cases, we deduce that for any $j\in
J_{+}$,

$\displaystyle\psi_{*}(D_{\rho_{j}}\cdot D_{+})$
$\displaystyle=-\psi_{*}\left(\sum_{\rho\in\Sigma(1)\setminus\Delta}\langle\nu_{j}^{*},u_{\rho}\rangle
D_{\rho}\cdot D_{+}\right)$
$\displaystyle=-\sum_{\rho\in\Sigma(1)\setminus\Delta}\langle\nu_{j}^{*},u_{\rho}\rangle
D^{\prime}_{\rho}\cdot D^{\prime}_{+}$
$\displaystyle=D^{\prime}_{\rho_{j}}\cdot D^{\prime}_{+}$

in $A^{\bullet}(X^{\prime})_{\mathbb{Q}}$. This concludes the proof of (4.4).

We can now compute the slopes. We have

$\displaystyle(L_{-\varepsilon})^{n-1}=(\phi^{*}L_{0})^{n-1}-(n-1)\varepsilon
D_{+}\cdot(\phi^{*}L_{0})^{n-2}+o(\varepsilon)\quad\text{and}$
$\displaystyle(L_{\varepsilon})^{n-1}=((\phi^{\prime})^{*}L_{0})^{n-1}+(n-1)\varepsilon
D_{+}\cdot((\phi^{\prime})^{*}L_{0})^{n-2}+o(\varepsilon).$

By the Projection formula [8, Proposition 2.3], for any $\rho\in\Sigma(1)$,
one has

$D_{\rho}\cdot(\phi^{*}L_{0})^{n-1}=(\phi_{*}D_{\rho})\cdot(L_{0})^{n-1}=\deg_{L_{0}}(D_{\rho})$

and

$D_{\rho}\cdot((\phi^{\prime})^{*}L_{0})^{n-1}=(\phi^{\prime}_{*}D_{\rho})\cdot(L_{0})^{n-1}=\deg_{L_{0}}(D_{\rho}).$

By (4.4), we get

$D_{\rho}\cdot D_{+}\cdot(\phi^{*}L_{0})^{n-2}=D_{\rho}\cdot
D_{+}\cdot((\phi^{\prime})^{*}L_{0})^{n-2}.$

Hence, for any coherent sheaf $\mathcal{E}$ on $X$,

$\mu_{L_{-\varepsilon}}(\mathcal{E})=\mu_{L_{0}}(\phi_{*}\mathcal{E})-\dfrac{c_{1}(\mathcal{E})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{E})}(n-1)\varepsilon+o(\varepsilon)$

and

$\mu_{L_{\varepsilon}}(\psi_{*}\mathcal{E})=\mu_{L_{0}}(\phi_{*}\mathcal{E})+\dfrac{c_{1}(\mathcal{E})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{E})}(n-1)\varepsilon+o(\varepsilon).$

Assume now $\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X))$ is given by the
family of filtrations $(E,E^{\rho}(j))$. By Proposition 4.2, to check
stability of $\mathcal{E}$, it is enough to compare the slope of $\mathcal{E}$
with the slope of any equivariant reflexive sheaf $\mathcal{F}$ given by the
family of filtrations $(F,F\cap E^{\rho}(j))$ for $F\subsetneq E$ a subspace.
As the set of vector subspaces $F\subset E$ that are necessary to test slopes
on is actually finite (see [18, Lemma 2.17]), we deduce from the above
$\varepsilon$-expansions for the slopes that there is $\varepsilon_{0}>0$ such
that for all $\varepsilon\in(0,\varepsilon_{0})$:

  * •

if $\phi_{*}\mathcal{E}$ is $L_{0}$-stable, then $\mathcal{E}$ (resp.
$\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-stable on $X$ (resp.
$L_{\varepsilon}$-stable on $X^{\prime}$);

  * •

and if $\phi_{*}\mathcal{E}$ is $L_{0}$-unstable, then $\mathcal{E}$ (resp.
$\psi_{*}\mathcal{E}$) is $L_{-\varepsilon}$-unstable on $X$ (resp.
$L_{\varepsilon}$-unstable on $X^{\prime}$).

We now consider the case where $\phi_{*}\mathcal{E}$ is $L_{0}$-semistable. We
first observe, as in the (un)stable case, that there is $\varepsilon_{1}>0$,
such that for all $\varepsilon\in(0,\varepsilon_{1})$ and for all
$\mathcal{F}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X_{0}))$ with
$\mathcal{F}\subsetneq\phi_{*}\mathcal{E}$ such that
$\mu_{L_{0}}(\mathcal{F})<\mu_{L_{0}}(\phi_{*}\mathcal{E})$ one has

$\mu_{L_{-\varepsilon}}((\phi^{*}\mathcal{F})^{\vee\vee})<\mu_{L_{-\varepsilon}}(\mathcal{E})\quad\text{and}\quad\mu_{L_{\varepsilon}}(((\phi^{\prime})^{*}\mathcal{F})^{\vee\vee})<\mu_{L_{\varepsilon}}(\psi_{*}\mathcal{E}).$

If $\mathcal{F}\subsetneq\phi_{*}\mathcal{E}$ is a sub object such that
$\mu_{L_{0}}(\mathcal{F})=\mu_{L_{0}}(\phi_{*}\mathcal{E})$, then there is
Jordan-Hölder filtration

$0=\mathcal{E}_{1}\subseteq\ldots\subseteq\mathcal{E}_{l}=\phi_{*}\mathcal{E}$

with $l\geq 1$ such that $\mathcal{F}=\mathcal{E}_{i}$ for some
$i\in\\{1,\ldots,l\\}$ (see [11, Section 1.6]) and we deduce that
$\mathcal{F}\in\mathfrak{E}$. Therefore, from the expansions of the slopes, to
get the points $(iii)$ and $(iv)$ of the theorem, it suffices to compare

$\dfrac{c_{1}(\mathcal{E})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{E})}\quad\text{and}\quad\dfrac{c_{1}((\phi^{*}\mathcal{F})^{\vee\vee})\cdot
D_{+}\cdot(\phi^{*}L_{0})^{n-2}}{\mathrm{rank}(\mathcal{F})}$

for any $\mathcal{F}\in\mathfrak{E}$. By uniqueness of the reflexive hull of
the graded object of a Jordan–Hölder filtration ([11, Theorem 1.6.7]),
$\mathfrak{E}$ is finite, and the result follows. ∎

######  Remark 4.6.

In the proof, we have shown that for any $\rho\in\Sigma(1)$,

$D_{\rho}\cdot D_{+}\cdot(\phi^{*}L_{0})^{n-2}=D^{\prime}_{\rho}\cdot
D^{\prime}_{+}\cdot((\phi^{\prime})^{*}L_{0})^{n-2}$

If $\ell\geq 2$, the equality

$D_{\rho}\cdot(D_{+})^{\ell}\cdot(\phi^{*}L_{0})^{n-1-\ell}=D^{\prime}_{\rho}\cdot(D^{\prime}_{+})^{\ell}\cdot((\phi^{\prime})^{*}L_{0})^{n-1-\ell}$

is no longer true in general (see for example Equations (4.5) and (4.6) in
Section 4.3). The arguments used to prove Theorem 4.4 are very close to those
used in [18]. One should be careful though that the results from [18] do not
imply directly Theorem 4.4, as $X_{0}$ is not $\mathbb{Q}$-factorial.

######  Remark 4.7.

While the case when $\phi_{*}\mathcal{E}$ is semistable on $X_{0}$ is not
fully covered by Theorem 4.4, items $(iii)$ and $(iv)$, one can easily adapt
the numerical criterion of [18, Theorem 1.3] to take into account higher order
terms in the $\varepsilon$-expansions of the $L_{-\varepsilon}$ and
$L_{\varepsilon}$ slopes, and obtain a full description of the stability
behaviour of $\mathcal{E}$ in terms of that of $\phi_{*}\mathcal{E}$, for the
considered polarisations.

Actually, Theorem 4.4 holds for some specific flat families of toric sheaves.
We recall that the characteristic function $\chi$ of an equivariant reflexive
sheaf $\mathcal{F}$ with family of filtrations $(F,\\{F^{\rho}(j)\\})$ is the
function

$\begin{array}[]{cccl}\chi(\mathcal{F}):&M&\longrightarrow&\mathbb{Z}^{\sharp\Sigma(n)}\\\
&m&\longmapsto&\left(\dim\left(\bigcap_{\rho\in\sigma(1)}F^{\rho}(\langle
m,u_{\rho}\rangle)\right)\right)_{\sigma\in\Sigma(n)}\end{array}.$

Let $S$ be a scheme of finite type over $\mathbb{C}$ and
$\mathcal{E}=(\mathcal{E}_{s})_{s\in S}$ be an $S$-family of equivariant
reflexive sheaves over $X$ (see [18, Section 3.5] for more details). We denote
by $(E_{s},E_{s}^{\rho}(i))$ the family of filtrations of $\mathcal{E}_{s}$.
There is a collection of increasing filtrations of reflexive sheaves

$(\mathcal{F},\\{\mathcal{F}_{m}^{\rho}:m\in M\\}_{\rho\in\Sigma(1)})$

such that for any $s\in S$ and all $m\in M$,

$E_{s}=\mathcal{F}(s)\quad\text{and}\quad E_{s}^{\rho}(\langle
m,u_{\rho}\rangle)=\mathcal{F}_{m}^{\rho}(s)$

where $\mathcal{F}(s)$ and $\mathcal{F}_{m}^{\rho}(s)$ are respectively the
fibers of $\mathcal{F}$ and $\mathcal{F}_{m}^{\rho}$ at $s$.

######  Lemma 4.8.

Let $X$ be a toric variety given by a simplicial fan $\Sigma$ and let
$\mathcal{E}=(\mathcal{E}_{s})_{s\in S}$ be an $S$-family of equivariant
reflexive sheaves over $X$ such that:

  1. (1)

$\mathcal{E}$ is locally free over $X\times S$, or

  2. (2)

the map $s\mapsto\chi(\mathcal{E}_{s})$ is constant.

Then, for all $\rho\in\Sigma(1)$ and $j\in\mathbb{Z}$, the map
$s\mapsto\dim(E_{s}^{\rho}(j))$ is constant.

###### Proof.

If $\mathcal{E}$ is locally free over $X\times S$, by [19, Proposition 3.13]
(Klyachko’s compatibility condition for $S$-families of locally free sheaves),
for any $\sigma\in\Sigma(n)$, there is a multiset $A_{\sigma}\subseteq M$ of
size $\mathrm{rank}(\mathcal{E})$ such that for any $m\in M$,
$\mathcal{F}_{m}^{\rho}$ is a locally free sheaf of rank

$\left|\\{\alpha\in A_{\sigma}:\langle\alpha,u_{\rho}\rangle\leq\langle
m,u_{\rho}\rangle\\}\right|.$

As for any $s\in S$ and $m\in M$,
$\dim(\mathcal{F}_{m}^{\rho}(s))=\mathrm{rank}(\mathcal{F}_{m}^{\rho})$, we
deduce that the map

$s\longmapsto\dim(E_{s}^{\rho}(\langle m,u_{\rho}\rangle))$

is constant.

We now assume that the map $s\mapsto\chi(\mathcal{E}_{s})$ is constant. For
any $\rho\in\Sigma(1)$, we denote by $i_{\rho}$ the smallest integer such that
for any $j\geq i_{\rho}$ and any $s\in S$,

$E_{s}^{\rho}(j)=E_{s}.$

Let $\sigma\in\Sigma(n)$. The set $\\{u_{\rho}:\rho\in\sigma(1)\\}$ is a
$\mathbb{Q}$-basis of $N\otimes_{\mathbb{Z}}\mathbb{Q}$; we denote by
$\\{u_{\rho}^{*}:\rho\in\sigma(1)\\}$ its dual basis. For any
$\rho^{\prime}\in\sigma(1)$, there is $m^{\prime}\in M$, such that $\langle
m^{\prime},u_{\rho^{\prime}}\rangle=j$. Let $m\in M$ be given by

$m=m^{\prime}+\sum_{\rho\in\sigma(1)\setminus\\{\rho^{\prime}\\}}a_{\rho}u^{*}_{\rho}$

where for any $\rho\in\sigma(1)\setminus\\{\rho^{\prime}\\}$,
$a_{\rho}\in\mathbb{Z}$ satisfies $a_{\rho}u^{*}_{\rho}\in M$ and
$a_{\rho}+\langle m^{\prime},u_{\rho}\rangle\geq i_{\rho}$. By construction of
$m$, one has

$\bigcap_{\rho\in\sigma(1)}E_{s}^{\rho}(\langle
m,u_{\rho}\rangle)=E_{s}^{\rho^{\prime}}(j).$

As $s\mapsto\chi(\mathcal{E}_{s})$ is constant, we deduce that the map
$s\mapsto\dim(E_{s}^{\rho^{\prime}}(j))$ is constant for any
$\rho^{\prime}\in\sigma(1)$ and any $j\in\mathbb{Z}$. ∎

If $\mathcal{E}$ is an $S$-family of equivariant reflexive sheaves on $X$
which satisfies the conditions of Lemma 4.8, then by [18, Lemma 3.12], for any
ample $\mathbb{Q}$-Cartier divisor $L$ on $X$, the set

$\left\\{\mu_{L}(\mathcal{G}_{s}):s\in S,\>\text{$\mathcal{G}_{s}$ is an
equivariant and saturated reflexive subsheaf of $\mathcal{E}_{s}$}\right\\}$

is finite. Therefore, in that case, the $\varepsilon_{0}$ in Theorem 4.4 can
be taken uniformly for $(\mathcal{E}_{s})_{s\in S}$.

###  4.3. An example

We denote by $(e_{1},e_{2},e_{3})$ the standard basis of $\mathbb{Z}^{3}$. Let

$u_{1}=e_{1},\quad u_{2}=e_{1}+e_{2}-e_{3},\quad u_{3}=e_{2},\quad
u_{4}=e_{3},\quad u_{0}=-(e_{1}+e_{2}+e_{3})$

and $\Sigma_{0}$ be a fan in $\mathbb{R}^{3}$ given by

$\Sigma_{0}=\\{\mathrm{Cone}(u_{1},u_{2},u_{3},u_{4})\\}\cup\bigcup_{i=1}^{4}\\{\mathrm{Cone}(A):A\subseteq\\{u_{0},u_{i},u_{i+1}\\}\\}$

where $u_{5}=u_{1}$. We denote by $\sigma_{0}$ the flipping cone
$\mathrm{Cone}(u_{1},u_{2},u_{3},u_{4})$; we have

$u_{2}+u_{4}-u_{1}-u_{3}=0.$

Let

$\Sigma=(\Sigma_{0}\setminus\\{\sigma_{0}\\})\cup\Sigma_{+}\quad\text{and}\quad\Sigma^{\prime}=(\Sigma_{0}\setminus\\{\sigma_{0}\\})\cup\Sigma_{-}$

where

$\displaystyle\Sigma_{+}$ $\displaystyle=\\{\mathrm{Cone}(u_{j}:j\in
J):J\subset\\{1,\ldots,4\\}\text{ and }\\{1,3\\}\nsubseteq
J\\}\quad\text{and}$ $\displaystyle\Sigma_{-}$
$\displaystyle=\\{\mathrm{Cone}(u_{j}:j\in J):J\subset\\{1,\ldots,4\\}\text{
and }\\{2,4\\}\nsubseteq J\\}.$

We denote by $D_{i}$ the torus invariant divisor associated to the ray
$\mathrm{Cone}(u_{i})$. By using (4.3) with $m\in\\{e_{1},e_{2},e_{3}\\}$, we
get the following linear equivalences of divisors on $X_{0}$, $X$ and
$X^{\prime}$:

$D_{1}\sim_{\rm lin}D_{3}\sim_{\rm lin}D_{0}-D_{2}\quad\text{and}\quad
D_{4}\sim_{\rm lin}D_{0}+D_{2}.$

By [6, Theorem 4.2.8.(d)], the divisor $D_{0}$ generates the set of invariant
Cartier divisors of $X_{0}$. As $\Sigma$ (resp. $\Sigma^{\prime}$) is
simplicial, by [6, Proposition 4.2.7] any invariant divisor of $X$ (resp.
$X^{\prime}$) is $\mathbb{Q}$-Cartier.

$\bullet$$\bullet$$u_{1}$$\bullet$$u_{2}$$\bullet$$u_{3}$$\bullet$$u_{4}$$\bullet$$u_{0}$
(a) Fan of $\Sigma$

$\bullet$$\bullet$$u_{1}$$\bullet$$u_{2}$$\bullet$$u_{3}$$\bullet$$u_{4}$$\bullet$$u_{0}$
(b) Fan of $\Sigma_{0}$

$\bullet$$\bullet$$u_{1}$$\bullet$$u_{2}$$\bullet$$u_{3}$$\bullet$$u_{4}$$\bullet$$u_{0}$
(c) Fan of $\Sigma^{\prime}$

Figure 1. Fans of varieties given in Section 4.3

######  Lemma 4.9 (Intersections of divisors).

  1. (1)

On $X$ and $X^{\prime}$, we have:

$\begin{array}[]{l}D_{1}\cdot D_{0}^{2}=\dfrac{1}{2}\qquad D_{2}\cdot
D_{0}^{2}=\dfrac{1}{4}\qquad D_{4}\cdot D_{0}^{2}=1\qquad D_{0}\cdot
D_{0}^{2}=\dfrac{3}{4}\\\ D_{1}\cdot(D_{0}\cdot D_{2})=\dfrac{1}{2}\qquad
D_{4}\cdot(D_{0}\cdot D_{2})=0\qquad D_{2}\cdot(D_{0}\cdot
D_{2})=-\dfrac{1}{4}.\end{array}$

  2. (2)

On $X$, we have

$D_{1}\cdot D_{2}^{2}=\dfrac{1}{2}\qquad D_{4}\cdot D_{2}^{2}=-1\qquad
D_{0}\cdot D_{2}^{2}=\dfrac{-1}{4}\qquad D_{2}\cdot D_{2}^{2}=\dfrac{-3}{4}.$

  3. (3)

On $X^{\prime}$, we have

$D_{1}\cdot D_{2}^{2}=\dfrac{-1}{2}\qquad D_{4}\cdot D_{2}^{2}=0\qquad
D_{0}\cdot D_{2}^{2}=\dfrac{-1}{4}\qquad D_{2}\cdot D_{2}^{2}=\dfrac{1}{4}.$

###### Proof.

The lemma follows from Formula (4.2) and Table 1.

$\mathbb{Z}u_{0}+\mathbb{Z}u_{1}+\mathbb{Z}u_{2}$ | $\mathbb{Z}u_{0}+\mathbb{Z}u_{2}+\mathbb{Z}u_{3}$ | $\mathbb{Z}u_{0}+\mathbb{Z}u_{3}+\mathbb{Z}u_{4}$ | $\mathbb{Z}u_{0}+\mathbb{Z}u_{4}+\mathbb{Z}u_{1}$
---|---|---|---
$2$ | $2$ | $1$ | $1$

$\mathbb{Z}u_{2}+\mathbb{Z}u_{4}+\mathbb{Z}u_{1}$ | $\mathbb{Z}u_{2}+\mathbb{Z}u_{4}+\mathbb{Z}u_{3}$ | $\mathbb{Z}u_{1}+\mathbb{Z}u_{3}+\mathbb{Z}u_{2}$ | $\mathbb{Z}u_{1}+\mathbb{Z}u_{3}+\mathbb{Z}u_{4}$
---|---|---|---
$1$ | $1$ | $1$ | $1$

Table 1. Sublattices and their index in $\mathbb{Z}^{3}$

We show the first point to illustrate the computations, the other intersection
numbers follow in the same way.

$\displaystyle D_{1}\cdot D_{0}^{2}$ $\displaystyle=D_{1}\cdot
D_{0}\cdot(D_{2}+D_{3})=D_{1}\cdot D_{0}\cdot D_{2}+D_{1}\cdot D_{0}\cdot
D_{3}=\dfrac{1}{2}$ $\displaystyle D_{2}\cdot D_{0}^{2}$
$\displaystyle=\dfrac{1}{2}D_{2}\cdot D_{0}\cdot(D_{3}+D_{4})=\dfrac{1}{4}$
$\displaystyle D_{4}\cdot D_{0}^{2}$ $\displaystyle=D_{4}\cdot
D_{0}\cdot(D_{2}+D_{3})=1$ $\displaystyle D_{0}\cdot D_{0}^{2}$
$\displaystyle=\dfrac{1}{2}(D_{3}+D_{4})\cdot
D_{0}^{2}=\dfrac{1}{4}+\dfrac{1}{2}=\dfrac{3}{4}.$

∎

We assume that $\mathcal{E}=\mathcal{T}_{X}$. By [7, Corollary 2.2.17], the
family of filtrations of the tangent sheaf $\mathcal{T}_{X}$ of $X$ is given
by:

$E^{\rho}(j)=\left\\{\begin{array}[]{ll}0&\text{if}~{}j<-1\\\
\mathrm{Span}(u_{\rho})&\text{if}~{}j=-1\\\
N\otimes_{\mathbb{Z}}\mathbb{C}&\text{if}~{}j>-1\end{array}\right..$

In this case, the inequalities of Proposition 4.2 become

$\dfrac{1}{\dim(F)}\sum_{u_{\rho}\in
F}\deg_{L}(D_{\rho})\stackrel{{\scriptstyle(\leq)}}{{<}}\dfrac{1}{n}\sum_{\rho\in\Sigma(1)}\deg_{L}(D_{\rho}).$

Let $L_{0}=D_{0}$ be an ample Cartier divisor on $X_{0}$. We have

$\mu_{L}(\mathcal{E})=1=\dfrac{1}{\dim(F_{1})}\sum_{u_{\rho}\in
F_{1}}\deg_{L}(D_{\rho})=\dfrac{1}{\dim(F_{2})}\sum_{u_{\rho}\in
F_{2}}\deg_{L}(D_{\rho})$

where $F_{1}=\mathrm{Span}(u_{4})$ and
$F_{2}=\mathrm{Span}(u_{0},u_{2},u_{4})$. As for any vector subspace
$F\subsetneq E$ such that $F\notin\\{F_{1},F_{2}\\}$, one has

$\dfrac{1}{\dim(F)}\sum_{u_{\rho}\in F}\deg_{L}(D_{\rho})<1,$

we deduce that $\mathcal{E}$ is semistable. We denote by $\mathcal{F}$ the
subsheaf of $\mathcal{E}$ given by the family of filtrations $(F_{1},F_{1}\cap
E^{\rho}(i))$.

Let

$D_{+}=D_{2}+D_{4}.$

We have

$D_{+}\sim_{\rm lin}D_{0}+2D_{2}.$

So

$L_{-\varepsilon}=\phi^{*}L_{0}-\varepsilon D_{+}\sim_{\rm
lin}(1-\varepsilon)D_{0}-2\varepsilon
D_{2}=(1-\varepsilon)\left(D_{0}-\dfrac{2\varepsilon}{1-\varepsilon}D_{2}\right)$

and

$L_{\varepsilon}=(\phi^{\prime})^{*}L_{0}+\varepsilon D_{+}\sim_{\rm
lin}(1+\varepsilon)D_{0}+2\varepsilon
D_{2}=(1+\varepsilon)\left(D_{0}+\dfrac{2\varepsilon}{1+\varepsilon}D_{2}\right).$

According to Lemma 4.9, on $X$ we have

(4.5)
$\displaystyle\deg_{L_{-\varepsilon}}(D_{1})=\dfrac{1}{2}-3\varepsilon+\dfrac{9}{2}\varepsilon^{2}$
$\displaystyle\deg_{L_{-\varepsilon}}(D_{2})=\dfrac{1}{4}+\dfrac{1}{2}\varepsilon-\dfrac{15}{4}\varepsilon^{2}$
$\displaystyle\deg_{L_{-\varepsilon}}(D_{4})=1-2\varepsilon-3\varepsilon^{2}$
$\displaystyle\deg_{L_{-\varepsilon}}(D_{0})=\dfrac{3}{4}-\dfrac{5}{2}\varepsilon+\dfrac{3}{4}\varepsilon^{2};$

and on $X^{\prime}$ we have

(4.6)
$\displaystyle\deg_{L_{-\varepsilon}}(D_{1})=\dfrac{1}{2}+3\varepsilon+\dfrac{1}{2}\varepsilon^{2}$
$\displaystyle\deg_{L_{-\varepsilon}}(D_{2})=\dfrac{1}{4}-\dfrac{1}{2}\varepsilon+\dfrac{1}{4}\varepsilon^{2}$
$\displaystyle\deg_{L_{-\varepsilon}}(D_{4})=1+2\varepsilon+\varepsilon^{2}$
$\displaystyle\deg_{L_{-\varepsilon}}(D_{0})=\dfrac{3}{4}+\dfrac{5}{2}\varepsilon+\dfrac{3}{4}\varepsilon^{2}.$

By Formulas (4.5), we get

$\mu_{L_{-\varepsilon}}(\mathcal{E})=1-\dfrac{10}{3}\varepsilon+\varepsilon^{2}\quad\text{and}\quad\mu_{L_{-\varepsilon}}(\mathcal{F})=1-2\varepsilon-3\varepsilon^{2}.$

Hence, there is $\varepsilon_{0}$ such that for any
$\varepsilon\in(0,\varepsilon_{0})\cap\mathbb{Q}$, the tangent sheaf
$\mathcal{T}_{X}$ is $L_{-\varepsilon}$-unstable. By Formulas (4.6), we have

$\mu_{L_{-\varepsilon}}(\mathcal{E})=1+\dfrac{10}{3}\varepsilon+\varepsilon^{2}\quad\text{and}\quad\mu_{L_{-\varepsilon}}(\mathcal{F})=1+2\varepsilon+\varepsilon^{2}.$

Hence, there is $\varepsilon_{0}$ such that for any
$\varepsilon\in(0,\varepsilon_{0})\cap\mathbb{Q}$, the tangent sheaf
$\mathcal{T}_{X^{\prime}}$ is $L_{\varepsilon}$-stable.

##  5\. Flips and stability for logarithmic subcategories

Consider a toric flip

${X}$${X^{\prime}}$${X_{0}}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$

as in Section 2. Fix $\Delta\subset\Sigma(1)$, and introduce the divisor

$D:=\sum_{\rho\in\Delta}D_{\rho}$

seen as a divisor on $X$, $X^{\prime}$ and $X_{0}$. We consider the equivalent
categories $\mathfrak{Ref}^{T}(X,D)$ and $\mathfrak{Ref}^{T}(X^{\prime},D)$ as
in Section 3.3. We will say that

$\psi_{*}:\mathfrak{Ref}^{T}(X,D)\to\mathfrak{Ref}^{T}(X^{\prime},D)$

preserves polystability for the pair of polarisations
$(\alpha,\alpha^{\prime})\in\mathrm{Pic}(X)_{\mathbb{Q}}\otimes\mathrm{Pic}(X^{\prime})_{\mathbb{Q}}$
if for any $\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D))$,
$\mathcal{E}$ is $\alpha$-polystable if and only if $\psi_{*}\mathcal{E}$ is
$\alpha^{\prime}$-polystable.

######  Theorem 5.1.

The following assertions are equivalent, for a pair of ample classes
$(\alpha,\alpha^{\prime})\in\mathrm{Pic}(X)_{\mathbb{Q}}\otimes\mathrm{Pic}(X^{\prime})_{\mathbb{Q}}$.

  1. (i)

The functor
$\psi_{*}:\mathfrak{Ref}^{T}(X,D)\to\mathfrak{Ref}^{T}(X^{\prime},D)$
preserves polystability for $(\alpha,\alpha^{\prime})$.

  2. (ii)

There is $c\in\mathbb{Q}_{>0}$ such that for all
$\mathcal{E}\in\mathfrak{Ref}^{T}(X,D)$,
$\mu_{\alpha}(\mathcal{E})=c\;\mu_{\alpha^{\prime}}(\psi_{*}\mathcal{E})$.

  3. (iii)

There is $c\in\mathbb{Q}_{>0}$ such that for all $\rho\notin\Delta$,
$\deg_{\alpha}D_{\rho}=c\;\deg_{\alpha^{\prime}}D_{\rho}^{\prime}$.

###### Proof.

Recall the formula

$\mu_{\alpha}(\mathcal{E})=-\frac{1}{\mathrm{rank}(\mathcal{E})}\sum_{\rho\in\Sigma(1)}\iota_{\rho}(\mathcal{E})\,\deg_{\alpha}(D_{\rho})$

for the slope, with

$\iota_{\rho}(\mathcal{E}):=\sum_{i\in\mathbb{Z}}i\left(\dim(E^{\rho}(i))-\dim(E^{\rho}(i-1))\right)$

where $(E,E^{\rho}(\bullet))_{\rho\in\Sigma(1)}$ stands for the family of
filtrations associated to $\mathcal{E}$. Then, by definition of the
logarithmic category $\mathfrak{Ref}^{T}(X,D)$, for any
$\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D))$, and any
$\rho\in\Delta$, we have

$\iota_{\rho}(\mathcal{E})=0,$

so the slope reads

$\mu_{\alpha}(\mathcal{E})=-\frac{1}{\mathrm{rank}(\mathcal{E})}\sum_{\rho\notin\Delta}\iota_{\rho}(\mathcal{E})\,\deg_{\alpha}(D_{\rho}).$

Note also that by construction, for any $\rho$,

$\iota_{\rho}(\mathcal{E})=\iota_{\rho}(\psi_{*}\mathcal{E}).$

Then, $(iii)\Rightarrow(ii)\Rightarrow(i)$ is straightforward. To prove
$(i)\Rightarrow(iii)$, we argue as in [4, proof of Proposition 4.8], and
consider for any pair $(\rho_{1},\rho_{2})\in\Sigma(1)^{2}$ the polystable
toric sheaf

$\mathcal{E}=\mathcal{O}_{X}(d\deg_{\alpha}(D_{\rho_{2}})D_{\rho_{1}})\oplus\mathcal{O}_{X}(d\deg_{\alpha}(D_{\rho_{1}})D_{\rho_{2}}),$

where $d$ is the common denominator of $\deg_{\alpha}(D_{\rho_{2}})$ and
$\deg_{\alpha}(D_{\rho_{1}})$. Its image by $\psi_{*}$ is

$\psi_{*}\mathcal{E}=\mathcal{O}_{X^{\prime}}(d\deg_{\alpha}(D_{\rho_{2}})D_{\rho_{1}}^{\prime})\oplus\mathcal{O}_{X^{\prime}}(d\deg_{\alpha}(D_{\rho_{1}})D_{\rho_{2}}^{\prime}).$

As $\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D))$ (see e.g. [7, Example
2.2.13] for the family of filtrations of rank one toric sheaves), if $\psi$
preserves polystability, we must have

$\deg_{\alpha}(D_{\rho_{2}})\deg_{\alpha^{\prime}}(D_{\rho_{1}}^{\prime})=\deg_{\alpha}(D_{\rho_{1}})\deg_{\alpha^{\prime}}(D_{\rho_{2}}^{\prime}).$

The result follows. ∎

######  Remark 5.2.

The reason for considering logarithmic subcategories is the following. If one
considers the full $\mathfrak{Ref}^{T}(X)$, and assume that $\psi_{*}$
preserves polystability for all
$\mathcal{E}\in\mathrm{Obj}(\mathfrak{Ref}^{T}(X,D))$, then by $(iii)$ of
Theorem 5.1, for any $\rho\in\Sigma(1)$, one must have

$\deg_{\alpha}D_{\rho}=c\;\deg_{\alpha^{\prime}}D_{\rho}^{\prime}.$

Up to scale, we can assume $c=1$. But then, a result due to Minkowski ([22,
p.455]), translated into the toric setting in [4, Proposition 5.3 and
Corollary 5.4], implies that the polytope associated to $(X,\alpha)$ equals
the polytope associated to $(X^{\prime},\alpha^{\prime})$, which is absurd, as
$X$ and $X^{\prime}$ are not isomorphic.

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