Lazarsfeld-Mukai Reflexive Sheaves and their Stability

TL;DR

This paper constructs and studies Lazarsfeld-Mukai reflexive sheaves on smooth projective varieties, proving their stability properties under specific conditions, thereby advancing understanding of their geometric and stability characteristics.

Contribution

It introduces a new construction of Lazarsfeld-Mukai reflexive sheaves via elementary transformations and establishes their $ ext{mu}_L$-(semi)stability under certain conditions.

Findings

Constructed reflexive sheaves via elementary transformations.

Proved $ ext{mu}_L$-(semi)stability of these sheaves.

Provided conditions ensuring stability properties.

Abstract

Consider an ample and globally generated line bundle $L$ on a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth divisor in the complete linear system of $L$. We construct reflexive sheaves on $X$ by an elementary transformation of a trivial bundle on $X$ along certain globally generated torsion-free sheaves on $D$. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the $\mu_L$-(semi)stability of such reflexive sheaves under certain conditions.

Equations

$$0⟶F⟶H^0(A)\otimes {\mathcal{O}}_X\stackrel{ev}{\to }{i}_{\ast }A⟶0.$$

$$0⟶{\mathcal{F}}_{D,A,V}⟶V\otimes {\mathcal{O}}_X⟶i_{\ast }(A\otimes {\mathcal{I}}_{Z(V)})⟶0.$$

$$0⟶M_{L,W}⟶W\otimes {\mathcal{O}}_X⟶L⟶0.$$

$$\text{sm}|L|=\{D\in |L|:D\text{ is smooth and irreducible}\}.$$

$${\mu }_L(F)=\frac{c_1(F)\cdot (L^{N-1})}{r}.$$

$$V\otimes {\mathcal{O}}_X\twoheadrightarrow V\otimes i_{\ast }{\mathcal{O}}_D⟶i_{\ast }(A\otimes {\mathcal{I}}_{Z(V)})⟶0.$$

$$0⟶{\mathcal{F}}_{D,A,V}⟶V\otimes {\mathcal{O}}_X⟶i_{\ast }(A\otimes {\mathcal{I}}_{Z(V)})⟶0.$$

$$0⟶V^{\vee }\otimes {\mathcal{O}}_X⟶{\mathcal{E}}_{D,A,V}⟶\mathcal{E}x{t}^1(i_{\ast }(A\otimes {\mathcal{I}}_{Z(V)}),{\mathcal{O}}_X)⟶0.$$

$${\mathcal{U}}_r=\{V\in G(r,H^0(D,A))|Z(V)\text{ has codimension }r\text{ in }D\}.$$

$$\frac{c_1(Q)\cdot (H^{N-1})}{s}\le \frac{H^N}{r}.$$

$$\stackrel{s}{\bigwedge }{\mathcal{O}}_X^r⟶\stackrel{s}{\bigwedge }\tilde{Q}\simeq \text{det}\tilde{Q}.$$

$$f:{\mathcal{O}}_X^s⟶\tilde{Q}.$$

$$0⟶K⟶{\mathcal{O}}_X^s\stackrel{f}{\to }\tilde{Q}⟶C⟶0.$$

$${\mathcal{O}}_X^s\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$D\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0⟶{\mathcal{F}}_{H,{\mathcal{O}}_H(m),V^{\prime }}⟶V^{\prime }\otimes {\mathcal{O}}_{{\mathbb{P}}^N}⟶j_{\ast }({\mathcal{O}}_H(m)\otimes {\mathcal{I}}_{Z(V^{\prime })})⟶0.$$

$$Z(V)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0⟶{\varphi }^{\ast }{\mathcal{F}}_{H,{\mathcal{O}}_H(m),V^{\prime }}⟶V\otimes {\mathcal{O}}_X⟶{\varphi }^{\ast }{j}_{\ast }({\mathcal{O}}_H(m)\otimes {\mathcal{I}}_{Z(V^{\prime })})⟶0.$$

$${\varphi }^{\ast }{j}_{\ast }({\mathcal{O}}_H(m)\otimes {\mathcal{I}}_{Z(V^{\prime })})\simeq i_{\ast }(\varphi {|}_D{)}^{\ast }({\mathcal{O}}_H(m)\otimes {\mathcal{I}}_{Z(V^{\prime })})\simeq i_{\ast }(A\otimes \varphi {|}_D^{\ast }{\mathcal{I}}_{Z(V^{\prime })}).$$

$$0⟶{\varphi }^{\ast }{\mathcal{F}}_{H,{\mathcal{O}}_H(m),V^{\prime }}⟶V\otimes {\mathcal{O}}_X⟶i_{\ast }(A\otimes {\mathcal{I}}_{Z(V)})⟶0.$$

$$k_E=-\frac{c_1(E)}{2}\text{ if }{c}_1(E)\text{ even, and }{k}_E=-\frac{c_1(E)+1}{2}\text{ for }{c}_1(E)\text{ odd}.$$

$$0⟶{\mathcal{F}}_{D,A,V}{|}_{\tilde{U}}⟶V\otimes {\mathcal{O}}_{\tilde{U}}⟶i_{\ast }(A\otimes {\mathcal{I}}_{Z(V)}){|}_{\tilde{U}}⟶0.$$

$$U\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0⟶{\mathcal{F}}_{D,A,V}{|}_{\tilde{U}}⟶V\otimes {\mathcal{O}}_{\tilde{U}}⟶(i{|}_U{)}_{\ast }(A{|}_U)⟶0.$$

$$0⟶K⟶{\mathcal{F}}_{D,A,V}{|}_D⟶V\otimes {\mathcal{O}}_D⟶A\otimes {\mathcal{I}}_{Z(V)}⟶0\text{ on }D.$$

$$0⟶A^{\vee }⟶V\otimes {\mathcal{O}}_D⟶A\otimes {\mathcal{I}}_{Z(V)}⟶0.$$

$$0⟶K⟶{\mathcal{F}}_{D,A,V}{|}_D⟶A^{\vee }⟶0.$$

$$\text{det}K\simeq A\otimes \text{det}{\mathcal{F}}_{D,A,V}{|}_D\simeq A\otimes (L{|}_D{)}^{\vee }.$$

$$A^{\vee }\simeq \mathcal{O}(-l){|}_D\text{ and }\text{det}K\simeq A\otimes (L{|}_D{)}^{\vee }\simeq \mathcal{O}(l){|}_D\otimes \mathcal{O}(-2l){|}_D\simeq \mathcal{O}(-l){|}_D.$$

$$0<l<\frac{d(r-1)}{r}.$$

Full text

Mathematics Classification numbers: 14C20, 14J60, 14M10

1. Introduction

Lazarsfeld-Mukai bundles were introduced by Lazarsfeld [15] and Mukai [21] in the 1980s. They are an important class of vector bundles obtained from certain elementary transformations and have found applications in studying syzygies and Brill-Noether theory. These bundles play a crucial role in Lazarsfeld’s proof of the Gieseker-Petri theorem [15] and Voisin’s proof of the generic Green’s conjecture [25, 26].

Suppose $X$ is a smooth projective surface over $\mathbb{C}$, and $C$ is a smooth, irreducible curve on $X$. Consider a globally generated line bundle $A$ on $C$. Denote by $i_{*}A$, the direct image of $A$ on $X$ where $i:C\hookrightarrow X$ is the inclusion. Then $i_{*}A$ is a globally generated coherent sheaf on $X$. We thus have the following exact sequence on $X$ where the kernel $F$ is a vector bundle:

[TABLE]

The dual of $F$ is called the Lazarsfeld-Mukai bundle on $X$ associated to the pair $(C,A)$. Lelli-Chiesa [17] has studied the (semi)stability of the Lazarsfeld-Mukai bundles on K3-surfaces, and similar results have been obtained by us on abelian surfaces [22]. Also, [2] and references therein give a general survey of Lazarsfeld-Mukai bundles with other applications.

In this article we generalize the above construction to higher dimensional varieties. We in fact obtain reflexive sheaves as kernels. We study their $\mu$-(semi)stability properties in various cases. This construction also enables us to obtain on any smooth projective variety $X$, semistable vector bundles $E$ with $\textrm{rank}\,E=\textrm{dim}\,X$.

Suppose $X$ is a smooth projective variety of dimension $N\geq 2$ over $\mathbb{C}$. Let $D\xhookrightarrow{i}X$ be a smooth, irreducible divisor on $X$ and $A$ be an ample and globally generated line bundle on $D$. Consider a general subspace $V\subset H^{0}(D,A)$ of dimension $r\geq 2$. Let $Z(V)\hookrightarrow D$ be the closed subscheme defined by the vanishing of sections of $V$ and $\mathcal{I}_{Z(V)}\subset\mathcal{O}_{D}$ be its ideal sheaf. Then $A\otimes\mathcal{I}_{Z(V)}$ is a globally generated torsion-free sheaf on $D$. We have the following short exact sequence, which defines the sheaf $\mathcal{F}_{D,A,V}$ associated to the triple $(D,A,V)$ on $X$:

[TABLE]

The kernel $\mathcal{F}_{D,A,V}$ is a reflexive sheaf of rank $r$ on $X$, whose dual is called the Lazarsfeld-Mukai reflexive sheaf (see $\mathcal{x}\,$3).

We remark that the same construction can be carried out under the weaker assumption that $D$ is just reduced and irreducible but not necessarily smooth. But for the purpose of this paper, we confine ourselves mainly with smooth and irreducible divisors $D$.

The $\mu$-(semi)stability properties of the sheaves $\mathcal{F}_{D,A,V}$ are studied in $\mathcal{x}\,$4. The first case we consider is that of a variety $X$ whose Picard group is cyclic.

Theorem 1.1**.**

Suppose $X$ is an irreducible smooth projective variety over $\mathbb{C}$ of dimension $N\geq 2$, such that $\emph{Pic}\,X=\mathbb{Z}\cdot[H]$, where $[H]$ is the class of an ample divisor. Let $D\in|\mathcal{O}_{X}(H)|$ and $A$ be an ample, globally generated line bundle on $D$. Consider $V\subset H^{0}(D,A)$, an $r$-dimensional subspace where $r\geq 2$. Then the reflexive sheaf $\mathcal{F}_{D,A,V}$ is $\mu_{H}$-stable.

Any $D\in|\mathcal{O}_{X}(H)|$ is reduced, irreducible and Cohen-Macaulay. Hence we can consider reflexive sheaves $\mathcal{F}_{D,A,V}$ for all such $D$. In the specific case when $X$ is the projective space, we have the following theorem.

Theorem 1.2**.**

Suppose $X=\mathbb{P}^{N}_{\mathbb{C}}$ for $N\geq 2$. Consider $L=\mathcal{O}(d)$ with $d>0$ on $X$. Let $D\in|L|$ be a general smooth, irreducible hypersurface and $A$ be an ample, globally generated line bundle on $D$. Suppose that $V$ is a general $r$-dimensional subspace of $H^{0}(D,A)$, where $r\geq 2$. Then the following table summarizes the conditions on $L$, $A$ and $r$ under which the sheaves $\mathcal{F}_{D,A,V}$ are $\mu_{\mathcal{O}(1)}$-(semi)stable.

[TABLE]

See $\mathcal{x}\,$4.3 for a proof. Part (a) of the above theorem follows from Theorem 1.1. Part (b) is proved by applying a lemma from [23]. In case of part (c), we prove that $\mathcal{F}_{D,A,V}|_{D}$ is $\mu_{\mathcal{O}(1)|_{D}}$-semistable, which implies our result. We prove part (d) of the theorem by proving it in general for any smooth, irreducible projective variety $X$, cf. Theorem 1.3. We remark that, in case (d) of the above theorem, if the condition $A=\mathcal{O}(md)|_{D}$ is weakened, the assertion is not necessarily true. Lemma 4.11 in $\mathcal{x}\,$4.3 gives a class of such examples. Note that parts (a) and (b) of the theorem hold for all reduced and irreducible $D$ in the linear system and all $r$-dimensional subspaces $V$.

We prove the following theorem on the $\mu$-(semi)stability of the reflexive sheaves $\mathcal{F}_{D,A,V}$ on arbitrary smooth projective varieties.

Theorem 1.3**.**

Suppose $X$ is an irreducible, smooth projective variety of dimension $N\geq 2$ over $\mathbb{C}$. Consider an ample, globally generated line bundle $L$ on $X$ and an irreducible, smooth $D\in|L|$. For $m>0$, let $V\subset H^{0}(D,L|_{D}^{\otimes m})$ be an $r$-dimensional subspace, where $2\leq r\leq{N-1+m\choose m}$. Then, for a general pair $(D,V)$, the reflexive sheaf $\mathcal{F}_{D,L|_{D}^{\otimes m},V}$ is $\mu_{L}$-semistable.

See $\mathcal{x}\,$4.2. The method of proof employed is the following. Consider an appropriate finite morphism $X\rightarrow\mathbb{P}^{N}$, where $N=\textrm{dim}\,X$. We prove the corresponding (semi)stability statement for the projective space. We know by [19, Lemma 1.17], that the pullback of a semistable torsion-free sheaf under a finite morphism is semistable; and that semistability is an open condition in flat families [13, Proposition 2.3.1]. We thus get the required result.

The same technique is applied to study the (semi)stability of some kernel bundles. A kernel bundle $M_{L,W}$ is defined as follows. Consider an ample and globally generated line bundle $L$ on a smooth, irreducible projective variety $X$ of dimension $n$. Let $W\subset H^{0}(X,L)$ be a subspace such that the linear system $\mathbb{P}W$ is base-point free. Hence we have the following short exact sequence where $M_{L,W}$ is the kernel vector bundle:

[TABLE]

Let $W\subset H^{0}(X,L)$ be a general $(n+1)$-dimensional subspace. We prove that the kernel bundle $M_{L,W}$ associated to $(L,W)$ is $\mu_{L}$-polystable, cf. Proposition 5.2. We mention that for curves, certain surfaces and projective spaces, the $\mu_{L}$-(semi)stability of $M_{L,W}$ has been obtained in [4, 5, 6, 7, 8, 9, 10, 20, 24], for $W=H^{0}(X,L)$ with certain assumptions on $L$.

Acknowledgements

I thank Dr. Jaya NN Iyer for her guidance during the course of this project. I also thank Dr. T. E. Venkata Balaji and Prof. D. S. Nagaraj for helpful discussions and their support. I also thank the referee and the editor for useful comments which helped make the exposition better.

2. Preliminaries

2.1. Definitions and Notations

Let $X$ be a smooth projective variety of dimension $N\geq 2$ over $\mathbb{C}$.

1. (1)

Let $L$ be an ample and globally generated line bundle on $X$. By Bertini’s theorem, the set $\textrm{sm}|L|$ as given below is a dense open set of $|L|$,

[TABLE] 2. (2)

Let $W$ be a vector space over $\mathbb{C}$. Then $G(m,W)$ ($1\leq m\leq\textrm{dim}\,W$) denotes the Grassmannian of $m$-dimensional subspaces of $W$.

2.2. Mumford-Takemoto (semi)stability

Let $L$ be an ample line bundle on $X$ (as above). Consider a torsion-free coherent sheaf $F$ of rank $r$ on $X$.

1. (1)

The slope of $F$ with respect to $L$ is defined as:

[TABLE] 2. (2)

The sheaf $F$ is said to be $\mu_{L}$-semistable (resp. $\mu_{L}$-stable), if for any coherent subsheaf $E\subset F$ of rank $s$ where $0<s<r$, one has $\mu_{L}(E)\leq\mu_{L}(F)$ (resp. $\mu_{L}(E)<\mu_{L}(F)$). 3. (3)

The torsion-free coherent sheaf $F$ is $\mu_{L}$-polystable if it is a direct sum of $\mu_{L}$-stable sheaves of the same slope.

3. Construction of Reflexive sheaves

Consider a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$, and an ample, globally generated line bundle $L$ on $X$. For a divisor $D\in\textrm{sm}|L|$, let $i:D\hookrightarrow X$ denote the inclusion. Let $A$ be an ample, globally generated line bundle on $D$.

By the Noether-Lefschetz Theorem, if $N\geq 4$, then $\text{Pic}\,X\longrightarrow\text{Pic}\,D$ is an isomorphism, cf. [16, Example 3.1.25]. Hence, the line bundle $A$ is the restriction of a line bundle from $X$.

Consider $G(r,H^{0}(D,A))$, the Grassmannian of $r$-dimensional subspaces of the space of global sections $H^{0}(D,A)$, where $2\leq r\leq h^{0}(D,A)$. For $V\in G(r,H^{0}(D,A))$, let $Z(V)$ denote the closed subscheme of $D$ defined by the vanishing of sections of $V$. Recall that $Z(V)$ has codimension at most $r$ in $D$. In fact, $Z(V)$ has codimension exactly $r$ in $D$ for a general $V\in G(r,H^{0}(D,A))$.

The base locus of the linear system $\mathbb{P}V$ corresponding to $(A,V)$ on $D$ is $Z(V)$. The ideal sheaf $\mathcal{I}_{Z(V)}$ of $Z(V)$ is the image of the morphism $V\otimes A^{\vee}\longrightarrow\mathcal{O}_{D}$ on $D$. This gives the surjective evaluation map $V\otimes\mathcal{O}_{D}\twoheadrightarrow A\otimes\mathcal{I}_{Z(V)}$ on $D$. Push-forward this morphism by the closed immersion $i$ to $X$, and consider the following composition:

[TABLE]

Let $\mathcal{F}_{D,A,V}$ denote the kernel of the composition. Thus, we get:

[TABLE]

Proposition 3.1**.**

The kernel sheaf $\mathcal{F}_{D,A,V}$ associated to $(D,A,V)$ has the following initial properties:

1. (a)

The sheaf $\mathcal{F}_{D,A,V}$ is reflexive of rank $r$. 2. (b)

For a general $V\in G(r,H^{0}(D,A))$, the sheaf $\mathcal{F}_{D,A,V}$ is locally free when $r\geq N$. 3. (c)

The determinant of $\mathcal{F}_{D,A,V}$ is $\mathcal{O}_{X}(-D)\simeq L^{\vee}$. 4. (d)

The sheaf $\mathcal{F}_{D,A,V}$ has no non-zero global sections, i.e. $H^{0}(X,\mathcal{F}_{D,A,V})=0$.

Proof.

For part (a), we note that in the exact sequence (1), $\mathcal{F}_{D,A,V}$ is the elementary transformation of a locally free sheaf by a torsion-free sheaf supported on the smooth divisor $D$. By [1, Lemma 2.4], $\mathcal{F}_{D,A,V}$ is a reflexive sheaf of rank $r$. When $r\geq N$, a general $V\in G(r,H^{0}(D,A))$ has $\textrm{codim}_{D}Z(V)=r\geq N$. Hence, $Z(V)$ is empty as $D$ is of dimension $N-1$. In this case, $A\otimes\mathcal{I}_{Z(V)}\,\simeq\,A$ and $\mathcal{F}_{D,A,V}$ is the usual elementary transformation, and is locally free. From (1), $\textrm{det}\,\mathcal{F}_{D,A,V}\simeq\textrm{det}\,i_{*}(A\otimes\mathcal{I}_{Z(V)})^{\vee}$. Since $Z(V)$ is of codimension at least 2 in $X$, we have $\textrm{det}\,i_{*}(A\otimes\mathcal{I}_{Z(V)})\simeq\textrm{det}\,i_{*}A\simeq\mathcal{O}_{X}(D).$ Part (d) can be proved by consider the long exact sequence of cohomology associated to (1). ∎

This construction gives us reflexive sheaves of rank $r\geq 2$ on smooth projective varieties of dimension $N\geq 2$. We call the dual sheaves $\mathcal{E}_{D,A,V}=\mathcal{F}_{D,A,V}^{\vee}$, the Lazarsfeld-Mukai reflexive sheaves. Dualizing the exact sequence (1) defining $\mathcal{F}_{D,A,V}$, we get:

[TABLE]

As with Lazarsfeld-Mukai bundles, the Lazarsfeld-Mukai reflexive sheaves are naturally equipped with an $r$-dimensional space of global sections $V^{\vee}\subset H^{0}(X,\mathcal{E}_{D,A,V})$.

Remark 3.2**.**

Suppose $X$ is a smooth, irreducible projective variety of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth and irreducible divisor on $X$ and $A$ be an ample and globally generated line bundle on $D$. Let $r$ be such that $2\leq r\leq h^{0}(D,A)$. Consider the dense open subscheme $\mathcal{U}_{r}\subset G(r,H^{0}(D,A))$ given by:

[TABLE]

If $V\in\mathcal{U}_{r}$, it is well-known that the ideal sheaves $\mathcal{I}_{Z(V)}$ form a flat family of sheaves parametrized by $\mathcal{U}_{r}$. Consequently, the torsion-free sheaves $\{A\otimes\mathcal{I}_{Z(V)}\}_{V\in\mathcal{U}_{r}}$ and the dual Lazarsfeld-Mukai reflexive sheaves $\{\mathcal{F}_{D,A,V}\}_{V\in\mathcal{U}_{r}}$ form flat families of sheaves parametrized by $\mathcal{U}_{r}$.

4. (Semi)stability of the sheaves $\mathcal{F}_{D,A,V}$

In this section we study the (semi)stability properties of the reflexive sheaves $\mathcal{F}_{D,A,V}$.

4.1. Varieties with cyclic Picard group

Suppose that $X$ is a smooth projective variety of dimension $N\geq 2$ such that $\textrm{Pic}\,X\simeq\mathbb{Z}\cdot[H]$. Here $[H]$ is the class of the ample generator the Picard group. Projective spaces, smooth hypersurfaces in $\mathbb{P}^{n}$ for $n\geq 4$, general complete intersections in higher dimensional projective spaces are some examples of such varieties. We now prove Theorem 1.1. When $X$ is a K3-surface with cyclic Picard group and $\mathcal{F}_{D,A,V}$ is a vector bundle, the $\mu_{H}$-stability and the simplicity of $\mathcal{F}_{D,A,V}$ is known, cf. [15, Lemma 1.3]. We generalize this to higher dimensional varieties when $\mathcal{F}_{D,A,V}$ is a reflexive sheaf.

Proof of Theorem 1.1..

Denote $\mathcal{E}:=\mathcal{E}_{D,A,V}=\mathcal{F}_{D,A,V}^{\vee}$. Recall the exact sequence (2) defining the Lazarsfeld-Mukai reflexive sheaf. Since the cokernel sheaf $\mathcal{E}xt^{1}(i_{*}(A\otimes\mathcal{I}_{Z(V)}),\mathcal{O}_{X})$ is supported only on $D$, there is a generically surjective morphism $\mathcal{O}_{X}^{r}\longrightarrow\mathcal{E}$.

Assume the contrary, i.e. $\mathcal{E}$ (equivalently $\mathcal{F}_{D,A,V}$) is not $\mu_{H}$-stable. Then, there is a torsion-free quotient $Q$ of $\mathcal{E}$, i.e. $\mathcal{E}\twoheadrightarrow Q$ of rank $s<r$, such that $\mu_{H}(\mathcal{E})\geq\mu_{H}(Q)$. From Proposition 3.1, $\textrm{det}\,(\mathcal{E})\simeq\mathcal{O}_{X}(H)$, thus

[TABLE]

Since $\textrm{Pic}\,X\simeq\mathbb{Z}$, we get $c_{1}(Q)\leq 0$.

Let $\widetilde{Q}=Q^{\vee\vee}$, the double dual of the torsion-free sheaf $Q$. The sheaf $\widetilde{Q}$ on $X$ is reflexive with $c_{1}(\widetilde{Q})=c_{1}(Q)$, and there is a natural inclusion $Q\hookrightarrow\widetilde{Q}$ which is generically an isomorphism. We have morphisms $\mathcal{O}_{X}^{r}\longrightarrow\mathcal{E}\longrightarrow Q\hookrightarrow\widetilde{Q}.$ As each morphism is generically surjective, we have a generically surjective morphism $\mathcal{O}_{X}^{r}\longrightarrow\widetilde{Q}\,.$ The $s$-th wedge product of the above morphism gives a generically surjective morphism:

[TABLE]

By [13, Proposition 1.2.7], we have $0=\mu_{H}(\bigwedge^{s}\mathcal{O}_{X}^{r})\leq\mu_{H}(\textrm{det}\,\widetilde{Q})=c_{1}(\textrm{det}\,\widetilde{Q})\cdot(H^{N-1})\,,$ implying that $c_{1}(\widetilde{Q})\geq 0$. Hence, $c_{1}(\widetilde{Q})=0$, i.e. $\textrm{det}\,Q\simeq\textrm{det}\,\widetilde{Q}\simeq\mathcal{O}_{X}$.

Let $\xi$ be the generic point of $X$. The generically surjective morphism $\mathcal{O}_{X}^{r}\longrightarrow\widetilde{Q}$, gives the surjective morphism of $\mathcal{O}_{\xi}$-vector spaces $(\mathcal{O}_{X}^{r})_{\xi}\twoheadrightarrow\widetilde{Q}_{\xi}$. Suppose the morphism $\mathcal{O}_{X}^{r}\longrightarrow\widetilde{Q}$ is given by global sections $t_{1},t_{2},\cdots,t_{r}$ of $\widetilde{Q}$, then the stalks of these at $\xi$ generate $\widetilde{Q}_{\xi}$ as an $\mathcal{O}_{\xi}$-vector space. Thereby, there are $s$ among these, say $t_{1},t_{2},\cdots,t_{s}$ which form a basis of $\widetilde{Q}_{\xi}$. These sections give a generically surjective morphism:

[TABLE]

The map $f$ is in fact an isomorphism. Indeed, if $K$ and $C$ denote the kernel and cokernel of $f$, we get:

[TABLE]

Since $f$ is generically surjective, $\textrm{rank}\,C=0$. Then, $\textrm{rank}\,K=0$, and thus $K=0$. Also, $\textrm{det}\,C\simeq\mathcal{O}_{X}$. Therefore, $C$ is supported on a closed set of codimension $\geq 2$ on $X$. Hence, $\mathcal{O}_{X}^{s}$ and $\widetilde{Q}$ are isomorphic on an open set whose complement has codimension $\geq 2$. By [12, Proposition 1.6 (iii)], $\widetilde{Q}\simeq\mathcal{O}_{X}^{s}$.

By [12, Corollary 1.2], $Q^{\vee}$ is reflexive. This gives $\mathcal{O}_{X}^{s}\simeq\widetilde{Q}^{\vee}\simeq Q^{\vee\vee\vee}\simeq Q^{\vee}$. From the surjection $\mathcal{E}\twoheadrightarrow Q$, we get $Q^{\vee}\simeq\mathcal{O}_{X}^{s}\hookrightarrow\mathcal{F}_{D,A,V}$. This contradicts $H^{0}(X,\mathcal{F}_{D,A,V})=0$ (Proposition 3.1 (d)). Hence, the kernel reflexive sheaf is $\mu_{H}$-stable. ∎

Remark 4.1**.**

From the proof of theorem, the following commutative diagram gives $Q\simeq\mathcal{O}_{X}^{s}$.

[TABLE]

The following result can be inferred from the above. Consider a smooth projective variety $X$. Let $Q$ be a torsion-free sheaf on $X$ with trivial determinant over an open set whose complement has codimension $\geq 2$. Suppose $Q$ admits a generically surjective morphism $\mathcal{O}_{X}^{r}\longrightarrow Q$, then the torsion-free sheaf $Q$ is itself trivial. We thank the referee for pointing this out.

Remark 4.2**.**

The proof of Theorem 1.1 proves essentially the following statement: Suppose a smooth projective variety $X$ has $\emph{Pic}\,X=\mathbb{Z}\cdot[H]$, where $H$ is ample. Let $F$ be a reflexive sheaf on $X$ such that

1. a.

the determinant $\emph{det}\,F=\mathcal{O}_{X}(-H)$, 2. b.

there is a generically surjective morphism from a trivial bundle on $X$ to $F^{\vee}$, 3. c.

the space of global sections of $F$, i.e. $H^{0}(X,F)=0$.

Then the reflexive sheaf $F$ is $\mu_{H}$-stable.

4.2. (Semi)stability in case of arbitrary smooth projective varieties

Suppose that $X$ is an irreducible, smooth projective variety of dimension $N\geq 2$ over $\mathbb{C}$. Let $L$ be an ample and globally generated line bundle on $X$.

Lemma 4.3**.**

For a general $D\in\emph{sm}|L|$, there is a finite, flat morphism $\phi:X\longrightarrow\mathbb{P}^{N}$ such that $D$ maps to the hyperplane $H=Z(x_{0})$ in $\mathbb{P}^{N}$, where $x_{0},x_{1},\cdots,x_{N}\in H^{0}(\mathbb{P}^{N},\mathcal{O}(1))$ are the homogeneous coordinates.

Proof.

Since $L$ is globally generated and $\textrm{dim}\,X=N$, any general collection of $N+1$ sections in $H^{0}(X,L)$ generate $L$. If $D\in\textrm{sm}|L|$ is general, then $D=Z(s_{0})$ for some $s_{0}\in H^{0}(X,L)$, and we can find $s_{1},s_{2},\cdots,s_{N}\in H^{0}(X,L)$ such that $\{s_{0},s_{1},s_{2},\cdots,s_{N}\}$ generate $L$. These sections give a morphism $\phi:X\longrightarrow\mathbb{P}^{N}$ such that $\phi^{*}(\mathcal{O}(1))=L$ and $s_{i}=\phi^{*}x_{i}$. If $H$ is the hyperplane $Z(x_{0})\subset\mathbb{P}^{N}$, we get the following commutative diagram.

[TABLE]

Since $X$ is a smooth projective variety over $\mathbb{C}$ and $\phi$ is defined by the sections of an ample line bundle $L$, the morphism $\phi$ is finite [16, Corollary 1.2.15]. This also implies that $\phi$ is surjective. Thereby, $\phi$ is a flat morphism [11, Exercise III.9.3 (a)]. ∎

We remark that, over positive characteristic we will obtain a finite map from $X$ to the projective space by choosing a very ample line bundle $L$.

Any ample, globally generated line bundle on the hyperplane $H$ is of the form $\mathcal{O}_{H}(m)\simeq\mathcal{O}_{\mathbb{P}^{N}}(m)|_{H}$ for some $m>0$. Consider $V^{\prime}\in G(r,H^{0}(H,\mathcal{O}_{H}(m))$ where $2\leq r\leq h^{0}(H,\mathcal{O}_{H}(m))$, such that $\textrm{codim}_{H}Z(V^{\prime})=r$. Then we have:

[TABLE]

By Theorem 1.1, $\mathcal{F}_{H,\mathcal{O}_{H}(m),V^{\prime}}$ is $\mu_{\mathcal{O}(1)}$-stable. We now have the following lemma.

Lemma 4.4**.**

Let $A=(L|_{D})^{\otimes m}$ on $D$. Then

1. (a)

the subspace $V=\phi|_{D}^{*}V^{\prime}\subset H^{0}(D,A)$ and $\emph{codim}_{D}Z(V)=r$, 2. (b)

the sheaf $\mathcal{F}_{D,A,V}\simeq\phi^{*}\mathcal{F}_{H,\mathcal{O}_{H}(m),V^{\prime}}$ and is $\mu_{L}$-semistable.

Proof.

Note that, $A=(L|_{D})^{\otimes m}\simeq\phi|_{D}^{*}\mathcal{O}_{H}(m)\,.$ This gives $V=\phi|_{D}^{*}V^{\prime}\subset\phi|_{D}^{*}H^{0}(H,\mathcal{O}_{H}(m))\subset H^{0}(D,A)\,.$ Since $V=\phi|_{D}^{*}V^{\prime}$, the closed subscheme $Z(V)$ maps to $Z(V^{\prime})$ under $\phi$, and we get the following commutative diagram.

[TABLE]

Since $\phi$ is a finite, surjective and flat morphism, so are $\phi|_{D}$ and $\phi|_{Z(V)}$. This implies that $Z(V)$ and $Z(V^{\prime})$ have the same dimension. Thus, $\textrm{codim}_{D}(Z(V))=\textrm{codim}_{H}(Z(V^{\prime}))=r$. This proves (a).

Consider the pullback of the exact sequence (3) by $\phi$:

[TABLE]

By [11, Proposition III.9.3],

[TABLE]

Note that $\phi|_{D}^{*}\,\mathcal{I}_{Z(V^{\prime})}\simeq\mathcal{I}_{Z(V)}\subset\mathcal{O}_{D}$. Hence, the short exact sequence (4) becomes

[TABLE]

Therefore, $\phi^{*}\mathcal{F}_{H,\mathcal{O}_{H}(m),V^{\prime}}\simeq\mathcal{F}_{D,A,V}$. Further, since $\mathcal{F}_{H,\mathcal{O}_{H}(m),V^{\prime}}$ is $\mu_{\mathcal{O}(1)}$-stable, and $\phi$ is a finite morphism, by [19, Lemma 1.17], $\mathcal{F}_{D,A,V}$ is $\mu_{L}$-semistable. ∎

We prove Theorem 1.3.

Proof of Theorem 1.3..

Let $D\in\textrm{sm}|L|$ be a divisor which is general in the linear system. By Lemma 4.3, there is a finite morphism $\phi:X\longrightarrow\mathbb{P}^{N}$ such that $\phi(D)$ is the hyperplane $H=Z(x_{0})$. Consider the line bundle $A=(L|_{D})^{\otimes m}$ on $D$ for any $m>0$. By Remark 3.2, there is a flat family of rank $r$ reflexive sheaves (where $2\leq r\leq h^{0}(D,A)$) parametrized by $V\in\mathcal{U}_{r}=\{V\in G(r,H^{0}(D,A))\,|\,\textrm{codim}_{D}Z(V)=r\}$. Each $V\in\mathcal{U}_{r}$ corresponds to the reflexive sheaf $\mathcal{F}_{D,A,V}\simeq\mathcal{F}_{D,L|_{D}^{\otimes m},V}$ .

Let $V=\phi^{*}V^{\prime}$, where $V^{\prime}\in G(s,H^{0}(H,\mathcal{O}_{H}(m))$ is such that $\textrm{codim}_{H}Z(V^{\prime})=s$. Then by Lemma 4.4, $V\in\mathcal{U}_{s}$ and $\mathcal{F}_{D,A,V}$ is $\mu_{L}$-semistable. Note that $s$ can only vary in the range $2\leq s\leq h^{0}(H,\mathcal{O}_{H}(m))={N-1+m\choose m}$. Hence, for any $r$ in the range $2\leq r\leq{N-1+m\choose m}$, we have a $V\in\mathcal{U}_{r}$ with the corresponding $\mathcal{F}_{D,A,V}$ being $\mu_{L}$-semistable. By [13, Proposition 2.3.1], semistability is an open condition in flat families. Therefore, for a general $V\in G(r,H^{0}(D,A))$ where $2\leq r\leq{N-1+m\choose m}$, $\mathcal{F}_{D,A,V}$ is $\mu_{L}$-semistable. ∎

4.3. (Semi)stability in case of projective space

Suppose ${X=\mathbb{P}^{N}}$ for $N\geq 2$.

4.3.1.

We first consider the case $L=\mathcal{O}(2)$ and $r=2$. Let $D\in\textrm{sm}|\mathcal{O}(2)|$ be a degree two hypersurface. Let $A$ be an ample, globally generated line bundle on $D$. We now discuss the $\mu_{\mathcal{O}(1)}$-stability of the rank 2 reflexive sheaf $\mathcal{F}_{D,A,V}$ where $V\in G(2,H^{0}(D,A))$.

We recall the concept of normalization of a torsion-free sheaf [23, Chapter II, $\mathcal{x}\,$1.2]. A torsion-free sheaf $E$ of rank 2 over $X=\mathbb{P}^{N}$ has a uniquely determined integer $k_{E}$ associated to it, namely,

[TABLE]

Note that, $c_{1}(E(k_{E}))\in\{0,-1\}$. We set $E_{\textrm{norm}}:=E(k_{E})$, and call $E$ normalized if $E=E_{\textrm{norm}}$.

Lemma 4.5**.**

[23, Lemma II.1.2.5]** A reflexive sheaf $E$ of rank 2 over $X=\mathbb{P}^{N}$ is stable if and only if $E_{\textrm{norm}}$ has no sections : $H^{0}(X,E_{\textrm{norm}})=0.$ If $c_{1}(E)$ is even, then $E$ is semistable if and only if $H^{0}(X,E_{\textrm{norm}}(-1))=0.$

Corollary 4.6**.**

Consider $L=\mathcal{O}(2)$ on $X=\mathbb{P}^{N}$ for $N\geq 2$. Let $D\in\emph{sm}|L|$, $A$ be an ample, globally generated line bundle on $D$ and $V\in G(2,H^{0}(D,A))$ be a 2-dimensional subspace. Then the associated rank two $\mathcal{F}_{D,A,V}$ is $\mu_{\mathcal{O}(1)}$-semistable.

Proof.

If $L=\mathcal{O}(2)$, then $c_{1}(\mathcal{F}_{D,A,V})=-2$. Then, $k_{\mathcal{F}_{D,A,V}}=1.$ Thus, ${\mathcal{F}_{D,A,V}}_{\textrm{norm}}=\mathcal{F}_{D,A,V}(1)$. Now, the result follows from the fact that $H^{0}(X,\mathcal{F}_{D,A,V})=0$ (Proposition 3.1 (d)). ∎

4.3.2.

We consider the case $L=\mathcal{O}(2l)$ and $r=2$.

Start with $L=\mathcal{O}(d)$ for any $d>0$. Consider $D\in\textrm{sm}|L|$ where $i:D\hookrightarrow X$ denotes the inclusion. Suppose $A$ is an ample, globally generated line bundle on $D$. For $r\geq 2$, let $V\in G(r,H^{0}(D,A))$ be such that $\textrm{codim}_{D}Z(V)=r$.

We restrict the exact sequence (1) to the open set $\widetilde{U}=X\setminus Z(V)$ to get:

[TABLE]

Let $U=D\setminus Z(V)$, an open subset of $D$. By [11, Proposition II.6.5], $U$ is a divisor in $\widetilde{U}$ and we have the following commutative diagram.

[TABLE]

By [11, Proposition III.9.3], $i_{*}(A\otimes\mathcal{I}_{Z(V)})|_{\widetilde{U}}\simeq(i|_{U})_{*}(A|_{U}\otimes\mathcal{I}_{Z(V)}|_{U})\simeq(i|_{U})_{*}(A|_{U})$. The short exact sequence (6) then becomes:

[TABLE]

Thus by [1, Lemma 1.1], $\mathcal{F}_{D,A,V}|_{\widetilde{U}}$ is an elementary transformation, and hence a locally free sheaf of rank $r$ on $\widetilde{U}$. Then, $(i|_{U})^{*}\mathcal{F}_{D,A,V}|_{\widetilde{U}}=\mathcal{F}_{D,A,V}|_{U}$ on $U\subset D$ is also locally free of rank $r$. Now, $U$ is an open subset of $D$ whose complement $Z(V)$ is of codimension $r\geq 2$ in $D$. This implies that the stalks of $\mathcal{F}_{D,A,V}|_{D}$ are torsion-free.

Remark 4.7**.**

The sheaf $\mathcal{F}_{D,A,V}|_{D}$ is torsion-free of rank $r$.

Assume now that $r=2$. Thus, $\mathcal{F}_{D,A,V}$ and $\mathcal{F}_{D,A,V}|_{D}$ have rank 2. Restrict the exact sequence (1) to $D$ to get:

[TABLE]

Here $K$ denotes the kernel, which is torsion-free by Remark 4.7. Note that $K$ is a sheaf of rank 1 on $D$. Consider the kernel $M$ of the surjective evaluation map $V\otimes\mathcal{O}_{D}\twoheadrightarrow A\otimes\mathcal{I}_{Z(V)}$. The sheaf $M$ is reflexive, cf. [12, Proposition 1.1]. Since $M$ is of rank 1, by [12, Proposition 1.9], $M$ is a line bundle. By comparing determinants, $M\simeq A^{\vee}$, and we get:

[TABLE]

From the exact sequences above, we get the following exact sequence on $D$:

[TABLE]

Remark 4.8**.**

As $\mathcal{F}_{D,A,V}|_{D}$ is torsion-free, $\emph{det}\,\mathcal{F}_{D,A,V}|_{D}\simeq(\emph{det}\mathcal{F}_{D,A,V})|_{D}\simeq(L|_{D})^{\vee}$.

Thus, from the exact sequence (7), the determinant of the rank 1 torsion-free sheaf $K$ is:

[TABLE]

Proposition 4.9**.**

Consider $L=\mathcal{O}(2l)$ for $l>0$. Let $D\in\emph{sm}|L|$, $A=\mathcal{O}(l)|_{D}$ and $V\in G(2,H^{0}(D,A))$ such that $\emph{codim}_{D}Z(V)=2$. Then the torsion-free sheaf $\mathcal{F}_{D,A,V}|_{D}$ on $D$ is $\mu_{\mathcal{O}(1)|_{D}}$-semistable. Thus $\mathcal{F}_{D,A,V}$ is $\mu_{\mathcal{O}(1)}$-semistable.

Proof.

Both $K$ and $A^{\vee}$ are rank one torsion-free sheaves on $D$ and hence are $\mu_{\mathcal{O}(1)|_{D}}$-stable. Further they have the same determinant. Indeed,

[TABLE]

Hence $\mathcal{F}_{D,A,V}|_{D}$ is a torsion-free sheaf which is an extension of $\mu_{\mathcal{O}(1)|_{D}}$-stable sheaves with the same slope. By [19, Lemma 1.10 (3)], $\mathcal{F}_{D,A,V}|_{D}$ is $\mu_{\mathcal{O}(1)|_{D}}$-semistable. This implies that $\mathcal{F}_{D,A,V}$ is $\mu_{\mathcal{O}(1)}$-semistable, cf. [18, Chapter 11]. ∎

We collect our observations to prove Theorem 1.2.

Proof of Theorem 1.2..

Part (a) of the theorem follows from Theorem 1.1. Part (b) of the theorem follows from Corollary 4.6. Proposition 4.9 proves part (c) of the theorem. Finally, part (d) follows from Theorem 1.3. ∎

Remark 4.10**.**

If $L=\mathcal{O}(d)$, then for all $r$ such that $(r,d)=1$, a $\mu_{\mathcal{O}(1)}$-semistable $\mathcal{F}_{D,A,V}$ is in fact stable. Indeed, if $D\in|\mathcal{O}(d)|$, then $\emph{deg}\,\mathcal{F}_{D,A,V}=c_{1}(\mathcal{F}_{D,A,V})\cdot(\mathcal{O}(1)^{N-1})=-d$ and $\emph{rank}\,\mathcal{F}_{D,A,V}=r$. Since the rank and degree are coprime, by [13, Lemma 1.2.14], semistability and stability coincide.

Theorem 1.2 shows that there are families of $\mu_{\mathcal{O}(1)}$-(semi)stable rank $r$ reflexive sheaves on the projective space with any prescribed $c_{1}$.

The following lemma shows that if we weaken our condition on $A$ in part (d) of Theorem 1.2, then we may not get the required (semi)stability.

Lemma 4.11**.**

Consider $L=\mathcal{O}_{X}(d)$ ($d>0$) on $X=\mathbb{P}^{N}$ for $N\geq 2$. Let $D\in|L|$ be a general smooth and irreducible hypersurface. Let $A=\mathcal{O}_{X}(l)|_{D}$ and $V\subset H^{0}(D,A)$ be a general $r$-dimensional subspace ($r\geq 2$) such that $r$ and $l$ satisfy the following condition:

[TABLE]

Then $\mathcal{F}_{D,A,V}$ is not $\mu_{\mathcal{O}(1)}$-semistable.

Note that if $l$ is an integer such that $0<l<\frac{d}{2}$, then for any $r\geq 2$, the above inequality is satisfied.

Proof.

Let $l$ and $r$ be integers that satisfy the given condition, i.e.

[TABLE]

Consider the line bundle $\mathcal{O}_{X}(l)$ on the projective space $X$. Since $D\in|\mathcal{O}_{X}(d)|$ and $A=\mathcal{O}_{X}(l)|_{D}$, we have $H^{0}(X,\mathcal{O}_{X}(l))\simeq H^{0}(D,A)$. Choose a general $V\in G(r,H^{0}(X,\mathcal{O}_{X}(l)))\simeq G(r,H^{0}(D,A))$.

Let $Z_{X}(V)$ (resp. $Z_{D}(V)$) denote the closed subscheme of $X$ (resp. $D$), defined by the vanishing of sections of $V\subset H^{0}(X,\mathcal{O}_{X}(l))$ (resp. $V\subset H^{0}(D,A)$). In fact $Z_{D}(V)=Z_{X}(V)\cap D$. Note that for a general $D$ and $V$, $\textrm{codim}_{X}Z_{X}(V)=\textrm{codim}_{D}Z_{D}(V)=r$. We get the following exact sequence on $X$:

[TABLE]

Here $M$ is a torsion-free (in fact reflexive) sheaf of rank $r-1$ and determinant $\mathcal{O}_{X}(-l)$. Thus $\mu_{\mathcal{O}_{X}(1)}(M)=\frac{-l}{r-1}$.

Also, corresponding to the triple $(D,A,V)$, we get the following exact sequence on $X$ (where $i:D\hookrightarrow X$ denotes the inclusion):

[TABLE]

Note that $\mathcal{F}_{D,A,V}$ is of rank $r$ and $\textrm{det}\,\mathcal{F}_{D,A,V}=\mathcal{O}_{X}(-d)$. We get $\mu_{\mathcal{O}_{X}(1)}(\mathcal{F}_{D,A,V})=\frac{-d}{r}$.

Comparing the above exact sequences, there is an inclusion $M\hookrightarrow\mathcal{F}_{D,A,V}$. But

[TABLE]

Hence $\mathcal{F}_{D,A,V}$ is not $\mu_{\mathcal{O}(1)}$-semistable. ∎

5. An application to Kernel bundles

In this section we see an application of the techniques used so far. As before, we work over the field of complex numbers.

Consider the line bundle $\mathcal{O}(d)$ ($d>0$) on $\mathbb{P}^{n}$, $n\geq 2$. We have the following exact sequence corresponding to the line bundle $\mathcal{O}(d)$ where $M_{\mathcal{O}(d)}$ is the kernel vector bundle.

[TABLE]

Flenner [10, Corollary 2.2] proved that the kernel bundles $M_{\mathcal{O}(d)}$ are $\mu_{\mathcal{O}(1)}$-semistable.

Remark 5.1**.**

In fact, in case of the line bundle $\mathcal{O}(1)$ on $\mathbb{P}^{n}$, the kernel bundle $M_{\mathcal{O}(1)}$ is stable. Indeed, note that $M_{\mathcal{O}(1)}$ is a locally free sheaf on $\mathbb{P}^{n}$ with $\emph{det}\,M_{\mathcal{O}(1)}=\mathcal{O}(-1)$. Further, there is a surjective morphism from a trivial bundle on $\mathbb{P}^{n}$ to $M_{\mathcal{O}(1)}^{\vee}$. For, dualizing (8) when $d=1$, we get the exact sequence:

[TABLE]

Finally, since $H^{0}(X,M_{\mathcal{O}(1)})=0$, by Remark 4.2, $M_{\mathcal{O}(1)}$ is $\mu_{\mathcal{O}(1)}$-stable.

Proposition 5.2**.**

Let $X$ be an irreducible, smooth projective variety of dimension $n$ over $\mathbb{C}$. Consider an ample and globally generated line bundle $L$ on $X$. For a general subspace $W\subset H^{0}(X,L)$ of dimension $n+1$, the kernel bundle $M_{L,W}$ associated to $(L,W)$ is $\mu_{L}$-polystable.

Proof.

For a general $W\in G(n+1,H^{0}(X,L))$, the corresponding linear system $\mathbb{P}W$ is basepoint-free and gives a finite surjective morphism $\psi_{W}:X\longrightarrow\mathbb{P}^{n}\,.$ Note that $\psi_{W}^{*}\mathcal{O}(1)\simeq L$ and that $\psi_{W}^{*}H^{0}(\mathbb{P}^{n},\mathcal{O}(1))\simeq W$. We have the kernel bundle $M_{\mathcal{O}(1)}$ on $\mathbb{P}^{n}$ defined by the exact sequence of the form (8) for $d=1$. Pullback this sequence to $X$, we get:

[TABLE]

Hence, $\psi_{W}^{*}M_{\mathcal{O}(1)}$ is the kernel bundle on $X$ associated to $(L,W)$, i.e. $\psi_{W}^{*}M_{\mathcal{O}(1)}\simeq M_{L,W}$. Again, since $M_{\mathcal{O}(1)}$ is $\mu_{\mathcal{O}(1)}$-stable, by [19, Lemma 1.17], $M_{L,W}$ is $\mu_{L}$-semistable. In fact, by a result of Kempf [14, Theorem 1], $M_{L,W}$ is $\mu_{L}$-polystable on $X$. ∎