System of Hodge Bundles and Generalized Opers on Smooth Projective Varieties

TL;DR

This paper establishes criteria for the stability of Hodge bundles and introduces generalized opers on smooth projective varieties, proving their associated Higgs bundles are semistable and linking oper structures to bundle stability.

Contribution

It defines generalized opers on smooth projective varieties and proves their associated Higgs bundles are semistable, also connecting partial oper structures with bundle stability.

Findings

Semistability criteria for Hodge bundles established

Generalized opers' associated Higgs bundles are semistable

Partial oper structures imply semistability of (E, ∇) pairs

Abstract

Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define notion of generalized opers on $X$, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle $E$ together with a connection $\nabla$ over $X$ implies semistability of the pair $(E, \nabla)$.

Equations

$${\mathcal{F}}^{\bullet }(E):E={\mathcal{F}}^0(E)⊋{\mathcal{F}}^1(E)⊋\cdots ⊋{\mathcal{F}}^{n-1}(E)⊋{\mathcal{F}}^n(E)=0$$

$$\theta {|}_{E_i}:E_i\stackrel{\simeq }{⟶}{E}_{i-1}\otimes {\Omega }_X^1,\forall i=1,\dots ,n$$

$${\mathcal{F}}^{\bullet }(E):0={\mathcal{F}}^n(E)⊊{\mathcal{F}}^{n-1}(E)⊊\cdots ⊊{\mathcal{F}}^1(E)⊊{\mathcal{F}}^0(E)=E$$

$$\nabla :E⟶E\otimes {\Omega }_X^1,$$

$$\nabla (f\cdot s)=s\otimes \mathrm{d}f+f\cdot \nabla (s),$$

$$\nabla :E\otimes {\Omega }_X^1⟶E\otimes {\Omega }_X^2$$

$$\kappa (\nabla ):=\nabla \circ \nabla \in H^0(X,\mathrm{End}(E)\otimes {\Omega }_X^2),$$

$${\mathcal{F}}^{\bullet }(E):E={\mathcal{F}}^0(E)⊋{\mathcal{F}}^1(E)⊋\cdots ⊋{\mathcal{F}}^{n-1}(E)⊋{\mathcal{F}}^n(E)=0,$$

$$\nabla ({\mathcal{F}}^i(E))\subseteq {\mathcal{F}}^{i-1}(E)\otimes {\Omega }_X^1,\forall i=1,\dots ,n-1.$$

$$\theta \wedge \theta :E\stackrel{\theta }{⟶}E\otimes {\Omega }_X^1\stackrel{\theta \otimes {\mathrm{Id}}_{{\Omega }_X^1}}{⟶}E\otimes {\Omega }_X^1\otimes {\Omega }_X^1\stackrel{{\mathrm{Id}}_E\otimes (-\wedge -)}{⟶}E\otimes {\Omega }_X^2.$$

$${\theta }_{\nabla }^i:{\mathrm{gr}}^i({\mathcal{F}}^{\bullet }(E))⟶{\mathrm{gr}}^{i-1}({\mathcal{F}}^{\bullet }(E))\otimes {\Omega }_X^1,$$

$${\theta }_{\nabla }:\mathrm{gr}({\mathcal{F}}^{\bullet }(E))⟶\mathrm{gr}({\mathcal{F}}^{\bullet }(E))\otimes {\Omega }_X^1,$$

$$\mathrm{gr}({\mathcal{F}}^{\bullet }(E))=\underset{i=0}{\overset{n-1}{⨁}}{\mathrm{gr}}^i({\mathcal{F}}^{\bullet }(E)).$$

$$\mathrm{deg}(E):=c_1(\det (E))\cdot [L{]}^{n-1},$$

$$\mu (F)\le \mu (E)(\text{respectively,}\mu (F)<\mu (E)).$$

$$n\left(\sum _{i=1}^n{a}_i{b}_i\right)\le \left(\sum _{i=1}^n{a}_i\right)\left(\sum _{j=1}^n{b}_j\right).$$

$$\left(\sum _{j=1}^n{b}_j\right)\left(\sum _{i=1}^n{a}_i\right)\le n\left(\sum _{i=1}^n{a}_i{b}_i\right).$$

$$\left(\sum _{i=0}^{r}i\cdot d^{i-1}\right)\left(\sum _{j=0}^n{d}^j\right)\le \left(\sum _{i=0}^{n}i\cdot d^{i-1}\right)\left(\sum _{j=0}^r{d}^j\right).$$

$$\mathrm{deg}(E_i)=i\cdot d^{i-1}\cdot \mathrm{deg}({\Omega }_X^1)\cdot \mathrm{rk}(E_0)+d^i\cdot \mathrm{deg}(E_0),$$

$$\mathrm{rk}(E_i)=d^i\cdot \mathrm{rk}(E_0),\forall i=0,\dots ,n.$$

$$\mu \left(\underset{i=0}{\overset{k}{⨁}}{E}_i\right)=\frac{\sum _{i=0}^k\mathrm{deg}(E_i)}{\sum _{i=0}^k\mathrm{rk}(E_i)}$$

$$\mu \left(\underset{i=0}{\overset{k}{⨁}}{E}_i\right)\le \mu (E),\forall k=0,\dots ,n.$$

$$\mu (F)>\mu (E).$$

$$F_i\cong \theta (F_i)\subseteq F_{i-1}\otimes {\Omega }_X^1,\forall i=0,1,\dots ,n.$$

$$0<\mathrm{rk}(F_r)\le \mathrm{rk}(F_{r-1})\le \cdots \le \mathrm{rk}(F_0).$$

$$\mathrm{deg}(F_i)\le \frac{\mathrm{rk}(F_i)\cdot \mathrm{deg}(E_i)/d^i}{\mathrm{rk}(E_0)},\forall i=0,1,\dots ,r.$$

$$\mathrm{deg}(F)\le \frac{\sum _{i=0}^r\mathrm{rk}(F_i)\mathrm{deg}(E_i)/d^i}{\mathrm{rk}(E_0)}\le \frac{\left(\sum _{i=0}^r\mathrm{rk}(F_i)\right)\left(\sum _{j=0}^r\mathrm{deg}(E_j)/d^j\right)}{(r+1)\mathrm{rk}(E_0)}$$

$$\mu (F)\le \frac{\left(\sum _{j=0}^r\mathrm{deg}(E_j)/d^j\right)}{(r+1)\mathrm{rk}(E_0)}\le \frac{\sum _{i=0}^r\mathrm{deg}(E_i)}{\mathrm{rk}(E_0)\cdot \sum _{i=0}^r{d}^i}=\mu \left(\underset{i=0}{\overset{r}{⨁}}{E}_i\right).$$

$$\mu (F)\le \mu (E),$$

$$\theta :=\left(\begin{array}{cc}0& \alpha \\ 0& 0\end{array}\right).$$

Full text

1. Introduction

Let $k$ be an algebraically closed field of characteristic $char(k)\geq 0$. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. Let $E$ be a vector bundle on $X$ together with a filtration

[TABLE]

where $\mathcal{F}^{i}(E)$ are subbundles of $E$, for all $i$. Suppose that $E$ admits a flat algebraic connection $\nabla:E\to E\otimes\Omega_{X}^{1}$ such that the filtration (1.1) is Griffiths transversal with respect to $\nabla$; meaning that $\nabla(\mathcal{F}^{i}(E))\subseteq\mathcal{F}^{i-1}(E)\otimes\Omega_{X}^{1}$, for all $i=1,\ldots,n-1$. Then $\nabla$ induces a Higgs field $\theta_{\nabla}$ on the associated vector bundle $\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)):=\bigoplus\limits_{i=0}^{n-1}\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E))$, where $\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E)):=\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)$, for all $i=0,1,\ldots,n-1$.

In [Si1], it is shown that given a flat connection $\nabla$ on $E$, there exists a Griffiths transversal filtration $\mathcal{F}^{\bullet}(E)$ such that the associated Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable. However, given a Griffiths transversal filtration $\mathcal{F}^{\bullet}(E)$ of $E$ with respect to $\nabla$, it is not known, in general, whether the associated Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable. When $X$ is a compact Riemann surface, in [Bi, p. 156], the following possible solution to this problem is proposed:

Let $E$ be a holomorphic vector bundle on $X$ whose all indecomposable components has degree zero. Let $\mathcal{F}^{\bullet}(E)$ be a filtration of $E$ by its subbundles on $X$. Then $E$ admits a holomorphic connection $\nabla$ such that $\mathcal{F}^{\bullet}(E)$ is Griffiths transversal with respect to $\nabla$ if and only if $\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))$ admits a holomorphic Higgs field $\theta$ such that

• •

$(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta)$ is semistable, and

• •

$\theta(\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E)))\subseteq\operatorname{\textnormal{gr}}^{i-1}(\mathcal{F}^{\bullet}(E))\otimes\Omega_{X}^{1}$, for all $i$.

Note that, the Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ admits a structure of a system of Hodge bundles on $X$; meaning that, $\theta_{\nabla}(\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E)))\subseteq\operatorname{\textnormal{gr}}^{i-1}(\mathcal{F}^{\bullet}(E))\otimes\Omega_{X}^{1}$, for all $i$. Therefore, it is natural to ask, more generally, when does a Higgs bundle (not necessarily of degree zero) on $X$ having a structure of a system of Hodge bundles is semistable and when it is stable. We give a criterion for this.

Fix an ample line bundle on $X$. We prove the following results :

Theorem 1.1**.**

Assume that $char(k)\geq 0$, and $\Omega_{X}^{1}$ is semistable with $\deg(\Omega_{X}^{1})\geq 0$. Let $(E,\theta)$ be a Higgs bundle on $X$ having a structure of a system of Hodge bundles: $E=\bigoplus\limits_{i=0}^{n}E_{i}$ such that $\theta|_{E_{i}}:E_{i}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}E_{i-1}\otimes\Omega_{X}^{1}$ is an isomorphism, for all $i=1,\ldots,n$. Then $(E,\theta)$ is semistable if $E_{i}$ is semistable, for all $i=0,1,\ldots,n$. The converse holds if $char(k)=0$.

Theorem 1.2**.**

Assume that $char(k)\geq 0$, and $\deg(\Omega_{X}^{1})>0$. The Higgs bundle $(E,\theta)$ in Theorem 1.1 is stable if $E_{i}$ is stable, for all $i=0,1,\ldots,n$. Converse holds if $\dim_{k}(X)=1$.

We give some examples to show that the isomorphism conditions

[TABLE]

and semistability of $E_{i}$, for all $i=0,1,\ldots,n$, in the Theorem 1.1 are crucial for semistability of $(E,\theta)$.

Finally, we give a criteria on the Griffiths transversal filtration for a flat connection, which we refer to as “generalized oper” so that the associated Higgs bundle becomes semistable. We define notion of semistability of connections, and prove the following :

Theorem 1.3**.**

Let $X$ be a polarized smooth projective variety over $k$, and let $\Omega_{X}^{1}$ be semistable of non-negative degree. Let $E$ be a vector bundle on $X$ together with a connection (not necessarily flat) $\nabla:E\longrightarrow E\otimes\Omega_{X}^{1}$. Let

[TABLE]

be a $\nabla$-Griffiths transversal filtration of $E$ by its subbundles such that the induced $\mathcal{O}_{X}$–module homomorphism $\theta_{\nabla}:\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))\longrightarrow\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))\otimes\Omega_{X}^{1}$ is a Higgs field on $\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))$ (i.e., $\theta_{\nabla}\wedge\theta_{\nabla}=0$ in $H^{0}(X,\operatorname{\mathcal{E}\!\textit{nd}}(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)))\otimes\Omega_{X}^{2})$). If the Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable (respectively, stable), then the pair $(E,\nabla)$ is semistable (respectively, stable).

Theorem 1.3 is a sort of converse of [LSYZ, Theorem 2.2] by Lan-Sheng-Yang-Zuo. This also generalize [JP, Proposition 3.4.4] of Joshi-Pauly proved for the case of curve in positive characteristic.

2. Griffiths Transversal Filtration

2.1. Preliminaries

Let $k$ be an algebraically closed field of characteristic $char(k)\geq 0$. Let $X$ be an irreducible smooth projective algebraic variety over $k$. Let $\mathcal{O}_{X}$ be the sheaf of regular functions on $X$. Let $\Omega_{X}^{1}$ be the cotangent bundle of $X$. Let $E$ be a coherent sheaf of $\mathcal{O}_{X}$–modules on $X$. The rank of $E$ is defined to be the dimension of the generic fiber of $E$. We denote it by $\operatorname{\mathrm{rk}}(E)$. Since $X$ is irreducible, this is well-defined. We say that $E$ is a vector bundle on $X$ if it is locally free and of finite rank on $X$. An $\mathcal{O}_{X}$–submodule $F$ of a vector bundle $E$ is said to be a subbundle of $E$ if $F$ is locally free and the quotient sheaf $E/F$ is torsion free on $X$.

Let $E$ be a vector bundle on $X$.

Definition 2.1**.**

A * connection* on $E$ is a $k$–linear sheaf homomorphism

[TABLE]

satisfying the following Leibniz rule:

[TABLE]

for every section $s\in E(U)$ and regular function $f\in\mathcal{O}_{X}(U)$, for any open subset $U\subset X$.

Let $\Omega_{X}^{2}:=\bigwedge^{2}\Omega_{X}^{1}$. Given a connection $\nabla$ on $E$, we can extend it to a $k$–linear sheaf homomorphism (denoted by the same symbol)

[TABLE]

satisfying $\nabla(s\otimes\omega)=s\otimes d\omega-\nabla(s)\wedge\omega$, for all local sections $s\in E(U)$ and $\omega\in\Omega_{X}^{1}(U)$. This defines an element

[TABLE]

called the curvature of $\nabla$. A connection $\nabla$ is said to be flat if $\kappa(\nabla)=0$.

Definition 2.2**.**

Let $E$ be a vector bundle on $X$ and let

[TABLE]

be a filtration of $E$ by its subbundles. The filtration $\mathcal{F}^{\bullet}(E)$ is said to be Griffiths transversal for a flat connection $\nabla$ on $E$ if it satisfies the following conditions:

[TABLE]

2.2. Associated Higgs Bundle

Definition 2.3**.**

A Higgs sheaf on $X$ is a pair $(E,\theta)$, where $E$ is a coherent sheaf of $\mathcal{O}_{X}$–modules on $X$ and $\theta:E\longrightarrow E\otimes\Omega_{X}^{1}$ is an $\mathcal{O}_{X}$–module homomorphism such that the following composite $\mathcal{O}_{X}$–module homomorphism vanishes identically:

[TABLE]

Consider a triple $(E,\mathcal{F}^{\bullet}(E),\nabla)$, where $E$ is a vector bundle on $X$ together with a flat connection $\nabla$, and a filtration $\mathcal{F}^{\bullet}(E)$ on $E$, as in (2.3), which is Griffiths transversal for $\nabla$ (see (2.4)). Then $\nabla$ induces an $\mathcal{O}_{X}$–linear homomorphism

[TABLE]

where ${\rm gr}^{i}(\mathcal{F}^{\bullet}(E))=\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)$, for all $0\leq i\leq n-1$, and ${\rm gr}^{-1}(\mathcal{F}^{\bullet}(E)):=0$; the $\mathcal{O}_{X}$–linearity of $\theta_{\nabla}^{i}$ follows from the Leibniz rule (2.2). Thus we have an $\mathcal{O}_{X}$–linear homomorphism

[TABLE]

where

[TABLE]

Note that, the flatness of $\nabla$ ensures that $\theta_{\nabla}\wedge\theta_{\nabla}=0$. Therefore, $(\operatorname{\textnormal{gr}}(\mathcal{E}^{\bullet}),\theta_{\nabla})$ is a Higgs bundle over $X$. Note that the Higgs field $\theta_{\nabla}$ satisfies $\theta_{\nabla}^{n}=0$, and hence is nilpotent in the graded $k$–algebra $\bigoplus\limits_{i=0}^{n}H^{0}\left(X\,,\,\operatorname{\mathcal{E}\!\textit{nd}}(E)\otimes\left(\Omega_{X}^{1}\right)^{\otimes i}\right)$, where $\operatorname{\mathcal{E}\!\textit{nd}}(E)$ is the sheaf of $\mathcal{O}_{X}$–module endomorphisms of $E$.

A polarization on $X$ is given by choice of an ample line bundle $L$ on it. Fix an ample line bundle $L$ on $X$. Let $E$ be a non-zero coherent sheaf of $\mathcal{O}_{X}$–modules on $X$. Then the degree of $E$ with respect to $L$ is defined by

[TABLE]

where $\det(E)$ is the determinant line bundle of $E$. If $\operatorname{\mathrm{rk}}(E)>0$, the ratio $\mu(E):=\deg(E)/\operatorname{\mathrm{rk}}(E)$ is called the slope of $E$.

Definition 2.4**.**

A torsion free Higgs sheaf $(E,\theta)$ on $X$ is said to be semistable (respectively, stable) if for any non-zero proper subsheaf $F\subset E$ with $0<\operatorname{\mathrm{rk}}(F)<\operatorname{\mathrm{rk}}(E)$ and $\theta(F)\subseteq F\otimes\Omega_{X}^{1}$, we have

[TABLE]

Remark 2.1**.**

A torsion free coherent sheaf $E$ on $X$ can be considered as a Higgs sheaf $(E,\theta)$ with zero Higgs field $\theta=0$ on $E$. Then the above notion of semistability and stability coincides with the corresponding notions for torsion free coherent sheaves.

Definition 2.5**.**

Let $(E_{1},\theta_{1})$ and $(E_{2},\theta_{2})$ be two Higgs sheaves on $X$. A Higgs homomorphism from $(E_{1},\theta_{1})$ to $(E_{2},\theta_{2})$ is given by an $\mathcal{O}_{X}$–module homomorphism $\varphi:E_{1}\longrightarrow E_{2}$ such that $\theta_{2}\circ\varphi=(\varphi\times\operatorname{\textnormal{Id}}_{\Omega_{X}^{1}})\circ\theta_{1}$.

Lemma 2.1**.**

Let $(E,\theta)$ and $(F,\phi)$ be two Higgs bundles on $X$. Let $\Theta=\theta\otimes\operatorname{\textnormal{Id}}_{F}+\operatorname{\textnormal{Id}}_{E}\otimes\phi$. If $(E\otimes F,\Theta)$ is semistable, then both $(E,\theta)$ and $(F,\phi)$ are semistable. Converse holds if the characteristic of $k$ is zero.

Proof.

Suppose that $(E\otimes F,\Theta)$ is semistable. If $(E,\theta)$ were not semistable, then there is a maximal destabilizing Higgs subsheaf $(E_{0},\theta|_{E_{0}})$ of $(E,\theta)$ with $\mu(E_{0})>\mu(E)$. Since the functor $-\otimes F$ is left exact, $(E_{0}\otimes F,\theta|_{E_{0}}\otimes\operatorname{\textnormal{Id}}_{F}+\operatorname{\textnormal{Id}}_{E_{0}}\otimes\phi)$ is a destabilizing subsheaf of $(E\otimes F,\Theta)$, with $\mu(E_{0}\otimes F)=\mu(E_{0})+\mu(F)>\mu(E)+\mu(F)=\mu(E\otimes F)$, contradicting Higgs semistability of $(E\otimes F,\Theta)$. Therefore, both $(E,\theta)$ and $(F,\phi)$ are semistable. For the converse part, see [Si2, Corollary 3.8, p. 38]. ∎

3. System of Hodge bundles and semistability

Let $X$ be an irreducible smooth projective algebraic variety over $k$ together with a fixed ample line bundle on it.

Definition 3.1**.**

A Higgs bundle $(E,\theta)$ is said to have a structure of a system of Hodge bundles if $E$ has a direct sum decomposition $E=\bigoplus\limits_{i=0}^{n}E_{i}$ by its subbundles $E_{i}$ such that $\theta(E_{i})\subseteq E_{i-1}\otimes\Omega_{X}^{1}$, for all $0\leq i\leq n$, with $E_{-1}=0$.

3.1. Criterion for semistability of a system of Hodge bundles

Now we give a criterion for semistability of a Higgs bundle having a structure of a system of Hodge bundles.

Theorem 3.1**.**

Assume that $\deg(\Omega_{X}^{1})\geq 0$. Let $(E,\theta)$ be a Higgs bundle on $X$ which admits a structure of a system of Hodge bundles $E=\bigoplus_{i=0}^{n}E_{i}$. Suppose that, $\theta|_{E_{i}}:E_{i}\longrightarrow E_{i-1}\otimes\Omega_{X}^{1}$ is an isomorphism of $\mathcal{O}_{X}$–modules, for all $i\in\{1,\ldots,n\}$. If $E_{i}$ is semistable, for all $i\in\{1,\ldots,n\}$, then $(E,\theta)$ is a semistable Higgs bundle.

To prove this theorem, we need the following useful inequalities:

Lemma 3.2** (Chebyshev’s sum inequalities).**

Let $\left(a_{i}\right)_{i=1}^{n}$ and $\left(b_{j}\right)_{j=1}^{n}$ be two finite sequence of real numbers.

1. (i)

If $a_{1}\geq a_{2}\geq\cdots\geq a_{n}$ and $b_{1}\leq b_{2}\leq\cdots\leq b_{n}$, then we have

[TABLE] 2. (ii)

If $a_{1}\leq a_{2}\leq\cdots\leq a_{n}$ and $b_{1}\leq b_{2}\leq\cdots\leq b_{n}$, then we have

[TABLE]

Lemma 3.3**.**

Let $d\geq 1$ be an integer. Then for any integers $r$ and $n$, with $0\leq r\leq n$, we have

[TABLE]

Proof of Theorem 3.1.

Since $E_{i}\cong E_{0}\otimes(\Omega_{X}^{1})^{\otimes i}$, for all $i\in\{0,1,\ldots,n\}$, we have,

[TABLE]

and

[TABLE]

Now for any integer $k\in\{0,1,\cdots,n\}$, by (3.4) and (3.5) we have,

[TABLE]

It follows from (3.6) and Lemma 3.3 that

[TABLE]

Suppose on the contrary that $(E,\theta)$ is not semistable. Let $F$ be the unique maximal semistable proper Higgs subsheaf of $(E,\theta)$ with

[TABLE]

It follows from [LSYZ, Lemma 2.4] that $F$ admits a structure of system of Hodge bundle; in particular, $F\cong\bigoplus\limits_{i=0}^{n}F_{i}$, with $F_{i}=F\cap E_{i}$, for all $i=0,1,\ldots,n$.

Since $\theta|_{E_{i}}$ is an isomorphism, we have

[TABLE]

Therefore, $F_{i}\neq 0$ implies $F_{i-1}\neq 0$, for all $1\leq i\leq n$. Let $r\in\{0,\cdots,n\}$ be the largest integer such that $F_{r}\neq 0$. Then $F=\bigoplus\limits_{i=0}^{r}F_{i}$. Now from (3.9), we have

[TABLE]

Since $F_{i}\neq 0$ and $E_{i}$ is semistable by assumption, using (3.5), we have

[TABLE]

Therefore, using (3.10) and (3.4), applying Lemma 3.2 (i), from (3.11), we have

[TABLE]

Now from (3.4) and (3.12), applying Lemma 3.2 (ii), we have

[TABLE]

Then from (3.13) and (3.7), we have

[TABLE]

which contradicts (3.8). Therefore, $(E,\theta)$ is semistable. ∎

Remark 3.1**.**

Note that, semistability of $E_{1}\cong E_{0}\otimes\Omega_{X}^{1}$ forces $\Omega_{X}^{1}$ to be semistable. It follows from the relation (3.4) and (3.5) that $E=\bigoplus_{i=0}^{n}E_{i}$, in Theorem 3.1, is semistable if and only if $\deg(\Omega_{X}^{1})=0$. Therefore, we get many examples of semistable Higgs bundles on $X$ whose underlying vector bundle is not semistable.

Remark 3.2**.**

If $char(k)>0$, it is expected that, if $\Omega_{X}^{1}$ and all $E_{i}$ are strongly semistable, then $(E,\theta)$ is strongly semistable; meaning that all the Frobenius pullbacks of $(E,\theta)$ are semistable.

We now give an example to show that a semistable Higgs bundle in Theorem 3.1 may not be stable, in general.

Example 3.1**.**

Let $X$ be an irreducible smooth complex projective algebraic curve of genus $g\geq 1$. Then $K_{X}:=\Omega_{X}^{1}$ is a line bundle of degree $2g-2$ on $X$. Let $Q=(\mathcal{O}_{X}(1))^{\otimes(g-1)}$ and set $E_{1}=Q\oplus Q$. Then $E_{1}$ is a rank $2$ strictly semistable vector bundle of degree $2g-2$ on $X$. Take $E_{0}:=E_{1}\otimes K_{X}^{-1}$ and define $E=E_{0}\oplus E_{1}$. Clearly, $\deg(E)=0$. Fix $\alpha\in\operatorname{\textnormal{Aut}}_{X}(E_{1})$ and consider it as an $\mathcal{O}_{X}$–module isomorphism $\alpha:E_{1}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}E_{0}\otimes K_{X}$. Define an $\mathcal{O}_{X}$–module homomorphism $\theta:E\longrightarrow E\otimes K_{X}$ by the matrix

[TABLE]

Then $(E,\theta)$ is a system of Hodge bundles on $X$ satisfying all conditions in Theorem 3.1. So $(E,\theta)$ is a semistable Higgs bundle on $X$. Since $E_{1}$ is not stable, there is a line subbundle $L_{1}$ of $E_{1}$ with $\deg(L_{1})=\mu(E_{1})=\deg(E_{1})/2$. Let $L_{0}=\theta(L_{1})\otimes K_{X}^{-1}\subset E_{0}$ and define $F:=L_{0}\oplus L_{1}$. Then $\theta(F)\subset F\otimes K_{X}$ and

[TABLE]

Therefore, $(E,\theta)$ is not stable.

Unless otherwise mentioned, from now on, we assume that $char(k)=0$.

Lemma 3.4**.**

Let $V$ be an unstable torsion free coherent sheaf of $\mathcal{O}_{X}$–modules on $X$. Let

[TABLE]

be the Harder-Narasimhan filtration of $V$. Then for any semistable vector bundle $W$ on $X$,

[TABLE]

is the Harder-Narasimhan filtration of $V\otimes W$.

Proof.

For each $i\in\{0,1,\cdots,m\}$, consider the exact sequence of coherent sheaves :

[TABLE]

Since $W$ is locally free, tensoring (3.14) with $W$, we get

[TABLE]

Then the result follows from the fact that $(V_{i}/V_{i+1})\otimes W$ is semistable (see e.g., [HL]) and $\mu((V_{i}/V_{i+1})\otimes W)=\mu((V_{i}/V_{i+1}))+\mu(W)$, for all $i=0,1,\ldots,m-1$. ∎

Lemma 3.5**.**

Let $E$ and $F$ be two isomorphic unstable torsion free coherent sheaf of $\mathcal{O}_{X}$–modules on $X$. Let $G\subset E$ be the maximal destabilizing subsheaf of $E$. Then for any two $\mathcal{O}_{X}$–module isomorphisms $f_{1},f_{2}:E\longrightarrow F$, we have $f_{1}(G)=f_{2}(G)$, and this is the maximal destabilizing subsheaf of $F$.

Proof.

This follows from the fact that the maximal destabilizing subsheaf is invariant under all $\mathcal{O}_{X}$–module automorphisms of the coherent sheaf. ∎

Theorem 3.6**.**

Assume that $\Omega_{X}^{1}$ is semistable with $\deg(\Omega_{X}^{1})\geq 0$. Let $(E,\theta)$ be a Higgs bundle on $X$ admitting a structure of a system of Hodge bundles given by $E=\bigoplus\limits_{i=0}^{n}E_{i}$ with $\theta|_{E_{i}}:E_{i}\longrightarrow E_{i-1}\otimes\Omega_{X}^{1}$ isomorphisms, for all $i=1,\ldots,n$. Then $(E,\theta)$ is semistable if and only if $E_{0}$ is semistable.

Proof.

Since $E_{p}\cong E_{0}\otimes(\Omega_{X}^{1})^{\otimes p}$, for all $p=0,1,\ldots,n$, and $\Omega_{X}^{1}$ is semistable, for any $p\in\{0,1,\cdots,n\}$ we have, $E_{p}$ is semistable if and only if $E_{0}$ is semistable. Therefore, if $E_{0}$ is semistable, then $(E,\theta)$ is semistable by Theorem 3.1. We now show the converse part.

Let $(E,\theta)$ be semistable. Tensoring $E$ with a sufficiently large degree line bundle, if required, we may assume that $\deg(E_{p})>0$, for all $p=0,1,\ldots,n$. Suppose that, $E_{0}$ is not semistable. Let $F_{p}\subset E_{p}$ be the maximal destabilizing subsheaf of $E_{p}$, for all $p=0,1,\ldots,n$. Since $\theta|_{E_{p}}:E_{p}\to E_{p-1}\otimes\Omega_{X}^{1}$ is an isomorphism, it follows from Lemma 3.4 and Lemma 3.5 that $\theta(F_{p})=F_{p-1}\otimes\Omega_{X}^{1}$, for all $p=0,1,\ldots,n$. Therefore, we have

[TABLE]

where $d=\operatorname{\mathrm{rk}}(\Omega_{X}^{1})=\dim(X)$. Clearly $F=\bigoplus\limits_{p=0}^{n}F_{p}$ is a Higgs subsheaf of $(E,\theta)$. Now from (3.15) we have,

[TABLE]

Therefore, $\mu(F)>\mu(E)$, which contradicts the fact that $(E,\theta)$ is semistable. ∎

3.2. Criterion for stability of a system of Hodge bundles

Let $char(k)\geq 0$.

Definition 3.2**.**

A Higgs bundle $(E,\theta)$ is said to be simple if any non-zero Higgs endomorphism of $(E,\theta)$ is an isomorphism.

Proposition 3.7**.**

Assume that $\deg(\Omega_{X}^{1})>0$. Let $(E,\theta)$ be a Higgs bundle on $X$ having a structure of a system of Hodge bundles: $E=\bigoplus\limits_{p=0}^{n}E_{p}$, with $\theta|_{E_{p}}:E_{p}\to E_{p-1}\otimes\Omega_{X}^{1}$ isomorphism, for all $p=1,\ldots,n$. If $E_{p}$ is stable, for all $p=0,1,\ldots,n$, then $(E,\theta)$ is simple.

Proof.

Since $\theta(E_{p})\subseteq E_{p-1}\otimes\Omega_{X}^{1}$, for all $p=0,1,\ldots,n$, the matrix of $\theta$ is strictly block-upper triangular and of the form :

[TABLE]

where $\theta_{ij}\in\mathrm{Iso}_{\mathcal{O}_{X}}(E_{j},E_{i}\otimes\Omega_{X}^{1})$, for all $i,j$. Let $\varphi:E\longrightarrow E$ be a non-zero $\mathcal{O}_{X}$–module homomorphism with

[TABLE]

For any $i,j\in\{0,1,\cdots,n\}$, let $\varphi_{ij}$ be the composite homomorphism

[TABLE]

where $\pi_{i}$ is the projection of $E$ onto the $i$-th factor. It follows from (3.16) that the matrix of $\varphi$ is block-upper triangular :

[TABLE]

Since $\mu(E_{i})=\mu(E_{0})+i\cdot\mu(\Omega_{X}^{1})$, and $E_{i}$ are stable by assumption, $\varphi_{ii}=\lambda_{i}\operatorname{\textnormal{Id}}_{E_{i}}$, for some $\lambda_{i}\in k$, for all $i=0,1,\ldots,n$, and $\varphi_{ij}=0$, for all $i<j$. Therefore, $\varphi$ is the diagonal matrix $\varphi=\mathrm{diag}(\varphi_{00},\cdots,\varphi_{nn})$. It follows from the relation (3.16) that $\lambda_{0}=\lambda_{1}=\cdots=\lambda_{n}$. Since $\varphi\neq 0$, we must have $\lambda_{i}\neq 0$, and hence $\varphi$ is an isomorphism. ∎

Definition 3.3**.**

A semistable Higgs bundle is said to be polystable if it is a direct sum of stable Higgs bundles.

Every semistable Higgs sheaf $(E,\theta)$ contains a unique maximal polystable Higgs subsheaf, called the socle of $(E,\theta)$. The socle of $(E,\theta)$ is invariant under all Higgs automorphisms of $E$.

Remark 3.3**.**

Note that, a simple polystable Higgs bundle is necessarily stable.

Theorem 3.8**.**

Assume that $char(k)\geq 0$, and $\deg(\Omega_{X}^{1})>0$. Let $(E,\theta)$ be a Higgs bundle on $X$ having a structure of a system of Hodge bundles: $E=\bigoplus\limits_{p=0}^{n}E_{p}$, with $\theta|_{E_{p}}:E_{p}\to E_{p-1}\otimes\Omega_{X}^{1}$ isomorphism, for all $p=1,\ldots,n$. If $E_{p}$ is stable, for each $p=0,1,\ldots,n$, then $(E,\theta)$ is stable.

Proof.

Clearly $(E,\theta)$ is semistable by Theorem 3.1. Suppose that $(E,\theta)$ is not stable. Then its socle $(F,\theta_{F})\subset(E,\theta)$ is the unique non-zero proper maximal polystable Higgs subsheaf with $\mu(F)=\mu(E)$. Clearly, $(F,\theta_{F})$ is invariant under the $\mathbb{G}_{m}$–action on $(E,\theta)$. Therefore, $(F,\theta_{F})$ admits a structure of a system of Hodge bundles, say $F=\bigoplus\limits_{i=0}^{n}F_{i}$, with $\theta_{F}(F_{i})\subseteq F_{i-1}\otimes\Omega_{X}^{1}$, for all $i=0,1,\ldots,n$. It follows from the proof of [LSYZ, Lemma 2.4] that, $F_{i}=F\cap E_{i}$, for all $i=0,1,\ldots,n$. Since $\theta|_{E_{p}}$ is an isomorphism, we have $F_{p}\cong\theta(F_{p})\subseteq F_{p-1}\otimes\Omega_{X}^{1}$, for all $p=1,\ldots,n$. Let $r\in\{0,1,\cdots,n\}$ be the largest integer such that $F_{r}\neq 0$. Then $F=\bigoplus\limits_{p=0}^{r}F_{p}$. Since $F$ is a proper subsheaf of $E$, there is at least one $p\in\{0,1,\cdots,r\}$ such that $F_{p}\neq E_{p}$. Since all $E_{p}$ are stable, we have,

[TABLE]

and the inequality (3.17) is strict for at least one $p\in\{0,1,\cdots,r\}$. Then from (3.17), following the inequality computations as in proof of Theorem 3.1, we have

[TABLE]

which contradicts the fact that $\mu(F)=\mu(E)$. Therefore, $(E,\theta)$ is polystable. Then by Proposition 3.7, $(E,\theta)$ is stable. ∎

Theorem 3.9**.**

Assume that $char(k)=0$. Let $X$ be an irreducible smooth projective algebraic curve of genus $g\geq 2$. Let $(E,\theta)$ be a Higgs bundle on $X$ admitting a structure of a system of Hodge bundles : $E=\bigoplus\limits_{p=0}^{n}E_{p}$ with $\theta|_{E_{p}}:E_{p}\longrightarrow E_{p-1}\otimes\Omega_{X}^{1}$ an isomorphism, for all $p=1,\ldots,n$. Then $(E,\theta)$ is stable if and only if $E_{p}$ is stable, for all $p=0,1,\ldots,n$.

Proof.

Suppose that, $(E,\theta)$ is stable. It follows from Theorem 3.6 that $E_{p}$ is semistable, for all $p=0,1,\ldots,n$. Since $K_{X}:=\Omega_{X}^{1}$ is a line bundle on $X$, for any $p\in\{0,1,\cdots,n\}$ we see that, $E_{p}$ stable if and only if $E_{0}$ is stable. Suppose that $E_{0}$ is not stable. Then there is a non-zero proper stable subsheaf $G_{0}\subset E_{0}$ with $\mu(G_{0})=\mu(E_{0})$. Since $\theta^{p}$ is an isomorphism of $E_{p}$ onto $E_{0}\otimes K_{X}^{\otimes p}$, for all $p$, there is a subsheaf $G_{p}\subset E_{p}$ such that $\theta^{p}:G_{p}\longrightarrow G_{0}\otimes K_{X}^{\otimes p}$ is isomorphism, for all $p=0,1,\ldots,n$. Then $G=\bigoplus\limits_{p=0}^{n}G_{p}$ is a Higgs subsheaf of $(E,\theta)$. Now a similar computation as in the proof of Theorem 3.6 shows that $\mu(G)=\mu(E)$, which contradicts the fact that $(E,\theta)$ is stable. ∎

Remark 3.4**.**

It is expected that for $\dim_{k}(X)\geq 2$ with $\Omega_{X}^{1}$ is stable and $\deg(\Omega_{X}^{1})>0$, if $(E,\theta)$ in Theorem 3.9 is stable then all $E_{p}$ are polystable.

Remark 3.5**.**

Note that, in the proofs of all Theorems in this Section, we have used only semistability (or stability) of $\Omega_{X}^{1}$ and the condition $\deg(\Omega_{X}^{1})\geq 0\,\,(\,>0)$. Therefore, with appropriate notion of semistability and stability of pairs $(E,\theta)$ with $\theta\in H^{0}(X,\operatorname{\mathcal{E}\!\textit{nd}}(E)\otimes V)$, all Theorems in this Section 3 hold if we replace $\Omega_{X}^{1}$ with any semistable (or stable) vector bundle $V$ on $X$ of degree $\geq 0$ (or, $>0$).

3.3. Examples of unstable system of Hodge bundles

We now give two examples to show that the isomorphism conditions in Theorem 3.1 are crucial.

Example 3.2**.**

Let $X$ be a smooth complex projective curve of genus $g\geq 2$. Let $L_{0}$ be a line bundle of degree $d>2g-2$ on $X$. Let $E_{1}$ be a non-trivial extension of $\mathcal{O}_{X}$ and $L_{0}\otimes K_{X}$. So we have a short exact sequence of $\mathcal{O}_{X}$–modules

[TABLE]

Then $E_{1}$ is semistable, but not necessarily stable. Let $E=E_{0}\bigoplus E_{1}$, where $E_{0}=L_{0}$. Then $\deg(E)=\deg(E_{0})+\deg(E_{1})=2d+2g-2$, and hence $\mu(E)=2(d+g-1)/3$. Since $d>2g-2$, we have $\mu(E)<d=\mu(L_{0})$. Define a Higgs field $\theta\in H^{0}(X,\operatorname{\mathcal{E}\!\textit{nd}}(E)\otimes K_{X})$ by

[TABLE]

Then $(E,\theta)$ is a Higgs bundle having a structure of a system of Hodge bundles on $X$. Note that, $\theta_{1}$** is surjective, but not isomorphism**. Since $L_{0}$ is a $\theta$–invariant, $(E,\theta)$ is not semistable.

Example 3.3**.**

Let $X$ be a smooth projective curve of genus $g\geq 2$ over $k$. Let $L_{0}$ be a line bundle on $X$ of positive degree. Let $L_{1}=L_{0}^{\vee}$ and $E=L_{0}\bigoplus L_{1}$. Since $\deg(\operatorname{\mathcal{H}\!\textit{om}}(L_{1},L_{0}\otimes K_{X}))=2\cdot\deg(L_{0})+(2g-2)$, choosing $L_{0}$ with $\deg(L_{0})$ sufficiently large, we can find a non-zero $\mathcal{O}_{X}$–module homomorphism $\theta_{1}:L_{1}\longrightarrow L_{0}\otimes K_{X}$. Note that, $\theta_{1}$** is injective**, because both $L_{1}$ and $L_{0}\otimes K_{X}$ are line bundles, but $\theta_{1}$** is not an isomorphism**. Define an $\mathcal{O}_{X}$–module homomorphism $\theta:E\longrightarrow E\otimes K_{X}$ by

[TABLE]

Then $(E,\theta)$ is a Higgs bundle having a structure of a system of Hodge bundles on $X$. Since $L_{0}$ is a $\theta$–invariant subbundle of positive degree, $(E,\theta)$ is not semistable.

We now shows that, if all $E_{p}$ are not semistable $(E,\theta)$ may fail to be semistable.

Example 3.4**.**

Let $X$ be a smooth complex projective curve of genus $g\geq 2$. Fix a square root $K^{1/2}$ of the cotangent bundle $K_{X}$ on $X$. Let $L_{0}$ be a positive degree line bundle on $X$. Consider the line bundles $L_{1}=\left(L_{0}^{\vee}\otimes K_{X}^{-1/2}\right)^{\otimes 2}$ and $L_{2}=L_{0}\otimes K_{X}$ on $X$. Let $E=E_{0}\bigoplus E_{1}$, where $E_{0}=L_{0}$ and $E_{1}=L_{1}\bigoplus L_{2}$. Clearly, $\deg(E)=0$. Consider the $\mathcal{O}_{X}$–module homomorphism $\theta:E\longrightarrow E\otimes K_{X}$ defined by

[TABLE]

where $\theta_{1}:E_{1}=L_{1}\bigoplus(L_{0}\otimes K_{X})\longrightarrow E_{0}\otimes K_{X}$ is the projection homomorphism onto the second factor. Then $(E,\theta)$ is a Higgs bundle of degree [math] having a structure of a system of Hodge bundles on $X$. Note that, $E_{1}$** is not semistable**, and $\theta|_{E_{1}}:E_{1}\to E_{0}\otimes K_{X}$ is surjective, but not isomorphism. Since $E_{0}$ is a $\theta$–invariant subbundle of positive degree, $(E,\theta)$ is not semistable.

4. Generalized Oper

4.1. Semistability of generalized oper

It is not know if the Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$, defined in Section 2.2 associated to a Griffiths transversal filtration $\mathcal{F}^{\bullet}(E)$ with respect to a flat connection $\nabla$ on $E$, is semistable or not. In this section, we give a criterion for semistability of $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$.

Definition 4.1**.**

An oper is a triple $(E,\mathcal{F}^{\bullet}(E),\nabla)$ consists of a vector bundle $E$ on $X$ together with a flat connection $\nabla$ and a Griffiths transversal filtration $\mathcal{F}^{\bullet}(E)$ (see (2.3)) of $E$ by its subbundles (see Definition 2.2), such that the quotients $\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)$ are line bundles on $X$ and the $\mathcal{O}_{X}$–linear homomorphisms $\theta_{\nabla}^{i}$ in (2.6) induced by $\nabla$ are isomorphisms, for all $i=1,\ldots,n-1$.

Let us first recall the following well-known result.

Proposition 4.1**.**

[Si1, p. 186]** Let $X$ be a connected smooth complex projective curve of genus $g\geq 1$. Let $(E,\mathcal{F}^{\bullet}(E),\nabla)$ be an oper on $X$. Then the associated Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ on $X$ is semistable.

We give a natural generalization of the above result for higher ranks and higher dimensional algebraic varieties.

Definition 4.2**.**

A generalized oper is a triple $(E,\mathcal{F}^{\bullet}(E),\nabla)$ consists of a vector bundle $E$ on $X$ together with a flat connection $\nabla$ and a Griffiths transversal filtration $\mathcal{F}^{\bullet}(E)$ (see (2.3)) of $E$ by its subbundles (see Definition 2.2), such that the quotients $\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E)):=\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)$ are semistable vector bundles on $X$ and the $\mathcal{O}_{X}$–linear homomorphisms $\theta_{\nabla}^{i}$ in (2.6) induced by $\nabla$ are isomorphisms, for all $i=1,\ldots,n-1$.

Remark 4.1**.**

Note that, if $(E,\mathcal{F}^{\bullet}(E),\nabla)$ is a generalized oper on $X$ and $\deg(\Omega_{X}^{1})>0$, then $\mathcal{F}^{\bullet}(E)$ is the Harder-Narasimhan filtration of $E$.

Theorem 4.2**.**

Let $(E,\mathcal{F}^{\bullet}(E),\nabla)$ be a generalized oper on $X$. If $\deg(\Omega_{X}^{1})\geq 0$, then the associated Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ on $X$ is semistable, where $\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))=\bigoplus\limits_{i=0}^{n-1}\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E))$.

Proof.

Since $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ has a structure of a system of Hodge bundles, all $\operatorname{\textnormal{gr}}^{i}(\mathcal{F}^{\bullet}(E))$ are semistable and all $\theta_{\nabla}^{i}$ are isomorphisms, by Theorem 3.1, $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable. ∎

Remark 4.2**.**

1. (1)

If $X$ is a smooth complex projective curve of genus $g\geq 1$, then we get Proposition 4.1 as a corollary to the Theorem 4.2. 2. (2)

With appropriate notion of logarithmic Higgs semistability, Theorem 4.2 holds for logarithmic connections singular over an effective divisor, using similar techniques.

4.2. Semistability of Connections

As before, let $X$ be a smooth polarized projective variety over and algebraically closed field $k$ of characteristic $p\geq 0$, and the cotangent bundle $\Omega_{X}^{1}$ is semistable and of non-negative degree.

Definition 4.3**.**

Let $E$ be a torsion free coherent sheaf on $X$ together with a connection $\nabla:E\to E\otimes\Omega_{X}^{1}$. Then the pair $(E,\nabla)$ is said to be semistable (respectively, stable) if for any non-zero proper $\mathcal{O}_{X}$–submodule $F\subset E$ with torsion free quotient sheaf $E/F$ on $X$ such that $\nabla(F)\subseteq F\otimes\Omega_{X}^{1}$, we have $\mu(F)\leq\mu(E)$ (respectively, $\mu(F)<\mu(E)$).

Definition 4.4**.**

A partial oper is a triple $(E,\mathcal{F}^{\bullet}(E),\nabla)$ consisting of a vector bundle $E$ on $X$ together with a flat connection $\nabla$ and a filtration $\mathcal{F}^{\bullet}(E)$ of $E$ by its subbundles on $X$ which is Griffiths transversal with respect to $\nabla$ such that the induced Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable on $X$.

Given a vector bundle $E$ on $X$ together with a semistable flat connection $\nabla$ on $E$, Lan-Sheng-Yang-Zuo proved that there is a filtration $\mathcal{F}^{\bullet}(E)$ of $E$ such that the triple $(E,\mathcal{F}^{\bullet}(E),\nabla)$ is a partial oper on $X$ (see [LSYZ, Theorem 2.2]). We now prove some sort of converse of the above result.

Theorem 4.3**.**

Let $E$ be a vector bundle on a smooth projective variety $X$ over an algebraically closed field $k$ of positive characteristic. Let $\nabla:E\longrightarrow E\otimes\Omega_{X}^{1}$ be a connection (not necessarily flat) on $E$. Let

[TABLE]

be a $\nabla$-Griffiths transversal filtration of $E$ by its subbundles such that the induced $\mathcal{O}_{X}$–module homomorphism $\theta_{\nabla}:\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))\longrightarrow\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))\otimes\Omega_{X}^{1}$ is a Higgs field on $\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E))$ (i.e., $\theta_{\nabla}\bigwedge\theta_{\nabla}=0$ in $H^{0}(X,\operatorname{\mathcal{E}\!\textit{nd}}(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)))\otimes\Omega_{X}^{2})$), and the Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable. Then the pair $(E,\nabla)$ is semistable.

Proof.

If $(E,\nabla)$ were not semistable, then there is a non-zero proper $\mathcal{O}_{X}$–submodule $F\subset E$ such that $\nabla(F)\subseteq F\otimes\Omega_{X}^{1}$ and

[TABLE]

The filtration $\mathcal{F}^{\bullet}(E)$ induces a filtration on $F$

[TABLE]

where $\mathcal{F}^{i}(F)=\mathcal{F}^{i}(E)\cap F$, for all $i$. Since $\nabla(F)\subseteq F\otimes\Omega_{X}^{1}$, the injective homomorphisms $\iota_{i}:\mathcal{F}^{i}(F)/\mathcal{F}^{i+1}(F)\hookrightarrow\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)$, induced by the inclusions $\mathcal{F}^{i}(F)\subseteq\mathcal{F}^{i}(E)$, fits into the following commutative diagram of $\mathcal{O}_{X}$–module homomorphisms

[TABLE]

where $\theta_{i}^{\prime}$ is the restriction of $\theta_{i}$, for all $i$. So $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(F)),\theta_{\nabla}^{\prime})$ is a non-zero Higgs subsheaf of the semistable Higgs bundle $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$. Now a simple degree rank computation shows that

[TABLE]

which contradicts the inequality (4.1). Therefore, $(E,\nabla)$ is semistable. ∎

Corollary 4.4**.**

Let $E$ be a vector bundle on $X$ together with a flat connection $\nabla:E\to E\otimes\Omega_{X}^{1}$. Suppose that $E$ admits a filtration by its subbundles

[TABLE]

which is Griffiths transversal with respect to $\nabla$, and the $\nabla$–induced $\mathcal{O}_{X}$–module homomorphisms $\theta_{i}:\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)\longrightarrow\left(\mathcal{F}^{i-1}(E)/\mathcal{F}^{i}(E)\right)\otimes\Omega_{X}^{1}$ are isomorphisms, for all $i=1,\ldots,n-1$. Then the pair $(E,\nabla)$ is semistable if $\mathcal{F}^{i}(E)/\mathcal{F}^{i+1}(E)$ is semistable, for all $i=0,1,\ldots,n-1$.

Proof.

If $p=char(k)$ is zero, then for any flat connection $\nabla$ on $E$, the pair $(E,\nabla)$ is automatically semistable. This follows from the fact that any non-zero coherent sheaf $F$ on $X$ admitting a flat connection has zero first Chern class, and hence has zero slope with respect to any polarization on $X$. This is not the case if $p>0$.

If $char(k)=p>0$, since $(\operatorname{\textnormal{gr}}(\mathcal{F}^{\bullet}(E)),\theta_{\nabla})$ is semistable by Theorem 3.1, we are done by using Theorem 4.3. ∎

Remark 4.3**.**

Corollary 4.4 was proved over smooth projective curve in positive characteristics by Joshi-Pauly (see [JP, Proposition 3.4.4]).

Acknowledgement

This work is supported by the Post-doctoral fellowship of the Institute of Mathematical Sciences (HBNI), Chennai, India.