Existence of nontrivial logarithmic co-Higgs structure on curves

TL;DR

This paper investigates the existence, properties, and parameter spaces of nontrivial logarithmic co-Higgs structures on unstable bundles over algebraic curves, extending some results to higher dimensions.

Contribution

It provides criteria for the existence and non-existence of such structures, describes their parameter spaces, and explores their invariants and higher-dimensional cases.

Findings

Criteria for existence and non-existence established

Parameter spaces of structures described

Higher-dimensional cases analyzed

Abstract

We study various aspects on nontrivial logarithmic co-Higgs structure associated to unstable bundles on algebraic curves. We check several criteria for (non-)existence of nontrivial logarithmic co-Higgs structures and describe their parameter spaces. We also investigate the Segre invariants of these structures and see their non-simplicity. In the end we also study the higher dimensional case, specially when the tangent bundle is not semistable.

Equations

$$0\to T_X(-\log \mathcal{D})\to {\mathcal{T}}_{\mathcal{X}}\to {\oplus }_{i=1}^m{{\epsilon }_i}_{\ast }{\mathcal{O}}_{{\mathcal{D}}_i}({\mathcal{D}}_i)\to 0,$$

$$\{0\}={\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \cdots \subset {\mathcal{F}}_s=\mathcal{E}$$

$${\mu }_{+}(\mathcal{E}):=\mu ({\mathcal{F}}_1),{\mu }_{-}(\mathcal{E}):=\mu ({\mathcal{F}}_s/{\mathcal{F}}_{s-1}).$$

$${\mu }_{-}(\mathcal{E})\le {\mu }_{+}(\mathcal{E})+{\mu }_{+}({\mathcal{T}}_{\mathcal{D}})={\mu }_{+}(\mathcal{E}\otimes {\mathcal{T}}_{\mathcal{D}}).$$

$$r=r_1+\cdots +r_s,d=d_1+\cdots +d_s.$$

$$\mathbb{U}={\mathbb{U}}_X(s;r_1,d_1;\dots ;r_s,d_s)$$

$$\{0\}={\mathcal{F}}_0\otimes {\mathcal{T}}_{\mathcal{D}}\subset {\mathcal{F}}_1\otimes {\mathcal{T}}_{\mathcal{D}}\subset \cdots \subset {\mathcal{F}}_s\otimes {\mathcal{T}}_{\mathcal{D}}$$

$$\mathcal{E}\otimes {\mathcal{T}}_{\mathcal{D}}\to \mathcal{E}\otimes {\mathcal{T}}_{\mathcal{D}}^{\otimes 2}$$

$$0\to {\mathcal{T}}_{\mathcal{D}}\to {\mathcal{T}}_{\mathcal{X}}\to {\oplus }_{i=1}^m{\mathbb{C}}_{p_i}\to 0.$$

$${\mu }_{-}(\mathcal{E})=\mu ({\mathcal{F}}_s/{\mathcal{F}}_{s-1})\le \gamma +\mu ({\mathcal{F}}_1)=\gamma +{\mu }_{+}(\mathcal{E}),$$

$$b(i):=\underset{i+1\le k\le s}{\min }\left\{k|\mu ({\mathcal{F}}_i/{\mathcal{F}}_{i-1})+\gamma \ge \mu ({\mathcal{F}}_k/{\mathcal{F}}_{k-1})\right\}$$

$$c(j):=\underset{1\le k\le j-1}{\max }\left\{k|\mu ({\mathcal{F}}_j/{\mathcal{F}}_{j-1})\le \gamma +\mu ({\mathcal{F}}_k/{\mathcal{F}}_{k-1})\right\}.$$

$$(s;r_1,\dots ,r_s;d_1,\dots ,d_s)\in {\mathbb{Z}}^{\oplus (2s+1)}$$

$$0\to {\mathcal{F}}_1\to {\mathcal{F}}_2\to {\mathcal{F}}_2/{\mathcal{F}}_1\to 0$$

$$0\to {\mathcal{G}}_1\to \mathcal{E}\to {\mathcal{G}}_2\to 0,$$

$$\mathcal{E}\cong {\mathcal{O}}_{{\mathbb{P}}^1}(b_1{)}^{\oplus r_1}\oplus \cdots \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(b_s{)}^{\oplus r_s},$$

$$\Delta :=\sum _{1\le i<j\le s}\max \{0,\gamma +1+b_i-b_j\}.$$

$$\mathcal{E}\cong {\mathcal{E}}_{+}\oplus {\mathcal{E}}_{-},\text{ with }{\mathcal{E}}_{+}\cong {\oplus }_{i=1}^e{\mathcal{O}}_{{\mathbb{P}}^1}(b_i{)}^{\oplus r_i}\text{ and }{\mathcal{E}}_{-}\cong {\oplus }_{i=e+1}^r{\mathcal{O}}_{{\mathbb{P}}^1}(b_i{)}^{\oplus r_i}.$$

$$0\to \mathcal{A}\stackrel{u}{\to }\mathcal{B}\to {\mathbb{C}}_p\to 0$$

$$\frac{d}{r}-\frac{a}{t}\ge \frac{(r-t)(g-1)}{r}.$$

$$\mu (\mathcal{B})-\mu ({\mathcal{G}}_t)$$

$$\ge \frac{t-r+(r-t)(g-1)}{rt}$$

$$=\frac{(r-t)(g-2)}{rt}\ge 0,$$

$${\mathbb{M}}_X(r_1,d_1)\times \cdots \times {\mathbb{M}}_X(r_s,d_s).$$

$${\mathbb{M}}_X(r_i,-d_i)\times {\mathbb{M}}_X(r_j,d_j+r_j\gamma ).$$

$$0\to {\mathcal{A}}_1\to {\mathcal{A}}_2\to {\mathcal{A}}_3\to 0,$$

$$\frac{a_2}{x_2}\le \frac{a_2-a_1}{x_2-x_1}\left(\text{resp. }\frac{a_2}{x_2}<\frac{a_2-a_1}{x_2-x_1}\right).$$

$${\mu }_{-}(\mathcal{E})=d_s/r_s\le d_1/r_1+\gamma ={\mu }_{-}(\mathcal{E}\otimes {\mathcal{T}}_{\mathcal{D}}),$$

$$d_2/r_2+\gamma <d_s/r_s\le d_1/r_1+\gamma .$$

$$0\to {\mathcal{F}}_i\to {\mathcal{F}}_{i+1}\to {\mathcal{E}}_{i+1}\to 0$$

$$\gcd (r_1,d_1)=\gcd (r_s,d_s)=1\text{ and }{d}_1/r_1+\gamma \ge d_s/r_s.$$

Full text

The first author is partially supported by MIUR and GNSAGA of INDAM (Italy). The second author is supported by Basic Science Research Program 2015-037157 through NRF funded by MEST and the National Research Foundation of Korea(KRF) 2016R1A5A1008055 grant funded by the Korea government(MSIP)

1. 1 Introduction

2. 2 Preliminary

3. 3 Curve case

4. 3.1 Projective line

5. 3.2 Elliptic curves

6. 3.3 Higher genus case

7. 4 Segre invariant

8. 4.1 Curve case

9. 4.2 Simplicity

10. 5 Higher dimensional case

11. 5.1 Case of $\cal{T}_{\cal{D}}$ semistable

12. 5.2 Case of $\cal{T}_{\cal{D}}$ not semistable

13. 5.2.1 Rank $2$ case

14. 5.2.2 Rank $3$ case

1. Introduction

A logarithmic co-Higgs sheaf on a complex manifold $X$ is a pair $(\cal{E},\Phi)$ with a torsion-free coherent sheaf $\cal{E}$ on $X$ and a morphism $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ satisfying the integrability condition $\Phi\wedge\Phi=0$, where $\cal{T}_{\cal{D}}$ is the logarithmic tangent bundle $X$ associated to an arrangement $\cal{D}$ of hypersurfaces with simple normal crossings. When $\cal{D}$ is empty, it is a co-Higgs sheaf in the usual sense, introduced and developed by Hitchin and Gualtieri; see [18, 15]. When $\cal{E}$ is locally free, it is a generalized vector bundle on $X$ considered as a generalized complex manifold, whose co-Higgs field vanishes in the normal direction to the support of $\cal{D}$.

It is observed in [4, Theorem 1.1] that the semistability of a co-Higgs bundle $(\cal{E},\Phi)$ on $X$ with nonnegative Kodaira dimension implies the semistability of $\cal{E}$. In case of negative Kodaira dimension, there are several works on description of moduli space of semistable co-Higgs bundles, including the case when the associated bundle is not stable; see [27] and [10].

Now the additional condition for a co-Higgs field to vanish in the normal direction to $\cal{D}$ with higher degree, forces the associated bundle to be unstable. So we are mainly interested in the logarithmic co-Higgs sheaves associated to the arrangement with high degree and assume that the length of Harder-Narasimhan filtration is at least two. We fix numeric data for the Harder-Narasimhan filtration of the sheaf in consideration, i.e. fix the length $s$ at least two of the filtration together with rank $r_{i}$ and degree $d_{i}$ of the successive quotients in the Harder-Narasimhan filtration (2). Setting $\gamma:=\deg\cal{T}_{\cal{D}}$ and $\mu_{i}:=d_{i}/r_{i}$, we always assume that $\mu_{s}-\mu_{1}\leq\gamma<0$ as the least requirement for the existence of the non-trivial co-Higgs field; see Corollary 3.9. Then we investigate the numeric criterion for the sheaf to admit a non-trivial co-Higgs field; see Proposition 3.4 and Theorem 3.7.

Theorem 1.1**.**

Fix the numeric data for the Harder-Narasimhan filtration and denote by $\mathbb{U}$ the set of the torsion-free sheaves on an algebraic curve $X$ with these data. Then the following hold:

• (i)

there exists an unstable sheaf in $\mathbb{U}$ with non-trivial co-Higgs field;

• (ii)

the inequality $\mu_{s}-\mu_{1}\geq\gamma+1-g$ implies the existence of an unstable sheaf in $\mathbb{U}$ with no non-trivial co-Higgs field;

• (iii)

the inequality $\mu_{s}-\mu_{1}<\gamma+1-g$ implies that every sheaf in $\mathbb{U}$ admits a non-trivial co-Higgs field.

The existence part is induced by explicit usage of positive elementary transformations and the positive answer to the Lange conjecture [28]. Furthermore we extend the notion of Segre invariant to the setting of logarithmic co-Higgs sheaves and show that it is well-defined over curves under the assumption that $\gamma<0$ and that this invariant is same as the usual Segre invariant under a certain condition; see Corollary 4.8 and Proposition 4.14.

Theorem 1.2**.**

For a logarithmic co-Higgs sheaf $(\cal{E},\Phi)$ on an algebraic curve $X$ with $\gamma<0$, the $k^{th}$-Segre invariant $s_{k}(\cal{E},\Phi)$ is well-defined. It is also equal to the Segre invariant $s_{k}(\cal{E})$ in the usual sense, if $\cal{E}$ admits the complete Harder-Narasimhan filtration, i.e. $r_{i}=1$ for all $i$.

Then we check in Proposition 4.14 that co-Higgs sheaves associated to unstable bundle are usually not stable, not even simple.

Over algebraic curves the bundle $\cal{T}_{\cal{D}}$ is automatically semistable. So, as the counterpart to the case of algebraic curves, in §5 we deal with the case when the dimension of $X$ is at least $2$ and $\cal{T}_{\cal{D}}$ is not semistable. Under the assumption that the biggest slope in the Harder-Narasimhan filtration of $\cal{T}_{\cal{D}}$ is negative, we give a recipe to construct all the pairs $(\cal{E},\Phi)$ with $\cal{E}$ reflexive of $\mathrm{rk}(\cal{E})=r\in\{2,3\}$ and non-trivial co-Higgs field $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$. When $r=2$ and in most cases with $r=3$, the map $\Phi$ is always $2$-nilpotent and so it is integrable. We also /point out exactly when we cannot guarantee the integrability.

2. Preliminary

Let $X$ be a smooth projective variety of dimension $n$ at least one with the tangent bundle $T_{X}$ over the field of complex numbers $\mathbb{C}$. We fix an ample line bundle $\cal{O}_{X}(1)$ and denote by $\cal{E}(t)$ the twist of $\cal{E}$ by $\cal{O}_{X}(t)$ for any coherent sheaf $\cal{E}$ on $X$ and $t\in\mathbb{Z}$. We also denote by $\cal{E}^{\vee}$ the dual of $\cal{E}$. The dimension of cohomology group $H^{i}(X,\cal{E})$ is denoted by $h^{i}(X,\cal{E})$ and we will skip $X$ in the notation, if there is no confusion. We define the slope $\mu(\cal{E})$ of a coherent sheaf $\cal{E}$ on $X$ with respect to $\cal{O}_{X}(1)$ to be $\deg\cal{E}/\mathrm{rk}(\cal{E})$.

Now consider an arrangement $\cal{D}=\{D_{1},\ldots,D_{m}\}$ of pairwise distinct, smooth and irreducible divisors $D_{i}$ on $X$, and if there is no confusion we also denote by $\cal{D}$ the divisor $D_{1}+\ldots+D_{m}$. We assume that the divisor $\cal{D}$ has simple normal crossings. Then the associated logarithmic tangent bundle $T_{X}(-\log\cal{D})$ is locally free and fits into the following exact sequence; see [13].

[TABLE]

where $\varepsilon_{i}:D_{i}\rightarrow X$ is the embedding. If there is no confusion, we will simply denote $T_{X}(-\log\cal{D})$ by $\cal{T}_{\cal{D}}$.

Definition 2.1**.**

[4] A $\cal{D}$-logarithmic co-Higgs sheaf on $X$ is a pair $(\cal{E},\Phi)$ where $\cal{E}$ is a torsion-free coherent sheaf on $X$ and $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ with $\Phi\wedge\Phi=0$. Here $\Phi$ is called the logarithmic co-Higgs field of $(\cal{E},\Phi)$ and the condition $\Phi\wedge\Phi=0$ is an integrability condition originating in the work of Simpson [29].

For a torsion-free coherent sheaf $\cal{E}$ on $X$, we consider its associated Harder-Narasimhan filtration:

[TABLE]

with the graduation $gr(\cal{E}):=\oplus_{i=1}^{s}\cal{F}_{i}/\cal{F}_{i-1}$ such that each $\cal{F}_{i}/\cal{F}_{i-1}$ is semistable and $\mu(\cal{F}_{i}/\cal{F}_{i-1})$ is strictly decreasing for all $i<s$. The integer $s$ is called the length of the filtration, and if $s=r$, then the filtration is said to be complete. We denote by $\mu_{+}(\cal{E})$ and $\mu_{-}(\cal{E})$ the maximal and minimal slopes in the filtration, respectively:

[TABLE]

Remark 2.2**.**

For two torsion-free sheaves $\cal{A}$ and $\cal{B}$ on $X$, let $\cal{A}\overline{\otimes}\cal{B}$ be the quotient of $\cal{A}\otimes\cal{B}$ by its torsion. If $\cal{A}$ and $\cal{B}$ are semistable, then $\cal{A}\overline{\otimes}\cal{B}$ is also semistable by [22, Theorem 2.5]. Applying this observation to the Harder-Narasimhan filtrations of $\cal{A}$ and $\cal{B}$, we get that $\mu_{+}(\cal{A}\overline{\otimes}\cal{B})=\mu_{+}(\cal{A})+\mu_{+}(\cal{B})$.

Lemma 2.3**.**

If $f:\cal{A}\rightarrow\cal{B}$ is a nonzero map between two torsion-free sheaves on $X$, then we have $\mu_{-}(\cal{A})\leq\mu_{+}(\cal{B})$

Proof.

Let $\{0\}=\cal{A}_{0}\subset\cal{A}_{1}\subset\cdots\subset\cal{A}_{a}=\cal{A}$ be the Harder-Narasimhan filtration of $\cal{A}$ and let $k\in\{1,\dots,a\}$ be the minimal integer such that $\cal{A}_{k}\nsubseteq\mathrm{ker}(f)$, i.e. the minimal integer such that $f_{|\cal{A}_{k}}\not\equiv 0$. Then we have $f_{|\cal{A}_{k-1}}\equiv 0$ and so $f_{|\cal{A}_{k}}$ induces a nonzero map $\widetilde{f}:\cal{A}_{k}/\cal{A}_{k-1}\rightarrow\cal{B}$.

Let $\{0\}=\cal{B}_{0}\subset\cal{B}_{1}\subset\cdots\subset\cal{B}_{b}=\cal{B}$ be the Harder-Narasimhan filtration of $\cal{B}$ and let $l$ be the minimal positive integer $l\leq b$ such that $\widetilde{f}(\cal{A}_{k}/\cal{A}_{k-1})\subseteq\cal{B}_{l}$. Then we have $\widetilde{f}(\cal{A}_{k}/\cal{A}_{k-1})\nsubseteq\cal{B}_{l-1}$ and so $\widetilde{f}$ induces a nonzero map $\hat{f}:\cal{A}_{k}/\cal{A}_{k-1}\rightarrow\cal{B}_{l}/\cal{B}_{l-1}$. Since $\cal{A}_{k}/\cal{A}_{k-1}$ and $\cal{B}_{l}/\cal{B}_{l-1}$ are semistable, we have $\mu(\cal{A}_{k}/\cal{A}_{k-1})\leq\mu(\cal{B}_{l}/\cal{B}_{l-1})$. By the definition of $\mu_{+}$ and $\mu_{-}$ in terms of the Harder-Narasimhan filtration we have $\mu(\cal{A}_{k}/\cal{A}_{k-1})\geq\mu_{-}(\cal{A})$ and $\mu(\cal{B}_{l}/\cal{B}_{l-1})\leq\mu_{+}(\cal{B})$, concluding the assertion. ∎

Remark 2.2 and Lemma 2.3 give the following whose assertion will be assumed throughout this article.

Corollary 2.4**.**

Assuming the existence of a nonzero map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$, we have

[TABLE]

Remark 2.5**.**

Assume that $\cal{E}$ is not semistable and so $s\geq 2$. If there exists a nonzero map $f\in\operatorname{Hom}(\cal{E}/\cal{F}_{s-1},\cal{F}_{s-1}\otimes\cal{T}_{\cal{D}})$, then we may composite the quotient map $\cal{E}\rightarrow\cal{E}/\cal{F}_{s-1}$ with it to get a nonzero $2$-nilpotent logarithmic co-Higgs field $\Phi_{f}$. Note that the associated co-Higgs field is uniquely determined by the choice of a map, i.e. if $f$ and $g$ are two different nonzero maps in $\operatorname{Hom}(\cal{E}/\cal{F}_{s-1},\cal{F}_{s-1}\otimes\cal{T}_{\cal{D}})$, then we get $\Phi_{f}\neq\Phi_{g}$.

If $n$ is at least two, we fix a polarization $\cal{O}_{X}(1)$ with respect to which we consider (semi-)stability. For most cases in this article we will mainly assume that $\cal{D}$ is of high degree so that $\cal{T}_{\cal{D}}$ is “sufficiently negative” and that $\cal{T}_{\cal{D}}$ is semistable with $\gamma=\deg\cal{T}_{\cal{D}}<0$, except in §5.2;

Assumption 2.6**.**

We always assume that $\gamma=\deg\cal{T}_{\cal{D}}$ is negative, if there is no specification.

Remark 2.7**.**

There are manifolds with $\Omega_{X}^{1}$ ample as in [11, 12], in which cases we may even take $\cal{D}=\emptyset$: If instead of logarithmic co-Higgs field we use the field $T_{X}(-\cal{D})\cong T_{X}\otimes\cal{O}_{X}(-\cal{D})$ vanishing on a divisor $\cal{D}$, then we may use the semistability of the tangent bundle of many Fano manifolds [26] and then take a very positive $\cal{D}$ to get $T_{X}(-\cal{D})$ negative and semistable.

We fix a triple of integers $(r,d,s)\in\mathbb{Z}^{\oplus 3}$ together with pairs $(r_{i},d_{i})\in\mathbb{Z}^{\oplus 2}$ for $1\leq i\leq s$ such that $r\geq 2$, $s\geq 1$, $r_{i}\geq 1$ and

[TABLE]

Assume further that $d_{i}/r_{i}>d_{i+1}/r_{i+1}$ for $i=1,\dots,s-1$. Then we denote by

[TABLE]

the set of all torsion-free coherent sheaves $\cal{E}$ of rank $r$ such that the Harder-Narasimhan filtration (2) of $\cal{E}$ with respect to $\cal{O}_{X}(1)$ has $(r_{1},d_{1};\cdots;r_{s},d_{s})$ as its numerical data, i.e. each quotient sheaf $\cal{F}_{i}/\cal{F}_{i-1}$ is semistable of rank $r_{i}$ and degree $d_{i}$. By [22] the filtration (2) tensored by $\cal{T}_{\cal{D}}$

[TABLE]

is the Harder-Narasimhan filtration of $\cal{E}\otimes\cal{T}_{\cal{D}}$ if $\cal{T}_{\cal{D}}$ is semistable. We also assume the existence of a nonzero co-Higgs field $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$: if $n$ is at least two, we do not assume for the moment the integrability condition $\Phi\wedge\Phi=0$, because in the most examples in this article it will follow from the other assumption, or from Lemma 2.8, where we assume that $s$ is at least two.

Denote by $\widetilde{\Phi}$ the following map:

[TABLE]

induced by $\Phi$. Comsiting the natural map $\cal{T}_{\cal{D}}^{\otimes 2}\rightarrow\wedge^{2}\cal{T}_{\cal{D}}$ with $\widetilde{\Phi}\circ\Phi$, we have $\Phi\wedge\Phi$ as an element in $\mathrm{Hom}(\cal{E},\cal{E}\otimes\wedge^{2}\cal{T}_{\cal{D}})$.

Lemma 2.8**.**

If $s$ is at least two, then we have $\widetilde{\Phi}\circ\Phi=0$, i.e. $\Phi$ is $2$-nilpotent. In particular, we have $\Phi\wedge\Phi=0$.

Proof.

Since we assume that $\gamma$ is negative, the sheaf $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}\subset\cal{E}\otimes\cal{T}_{\cal{D}}$ is the Harder-Narasimhan filtration of $\cal{E}\otimes\cal{T}_{\cal{D}}$ and $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}^{\otimes 2}\subset\cal{E}\otimes\cal{T}_{\cal{D}}^{\otimes 2}$ is the Harder-Narasimhan filtration of $\cal{E}\otimes\cal{T}_{\cal{D}}^{\otimes 2}$. Thus we have $\Phi(\cal{E})\subseteq\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ and $\widetilde{\Phi}(\cal{F}_{1}\otimes\cal{T}_{\cal{D}})=0$, implying that $\widetilde{\Phi}\circ\Phi=0$. ∎

3. Curve case

Assume that $X$ is a smooth algebraic curve of genus $g$ and take $\cal{D}=\{p_{1},\ldots,p_{m}\}$ a set of $m$ distinct points. Then we have $\cal{T}_{\cal{D}}\cong T_{X}\otimes\cal{O}_{X}(-\cal{D})$ with degree $\gamma:=2-2g-m$. We assume that $\gamma$ is negative so that we are not in the set-up of [24]. The sequence (1) turns into the following

[TABLE]

Another feature of the case $n=1$ is that all logarithmic co-Higgs fields automatically satisfy the integrability condition.

Consider a vector bundle $\cal{E}$ of rank $r$ with the Harder-Narasimhan filtration (2) and we assume

[TABLE]

which is a necessary condition for the existence of a nonzero map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$; see Corollary 3.9.

Remark 3.1**.**

For each $i\in\{1,\dots,s\}$ with $\mu(\cal{F}_{i}/\cal{F}_{i-1})+\gamma\geq\mu_{-}(\cal{E})$, define

[TABLE]

and then the map $\Phi$ induces a map $\Phi^{i}:\cal{F}_{b(i)}/\cal{F}_{b(i)-1}\rightarrow\left(\cal{F}_{i}/\cal{F}_{i-1}\right)\otimes\cal{T}_{\cal{D}}$. Similarly, for each $j\in\{2,\dots,s\}$ with $\mu(\cal{F}_{j}/\cal{F}_{j-1})\leq\gamma+\mu_{+}(\cal{E})$, define

[TABLE]

The map $\Phi$ induces a map $\Phi_{j}:\cal{F}_{j}/\cal{F}_{j-1}\rightarrow\left(\cal{F}_{c(j)}/\cal{F}_{c(j)-1}\right)\otimes\cal{T}_{\cal{D}}$. Note that these maps $\Phi^{i}$ and $\Phi_{j}$ are not necessarily nonzero.

Now fix the following numeric data

[TABLE]

with $s,r_{i}>0$ for each $i$ such that $d_{i}/r_{i}>d_{i+1}/r_{i+1}$ for all $i$; if $g=0$, we also assume $a_{i}/r_{i}\in\mathbb{Z}$ for each $i$. Recall that we denote by $\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ the set of all vector bundles $\cal{E}$ of rank $r:=\sum_{i=1}^{s}r_{i}$ on $X$ with the Harder-Narasimhan filtration (2) such that $\mathrm{rk}(\cal{F}_{i}/\cal{F}_{i-1})=r_{i}$ and $\deg\cal{F}_{i}/\cal{F}_{i-1}=d_{i}$ for each $i$. The conditions just given above for $s$, $r_{i}$ and $d_{i}$ are the necessary and sufficient conditions for the existence of a vector bundle $\cal{E}$ on $X$ with rank $r$ and degree $d:=d_{1}+\cdots+d_{s}$.

Indeed, for the existence part, in case $g\geq 2$ we may even take a stable bundle $\cal{F}_{i}/\cal{F}_{i-1}$, while in case $g=1$ by Atiyah’s classification of vector bundles on elliptic curves, we may take as $\cal{F}_{i}/\cal{F}_{i-1}$ a semistable bundle; we can choose either indecomposable one or polystable one, depending on our purpose.

To get parameters spaces we first get parameter spaces for the sheaves $\cal{E}$, then for a fixed sheaf $\cal{E}$ we study all logarithmic co-Higgs fields $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ and then we put together the informations. We have several problems coming from the sheaves $\cal{E}$ such as non-separatedness or often reducibility of moduli of sheaves, and then more problems bring the logarithmic co-Higgs field into the picture.

First of all, we fix enough numerical invariant to get a bounded family of pairs $(\cal{E},\Phi)$. Fixing an ample line bundle $\cal{O}_{X}(1)$, we consider sheaves $\cal{E}$ with a Harder-Narasimhan filtration (2) and we fix the Hilbert function of each subquotient $\cal{F}_{i}/\cal{F}_{i-1}$. Since each $\cal{F}_{i}/\cal{F}_{i-1}$ is assumed to be semistable, the family of all $\cal{F}_{i}/\cal{F}_{i-1}$ are bounded. We first see that the $\operatorname{Ext}^{1}$-groups involved in the extensions

[TABLE]

are upper bounded and that the set of all $\cal{F}_{2}$ is bounded. Then we consider the set of all $\cal{F}_{3}$ and so on, inductively. We may get relative $\operatorname{Ext}^{1}$-groups as parameter spaces, but these parameter spaces usually do not parametrizes one-to-one isomorphism classes of sheaves, even by taking into account that proportional extensions gives isomorphic sheaves.

For the relative $\operatorname{Ext}^{1}$ we need to have universal family parametrizing all $\cal{F}_{i}/\cal{F}_{i-1}$ and we usually need to work with parameter spaces of sheaves which do not parametrizes one-to-one isomorphic classes. Note that there is a flat family with isomorphic sheaves $\cal{E}$ whose flat limit is $gr(\cal{E})=\oplus_{i=1}^{s}\cal{F}_{i}/\cal{F}_{i-1}$. Thus there is no hope of one-to-one parametrization of isomorphism classes of sheaves; when the numerology allows that some $\cal{F}_{i}/\cal{F}_{i-1}$ is strictly semistable, then this phenomenon occurs even for the graded subquotient $\cal{F}_{i}/\cal{F}_{i-1}$. Algebraic stacks of course do not parametrize isomorphism classes of sheaves, not even of vector bundles; see [14]. In the case $n=1$ with $X=\mathbb{P}^{1}$, we have a unique bundle, $\cal{E}$ for any fixed parameter space $\mathbb{U}_{\mathbb{P}^{1}}(s;r_{1},d_{1};\cdots;r_{s},d_{s})$ and so the parameter space for $(\cal{E},\Phi)$ is the vector space $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})$, which parametrizes one-to-one the isomorphism classes of pairs $(\cal{E},\Phi)$. See Remark 3.12 for the case $n=1$ and $X$ a curve of genus $g\geq 2$.

Remark 3.2**.**

In the case $s=2$, the datum of $(\cal{E},\Phi)$ with $[\cal{E}]\in\mathbb{U}_{X}(2;r_{1},d_{1};r_{2},d_{2})$ and $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ induces a holomorphic triple $\psi:\cal{E}/\cal{F}_{1}\xrightarrow{\ \ }\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ in the sense of [9] and we may study the stability of the holomorphic triple. Conversely, for every holomorphic triple $f:\cal{G}_{2}\rightarrow\cal{G}_{1}\otimes\cal{T}_{\cal{D}}$ such that $\cal{G}_{1}$ and $\cal{G}_{2}$ are semistable with $\mathrm{rk}(\cal{G}_{i})=r_{i}$ and $\deg\cal{G}_{i}=d_{i}$, $i=1,2$, and for any extension class

[TABLE]

we get $[\cal{E}]\in\mathbb{U}_{X}(2;r_{1},d_{1};r_{2},d_{2})$ with $0\subset\cal{G}_{1}\subset\cal{E}$ as its Harder-Narasimhan filtration and a $2$-nilpotent map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ induced by $f$. Two sheaves, say $\cal{E}$ and $\cal{E}^{\prime}$, fitting as middle bundles in (6) for the same $\cal{G}_{1}$ and $\cal{G}_{2}$ are isomorphic if and only if their associated extensions are proportional, because $\cal{G}_{1}$ and $\cal{G}_{2}$ are assumed to be semistable with $d_{1}/r_{1}>d_{2}/r_{2}$ and so (6) is the Harder-Narasimhan filtration of the bundle in the middle.

This argument fits very well in §5.1, where $\cal{T}_{\cal{D}}$ is assumed to be semistable, because $\cal{F}_{i}\otimes\cal{T}_{\cal{D}}$ would be in the Harder-Narasimhan filtration of $\cal{E}\otimes\cal{T}_{\cal{D}}$; in this case we only require that $\cal{E}$ is torsion-free and then define $\mathbb{U}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with Mumford’s (slope-)semistability.

3.1. Projective line

We take $X=\mathbb{P}^{1}$ and then we have $\cal{T}_{\cal{D}}\cong\cal{O}_{\mathbb{P}^{1}}(\gamma)$ with $\gamma<0$. Any vector bundle $\cal{E}\cong\oplus_{i=1}^{r}\cal{O}_{\mathbb{P}^{1}}(a_{i})$ on $\mathbb{P}^{1}$ with $a_{1}\geq\cdots\geq a_{r}$ can be rewritten as

[TABLE]

with $\sum_{i=1}^{s}r_{i}=r$ and $b_{1}>\cdots>b_{s}$, i.e. in the Harder-Narasimhan filtration (2) associated to $\cal{E}$, we have $\cal{F}_{i}/\cal{F}_{i-1}\cong\cal{O}_{\mathbb{P}^{1}}(b_{i})^{\oplus r_{i}}$. Now consider $\mathbb{U}_{\mathbb{P}^{1}}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with $b_{i}:=d_{i}/r_{i}$ and then it is a single point set, consisting only of $\cal{E}$. Set

[TABLE]

Then we have $h^{0}(\mathcal{H}om(\cal{E},\cal{E}(\gamma)))=\Delta$. So the parameter space is a well-defined vector space, or its associated projective space if we consider nonzero co-Higgs fields up to scalar multiplication. We have $\Delta>0$ if and only if $b_{1}+\gamma\geq b_{s}$.

For any $\Phi\in\operatorname{Hom}(\cal{E},\cal{E}(\gamma))$ and any positive integer $i$, let $\Phi^{(i)}:\cal{E}\rightarrow\cal{E}(i\gamma)$ be the map obtained by iterating $i$ times a shift of $\Phi$. If $b_{1}+i\gamma<b_{s}$ for some $i$, then we have $\Phi^{(i)}=0$ and so $\Phi$ is a nilpotent logarithmic co-Higgs field. In particular, if $b_{1}+2\gamma<b_{s}\leq b_{1}+\gamma$, then all logarithmic co-Higgs fields are $2$-nilpotents and so we have the following.

Proposition 3.3**.**

For the bundle $\cal{E}$ in (7) with $2\gamma\leq b_{s}-b_{1}<\gamma$, the set of its co-Higgs structures is identified with a $\Delta$-dimensional vector space.

Now the assumption in (5) is simply $b_{1}+\gamma\geq b_{s}$ and let $e$ be the last integer $i$ such that $b_{i}>\gamma+b_{1}$. Then we may write

[TABLE]

It is possible to have $e=0$ and so $\cal{E}_{+}$ is trivial. Then we have $H^{0}(\mathcal{E}nd(\cal{E})(\gamma))=H^{0}(\mathcal{H}om(\cal{E}_{-},\cal{E})(\gamma))$. Thus in case of $\mathbb{P}^{1}$ we may rephrase our question in the set-up of holomorphic triples $(\cal{E}_{1},\cal{E}_{2},f)$ with $\cal{E}_{1}=\cal{E}_{-}$, $\cal{E}_{2}=\cal{E}(\gamma)$ and $f:\cal{E}_{1}\rightarrow\cal{E}_{2}$. Here, $\cal{E}_{1}$ and $\cal{E}_{2}$ are related in a sense that $\cal{E}_{1}$ is a twist of a factor of $\cal{E}_{2}$. So our general problem concerning nonzero maps $\Phi:\cal{E}\rightarrow\cal{E}(\gamma)$ is equivalent to a problem about nonzero maps $\Phi:\cal{E}_{-}\rightarrow\cal{E}(\gamma)$.

3.2. Elliptic curves

Let $X$ be an elliptic curve and use the classification of vector bundles on elliptic curves due to M. Atiyah in [1]. We have $\cal{T}_{\cal{D}}\cong\cal{O}_{X}(-\cal{D})$.

Proposition 3.4**.**

Fix an integer $s\geq 2$ and consider $\mathbb{U}:=\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with $d_{s}/r_{s}\leq d_{1}/r_{1}+\gamma$.

• (i)

There exists $[\cal{E}]\in\mathbb{U}$ with $\operatorname{Hom}(\cal{E},\cal{E}(-\cal{D}))\neq 0$.

• (ii)

If $d_{s}/r_{s}=d_{1}/r_{1}+\gamma$, there is $[\cal{E}]\in\mathbb{U}$ with $\operatorname{Hom}(\cal{E},\cal{E}(-\cal{D}))=0$.

• (iii)

If $d_{s}/r_{s}<d_{1}/r_{1}+\gamma$, then we have $\operatorname{Hom}(\cal{E},\cal{E}(-\cal{D}))\neq 0$ for all $[\cal{E}]\in\mathbb{U}$.

• (iv)

If $e$ is the maximal integer such that $d_{s}/r_{s}\leq d_{1}/r_{1}+e\gamma$, then we have $\Phi^{(e+1)}=0$ for every $[\cal{E}]\in\mathbb{U}$ and $\Phi\in\operatorname{Hom}(\cal{E},\cal{E}(-\cal{D}))=0$.

Proof.

Take $[\cal{E}]\in\mathbb{U}$ and set $\cal{E}_{s}:=\cal{F}_{s}/\cal{F}_{s-1}$. In the set-up of part (iii) we have $\mu(\cal{E}_{s}^{\vee}\otimes\cal{F}_{1}(-\cal{D}))>0$ and so Riemann-Roch gives $h^{0}(\cal{E}_{s}^{\vee}\otimes\cal{F}_{1}(-\cal{D}))>0$. Take as $\Phi$ the composition of the surjection $\cal{E}\rightarrow\cal{E}_{s}$ with a nonzero map $\cal{E}_{s}\rightarrow\cal{F}_{1}(-\cal{D})$ and then the inclusion $\cal{F}_{1}(-\cal{D})\hookrightarrow\cal{E}(-\cal{D})$, proving (iii).

Now assume $d_{s}/r_{s}=d_{1}/r_{1}+\gamma$. Take as $\cal{E}_{i}$ any semistable bundle with prescribed numeric data so that $\cal{E}_{1}$ and $\cal{E}_{s}$ are polystable and no factor of $\cal{E}_{1}(-\cal{D})$ is isomorphic to a factor of $\cal{E}_{s}$. Set $\cal{E}:=\oplus_{i=1}^{s}\cal{E}_{i}$. Due to the slope, we have $\operatorname{Hom}(\cal{E}_{i},\cal{E}_{j}(-\cal{D}))=0$ if $(i,j)\neq(s,1)$. Since every nonzero map between stable bundles with the same slope is an isomorphism, we have $\operatorname{Hom}(\cal{E}_{s},\cal{E}_{1}(-\cal{D}))=0$ and so $\operatorname{Hom}(\cal{E},\cal{E}(-\cal{D}))=0$, proving part (ii).

Under the same situation, set $t:=\gcd(|d_{s}|,r_{s})$ and write $r_{s}=at$ and $d_{s}=bt$. Then each indecomposable factor of $\cal{E}_{s}$ has rank $a$ and degree $b$, which is also stable. Pick one of these indecomposable factors, say $\cal{A}$. Now from $d_{s}/r_{s}=d_{1}/r_{1}+\gamma$, we see that $a$ divides $r_{1}$. Then we have $r_{1}/a\in\mathbb{Z}$ and it also divides $d_{1}$, say $r_{1}=ap$ and $d_{1}=qp$. We also see that $\gcd(a,q)=\gcd(a,b)=1$ and so $\cal{E}_{1}$ is a polystable bundle whose factors have rank $a$ and degree $q=b-a\gamma$. Let $\cal{G}$ be any polystable vector bundle of rank $r_{1}$ and degree $d_{1}$ with $\cal{A}\otimes\cal{O}_{X}(\cal{D})$ as one of its factors. Set $\cal{F}:=\cal{G}\oplus\left(\oplus_{i=2}^{s}\cal{E}_{i}\right)$ and then we have $[\cal{F}]\in\mathbb{U}$. Since $\operatorname{Hom}(\cal{E}_{s},\cal{G}(-\cal{D}))\neq 0$, we have $\operatorname{Hom}(\cal{F},\cal{F}(-\cal{D}))\neq 0$, proving part (i).

Part (iv) is obvious. ∎

Remark 3.5**.**

In parts (i) and (iii) of Proposition 3.4 the proof gives the existence of a nonzero $2$-nilpotent co-Higgs field $\Phi$.

3.3. Higher genus case

Assume that $X$ has genus $g\geq 2$. Note that $\gamma\leq 2-2g$. For the pairs of integers $(r,d)$ with $r>0$, denote by $\mathbb{M}_{X}(r,d)$ the moduli space of the stable vector bundles of rank $r$ on $X$ with degree $d$. It is known to be a non-empty, smooth and irreducible quasi-projective variety of dimension $r^{2}(g-1)+1$.

Fix a point $p\in X$ and take any exact sequence on $X$:

[TABLE]

with $\cal{A}$ and $\cal{B}$ locally free. Note that $\mathrm{rk}(\cal{A})=\mathrm{rk}(\cal{B})$ and that $\deg\cal{B}=\deg\cal{A}+1$. Then we say that $\cal{B}$ is obtained from $\cal{A}$ by applying a positive elementary transformation at $p$ and that $\cal{A}$ is obtained from $\cal{B}$ by applying a negative elementary transformation at $p$. For a fixed $\cal{A}$ (resp. $\cal{B}$) the set of all extensions (9) is parametrized by a vector space of dimension $\mathrm{rk}(\cal{A})$ (resp. $\mathrm{rk}(\cal{B})$); since it is an irreducible variety, we may speak about the general positive elementary transformation of $\cal{A}$ (resp. a general negative elementary transformation of $\cal{B}$).

Lemma 3.6**.**

For $(r,d,k)\in\mathbb{Z}^{\oplus 3}$ with $r,k>0$, fix a general bundle $[\cal{A}]\in\mathbb{M}_{X}(r,d)$. If $\cal{B}$ is obtained from $\cal{A}$ by applying $k$ positive elementary transformations, then it is stable.

Proof.

Since the statement is trivial for $r=1$, we may assume $r\geq 2$.

(a) First assume $k=1$ with the sequence (9) and that $\cal{B}$ is not stable so that there exists a subbundle $\cal{G}_{t}\subset\cal{B}$ of rank $t\in\{1,\dots,r-1\}$ with $\deg\cal{G}_{t}/t\geq(d+1)/r$. Let $\cal{C}\subset\cal{A}$ be the saturation of $u^{-1}(\cal{G})$ and set $a:=\deg\cal{C}$. Then we have $a\geq\deg u^{-1}(\cal{G})\geq\deg\cal{G}-1$. Since $\cal{A}$ is general, we get by [21, Theorem 3.10] or [7, Theorem 2] that $\mu(\cal{A}/\cal{C})-\mu(\cal{C})\geq g-1$, from which we get

[TABLE]

Using this with $\deg(\cal{G}_{t})\leq a+1$, we get

[TABLE]

The equality holds if and only if $g=2$ and $\deg\cal{G}_{t}=a+1$. Let $\widetilde{a}$ be the maximal degree of a rank $t$ subbundle of $\cal{A}$. For arbitrary $t$ and $g$, Mukai and Sakai proved in [23] that $td-\widetilde{a}r\leq t(r-t)g$, while the quoted results also said that $td-\widetilde{a}r\geq t(r-t)(g-1)$. The precise value of $\widetilde{a}$ is known by an unpublished result of A. Hirschowitz in [17] and [21, Remark 3.14], which says that $td-\widetilde{a}r=t(r-t)(g-1)+\varepsilon$, where $\varepsilon$ is the only integer such that $0\leq\varepsilon<r$ and $\varepsilon+t(r-t)(g-1)\equiv td\pmod{r}$.

Now assume $g=2$. We conclude unless $\varepsilon=0$, $a=\widetilde{a}$ and $\deg\cal{G}_{t}=a+1$. In this case we use that we take a general positive elementary transformation of $\cal{A}$. Since $\varepsilon=0$ and $\cal{A}$ is general, $\cal{A}$ has only finitely many rank $t$ subbundles of maximal degree $a=\widetilde{a}$, say $\cal{N}_{i}$ for $1\leq i\leq\delta$; see [25] and [30]. The fiber $\cal{N}_{i|\{p\}}$ of $\cal{N}_{i}$ at $p$ is a $t$-dimensional linear subspace of the fiber $\cal{A}_{|\{p\}}$ of $\cal{A}$ at $p$, which is an $r$-dimensional vector space. The union of these $t$-dimensional linear subspaces $\cal{N}_{i}$ for $1\leq i\leq\delta$, is a proper subset of $\cal{A}_{|\{p\}}$. Thus, for a general positive elementary transformation $\cal{B}$ of $\cal{A}$ at $p$, the saturation $\cal{M}_{i}$ of $\cal{N}_{i}$ is just $\cal{N}_{i}$ for all $i$, i.e. $\deg u^{-1}(\cal{M}_{i})=\deg\cal{M}_{i}$ for all $i$, contradicting the assumptions $a=\widetilde{a}$ and $\deg\cal{G}_{t}=a+1$.

(b) Now assume $k\geq 2$. The case $k=1$ proves that a general positive elementary transformation of a stable bundle is stable. Similarly a general negative transformation of a stable bundle is also stable, and so we may apply the step (a) $k$ times to get the assertion. ∎

Theorem 3.7**.**

Fix an integer $s\geq 2$ and consider $\mathbb{U}:=\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with $d_{s}/r_{s}\leq d_{1}/r_{1}+\gamma$.

• (i)

There exists $[\cal{E}]\in\mathbb{U}$ with $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})\neq 0$.

• (ii)

If $d_{s}/r_{s}\geq d_{1}/r_{1}+\gamma+1-g$, there is $[\cal{E}]\in\mathbb{U}$ with $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})=0$.

• (iii)

If $d_{s}/r_{s}<d_{1}/r_{1}+\gamma+1-g$, then $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})\neq 0$ for all $[\cal{E}]\in\mathbb{U}$.

Proof.

Take $[\cal{E}]\in\mathbb{U}$ and set $\cal{E}_{s}:=\cal{F}_{s}/\cal{F}_{s-1}$. Since $\cal{E}_{s}$ and $\cal{F}_{1}$ are semistable, the bundle $\cal{E}_{s}^{\vee}\otimes\cal{F}_{1}(-D)$ is also semistable. In the set-up of part (iii) we have $\mu(\cal{E}_{s}^{\vee}\otimes\cal{F}_{1}\otimes\cal{T}_{\cal{D}})>g-1$ and so Riemann-Roch gives $h^{0}(\cal{E}_{s}^{\vee}\otimes\cal{F}_{1}\otimes\cal{T}_{\cal{D}})>0$. Take as $\Phi$ the composition of the surjection $\cal{E}\rightarrow\cal{E}_{s}$ with a nonzero map $\cal{E}_{s}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ and then the inclusion $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}\hookrightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$, proving (iii).

Now assume the set-up of (ii) and pick a general element $(\cal{E}_{1},\dots,\cal{E}_{s})$ in

[TABLE]

In particular, each $\cal{E}_{i}$ is a general stable bundle in $\mathbb{M}_{X}(r_{i},d_{i})$. Set $\cal{E}:=\oplus_{i=1}^{s}\cal{E}_{i}$ and then it is sufficient to prove the following claim for (ii).

Claim 1: We have $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})=0$.

Proof of Claim 1: Since $\cal{E}:=\oplus_{i=1}^{s}\cal{E}_{i}$, it is enough to prove that $H^{0}(\cal{E}_{i},\cal{E}_{j}\otimes\cal{T}_{\cal{D}})=0$ for all $i,j$. We have $H^{0}(\cal{E}_{i},\cal{E}_{i}\otimes\cal{T}_{\cal{D}})=0$ for each $i$, because $\cal{E}_{i}$ is stable and $\gamma<0$. Now assume $i\neq j$. Note that $(E_{i}^{\vee},E_{j}\otimes\cal{T}_{\cal{D}})$ is a general element of

[TABLE]

We have $\mu(\cal{E}_{i}^{\vee}\otimes\cal{E}_{j}\otimes\cal{T}_{\cal{D}})=-\mu(\cal{E}_{i})+\mu(\cal{E}_{j})+\gamma\leq g-1$. By a theorem of A. Hirschowitz in [28, Theorem 1.2], we have $h^{0}(\cal{E}_{i}^{\vee}\otimes\cal{E}_{j}\otimes\cal{T}_{\cal{D}})=0$, concluding the proof of Claim 1.

Now we prove part (i). Let $\cal{B}_{i}$ be a semistable bundle of rank $r_{i}$ on $X$ with degree $d_{i}$ for each $i$, and let $\cal{E}:=\oplus_{i=1}^{s}\cal{B}_{i}$. Our strategy is to find appropriate $\cal{B}_{1}$ and $\cal{B}_{s}$ with the additional condition $\operatorname{Hom}(\cal{B}_{s},\cal{B}_{1}\otimes\cal{T}_{\cal{D}})\neq 0$, which would imply part (i).

(a) Assume $r_{s}<r_{1}$. Setting $r^{\prime}:=r_{1}-r_{s}$ and $d^{\prime}:=d_{1}+\gamma r_{1}-d_{s}$, it is enough to show the existence of an exact sequence of vector bundles on $X$:

[TABLE]

with $\cal{A}_{1},\cal{A}_{2}$ semistable and $\cal{A}_{1}$ of rank $r_{s}$ and degree $d_{s}$, $\cal{A}_{2}$ of rank $r_{1}$ and degree $d_{1}+\gamma r_{1}$. Then $\cal{A}_{3}$ would be of rank $r^{\prime}$ and degree $d^{\prime}$, and we may take $\cal{B}_{s}:=\cal{A}_{1}$ and $\cal{B}_{1}:=\cal{A}_{2}\otimes\cal{T}_{\cal{D}}^{\vee}$.

Note that for a quadruple of integers $(x_{1},x_{2},a_{1},a_{2})\in\mathbb{Z}^{\oplus 4}$ with $x_{2}>x_{1}>0$ and $a_{1}/x_{1}\leq a_{2}/x_{2}$ (resp. $a_{1}/x_{1}<a_{2}/x_{2}$), we have

[TABLE]

Using the above to $(x_{1},x_{2},a_{1},a_{2})=(r_{s},r_{1},d_{s},d_{1}+\gamma r_{1})$, together with

[TABLE]

we have $d_{1}/r_{1}+\gamma\leq d^{\prime}/r^{\prime}$ with equality if and only if $d_{s}/r_{s}=d_{1}/r_{1}+\gamma$. When the equality holds, we may take as $\cal{A}_{1}$ and $\cal{A}_{3}$ arbitrary semistable bundles with the prescribed ranks and degrees and then take $\cal{A}_{2}:=\cal{A}_{1}\oplus\cal{A}_{3}$. Now assume $d_{s}/r_{s}<d_{1}/r_{1}+\gamma$ and so $d_{1}/r_{1}+\gamma<d^{\prime}/r^{\prime}$. In this case by the positive answer to the conjecture of Lange, there is an exact sequence (10) of vector bundles on $X$ with the prescribed ranks and degrees and with stable $\cal{A}_{1}$, $\cal{A}_{2}$ and $\cal{A}_{3}$; see [28, Introduction].

(b) Assume $r_{s}>r_{1}$. Similarly as in (a) we set $r^{\prime\prime}:=r_{s}-r_{1}$ and $d^{\prime\prime}:=d_{1}-\gamma r_{1}-d_{s}$. By taking $\cal{B}_{s}:=\cal{A}_{2}$ and $\cal{B}_{1}:=\cal{A}_{3}\otimes\cal{T}_{\cal{D}}^{\vee}$, it is sufficient to find an exact sequence (10) with $\cal{A}_{2}$ and $\cal{A}_{3}$ semistables, $\cal{A}_{1}$ of rank $r^{\prime\prime}$ and degree $d^{\prime\prime}$, $\cal{A}_{2}$ of rank $r_{s}$ and degree $d_{s}$ and $\cal{A}_{3}$ or rank $r_{1}$ and degree $d_{1}+\gamma r_{1}$.

First assume $d_{s}/r_{s}=d_{1}/r_{1}+\gamma$. In this case we have $d^{\prime\prime}/r^{\prime\prime}=d_{s}/r_{s}$ and we take as $\cal{A}_{1}$ and $\cal{A}_{3}$ arbitrary semistable bundles with prescribed ranks and degrees and set $\cal{A}_{2}:=\cal{A}_{1}\oplus\cal{A}_{3}$. Now assume $d_{s}/r_{s}<d_{1}/r_{1}+\gamma$ and so $d_{1}/r_{1}+\gamma>d^{\prime\prime}/r^{\prime\prime}$. Again by the conjecture of Lange proved in [28] we may take $\cal{A}_{1}$, $\cal{A}_{2}$, $\cal{A}_{3}$ with the prescribed ranks and degree and stable.

(c) Assume $r_{s}=r_{1}$. First assume $d_{s}/r_{s}=d_{1}/r_{1}+\gamma$, i.e. $d_{1}=d_{s}-\gamma r_{1}$. In this case we take as $\cal{B}_{s}$ any semistable bundle with rank $r_{s}$ and degree $d_{s}$ and set $\cal{B}_{1}:=\cal{B}_{s}\otimes\cal{T}_{\cal{D}}$. Now assume $k:=d_{1}+r_{1}\gamma-d_{s}>0$. We take as $\cal{B}_{s}$ a general stable bundle of rank $r_{s}$ and degree $d_{s}$. Then $\cal{B}_{s}\otimes\cal{T}_{\cal{D}}$ is a general element of $\mathbb{M}_{X}(r_{1},d_{1}-t)$. We take as $\cal{B}_{1}$ a bundle obtained from $\cal{B}_{s}\otimes\cal{T}_{\cal{D}}$ by applying $k$ general positive elementary transformations. ∎

Remark 3.8**.**

Consider a smooth algebraic curve $X$ of an arbitrary genus $g\geq 0$ and assume $s\geq 3$ together with

[TABLE]

By Theorem 3.7 in case $g\geq 2$ and Proposition 3.4 for $g=1$, there is $(\cal{E},\Phi)$ with $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ and a nonzero map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$. Take any $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with the Harder-Narasimhan filtration (2). Note that we have $\operatorname{Hom}(\cal{A},\cal{B})=0$ for any semistable bundles $\cal{A}$ and $\cal{B}$ with $\mu(\cal{A})>\mu(\cal{B})$. Claim 1 in the proof of Theorem 3.7 applied to $\cal{E}/\cal{F}_{1}$ shows that any map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ is uniquely determined by $f:\cal{E}/\cal{F}_{s-1}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$; moreover we get $\mathrm{Im}(\Phi)=\mathrm{Im}(f)$ and $\mathrm{ker}(\Phi)$ is the inverse image of $\mathrm{ker}(f)$ under the surjection $\cal{E}\rightarrow\cal{E}/\cal{F}_{s-1}$.

Conversely, for $1\leq i\leq s$, choose arbitrary semistable bundles $\cal{E}_{i}$ with $\mathrm{rk}(\cal{E}_{i})=r_{i}$ and $\deg\cal{E}_{i}=d_{i}$, and a map $f:\cal{E}_{s}\rightarrow\cal{E}_{1}\otimes\cal{T}_{\cal{D}}$. To get a vector bundle $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$, we only need to consider $(s-1)$ extension classes

[TABLE]

for $i=1,\dots,s-1$, where $\cal{F}_{1}:=\cal{E}_{1}$. Once $\cal{E}$ is chosen, the map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ is uniquely determined by $f$.

In case of curves, we sometimes may improve Remark 2.3 to a strict inequality in the following way.

Remark 3.9**.**

Take $[\cal{E}]\in\mathbb{U}:=\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with the Harder-Narasimhan filtration (2). Assume that $s\geq 2$ with $r_{s}\neq r_{1}$,

[TABLE]

Since $\gcd(r_{1},d_{1})=1$, the sheaf $\cal{F}_{1}$ is stable and so $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ is stable. Since $\gcd(r_{s},d_{s})=1$, the sheaf $\cal{F}_{s}/\cal{F}_{s-1}$ is also stable. From $r_{1}\neq r_{s}$ we get that $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ and $\cal{F}_{s}/\cal{F}_{s-1}$ are not isomorphic and so we have $\operatorname{Hom}(\cal{F}_{s}/\cal{F}_{s-1},\cal{F}_{1}\otimes\cal{T}_{\cal{D}})=0$. If $s\geq 3$, we obviously have $\operatorname{Hom}(\cal{F}_{i}/\cal{F}_{i-1},\cal{F}_{j}\otimes\cal{T}_{\cal{D}})=0$ for all $i,j\in\{1,\dots,s\}$ with $(i,j)\neq(s,1)$, even without the assumptions $r_{s}\neq r_{1}$ and $\gcd(r_{1},d_{1})=\gcd(r_{s},d_{s})=1$. Thus we have $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})=0$.

Remark 3.10**.**

In the following exact sequence

[TABLE]

of vector bundles on $X$ with $\cal{A}$ and $\cal{B}$ semistable, if we have $\mu(\cal{A})+2-2g>\mu(\cal{B})$, then we have $h^{1}(\cal{A}\otimes\cal{B}^{\vee})=0$ and so (11) splits. Thus if we have

[TABLE]

for all $i$, then we have $\cal{E}\cong gr(\cal{E})$ for all $[\cal{E}]\in\mathbb{U}:=\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$.

Now assume $d_{i}/r_{i}+2-2g\geq d_{i+1}/r_{i+1}$ for all $i$. If the equality holds for some $i$, i.e. $d_{i}/r_{i}=d_{i+1}/r_{i+1}+2g-2$, then we have $r_{i}\neq r_{i+1}$ and that $r_{h}$ and $d_{h}$ are coprime, where $h\in\{i,i+1\}$ is the index with higher rank $r_{h}=\max\{r_{i},r_{i+1}\}$. As in Remark 3.9 we get $\cal{E}\cong gr(\cal{E})$ for all $[\cal{E}]\in\mathbb{U}$.

For example, take $s=2$. We just proved that $\cal{E}\cong\cal{F}_{1}\oplus\cal{F}_{2}/\cal{F}_{1}$ for all $[\cal{E}]\in\mathbb{U}(2;1,d_{1};1,d_{2})$ with a nonzero map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ and either $\cal{D}\neq 0$ or $r_{1}\neq r_{2}$, and $d_{h},r_{h}\in\mathbb{Z}$, where $r_{h}=\max\{r_{1},r_{2}\}$.

Example 3.11**.**

Assume $d_{1}>d_{2}-\gamma$. For a fixed $\cal{R}\in\mathrm{Pic}^{d_{2}}(X)$, consider the set

[TABLE]

where the equivalent relation $\sim$ is given by $(\cal{F}_{1},\psi)\sim(\cal{F}_{1},c\psi)$ for all $c\in\mathbb{C}^{\ast}$. $\mathbb{E}$ is the set of all effective divisors of $X$ with degree $d_{1}+\gamma-d_{2}$ and so $\mathbb{E}$ is isomorphic to a symmetric product of $d_{1}+\gamma-d_{2}$ copies of $X$ and in particular it is irreducible. By Remark 3.10 we have $\cal{E}\cong\cal{F}_{1}\oplus\cal{F}_{2}/\cal{F}_{1}$ for all $[\cal{E}]\in\mathbb{U}_{X}(2;1,d_{2}+\gamma;1,d_{2})$.

Example 3.12**.**

Assume $g\geq 2$ and take $\cal{D}=\emptyset$ so that $\gamma=2-2g$. Fix any $d\in\mathbb{Z}$ and consider $\cal{E}\in\mathbb{U}_{X}(2;1,d+2g-2;1,d)$ with a nonzero map $\Phi:\cal{E}\rightarrow\cal{E}\otimes T_{X}$. Set $\cal{R}:=\cal{F}_{2}/\cal{F}_{1}\in\mathrm{Pic}^{d}(X)$ and then $\Phi$ is induced by a nonzero map $\psi:\cal{R}\rightarrow\cal{F}_{1}\otimes T_{X}$. Since $\cal{F}_{1}$ is in $\mathrm{Pic}^{d+2g-2}(X)$, the map $\psi$ is an isomorphism. Thus we get $\cal{F}_{1}\cong\cal{R}\otimes\omega_{X}$ and that for a fixed $\cal{E}$ the set of all nonzero map $\Phi$ is parametrized by a nonzero scalar. From $h^{1}(\omega_{X})=1$ we see that there are, up to isomorphism, exactly two vector bundles $\cal{E}$ fitting into an exact sequence

[TABLE]

that is, $\left(\cal{R}\otimes\omega_{X}\right)\oplus\cal{R}$ and an indecomposable bundle. Thus the set of all $(\cal{E},\Phi)$, up to isomorphisms, with nonzero $\Phi$, is parametrized one-to-one by the disjoint union of two copies of $\mathrm{Pic}^{d}(X)\times\mathbb{C}^{\ast}$. Thus no one-to-one parameter space is irreducible. We get another irreducible parameter space that is not one-to-one, by taking as parameter space, up to a nonzero constant, the relative $\operatorname{Ext}^{1}$ group of (12) over $\mathrm{Pic}^{d}(X)$; each indecomposable bundle $\cal{E}$ appears $\infty^{1}$-times and it has $gr(\cal{E})\cong\left(\cal{R}\otimes\omega_{X}\right)\oplus\cal{R}$ as its limit inside the parameter space.

Now for $s$ at least two let us define the set $\mathbb{U}^{\mathrm{co}}=\mathbb{U}_{X}^{\mathrm{co}}(s;r_{1},d_{1};\ldots;r_{s},d_{s})$ of certain co-Higgs bundles associated to $\mathbb{U}=\mathbb{U}_{X}(s;r_{1},d_{1};\ldots;r_{s},d_{s})$ as follows.

[TABLE]

Denote by $\Gamma\subseteq\mathbb{M}_{X}(r_{1},d_{1})\times\mathbb{M}_{X}(r_{s},d_{s})$ the set of all pairs $(\cal{F}_{1},\cal{F}_{s}/\cal{F}_{s-1})$ obtained from $\mathbb{U}^{\mathrm{co}}$ and call the projection from $\Gamma$ to each factor by $\pi_{1}$ and $\pi_{2}$, respectively.

Proposition 3.13**.**

Assume that

• •

each $d_{i}$ is positive such that $d_{i}/r_{i}>d_{i+1}/r_{i+1}$ for all $i$, and

• •

$d_{1}/r_{1}+\gamma\geq d_{s}/r_{s}$.

Then we have the following assertions.

• (i)

If $r_{1}=r_{s}$, then $\pi_{1}$ and $\pi_{2}$ are dominant.

• (ii)

If $r_{1}<r_{s}$ (resp. $r_{1}>r_{s}$) and $d_{1}/r_{1}+\gamma>d_{s}/r_{s}$, then $\pi_{1}$ (resp. $\pi_{2}$) is dominant.

• (iii)

Assume $d_{1}/r_{1}+\gamma>g-1+d_{s}/r_{s}$. Then $\Gamma$ contains a non-empty open subset of $\mathbb{M}_{X}(r_{1},d_{1})\times\mathbb{M}_{X}(r_{s},d_{s})$; if $r_{1}\geq r_{s}$, then we have $\mathrm{ker}(\Phi)=\cal{F}_{s-1}$.

Proof.

Fix a point $([\cal{A}_{1}],[\cal{A}_{s}])\in\mathbb{M}_{X}(r_{1},d_{1})\times\mathbb{M}_{X}(r_{s},d_{s})$. For arbitrary $[\cal{A}_{i}]\in\mathbb{M}_{X}(r_{i},d_{i})$, $i=2,\cdots,s-1$, we consider $\cal{E}:=\oplus_{i=1}^{s}\cal{A}_{i}$ with the Harder-Narasimhan filtration (2) such that $\cal{F}_{1}\cong\cal{A}_{1}$ and $\cal{F}_{s}/\cal{F}_{s-1}\cong\cal{A}_{s}$. We take a map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ with $\Phi(\cal{F}_{s-1})=0$, which is induced by a map $\psi:\cal{A}_{s}\rightarrow\cal{A}_{1}\otimes\cal{T}_{\cal{D}}$ whose existence is guaranteed by the assumptions.

First assume $r_{s}=r_{1}$. We need to prove the existence of a map $\psi$ of rank $r_{1}$ when $\cal{A}_{1}$ is general in $\mathbb{M}_{X}(r_{1},d_{1})$ and $\cal{A}_{s}$ is general in $\mathbb{M}_{X}(r_{s},d_{s})$; we do not claim here that $([\cal{A}_{1}],[\cal{A}_{2}])$ is general in $\mathbb{M}_{X}(r_{1},d_{1})\times\mathbb{M}_{X}(r_{s},d_{s})$. The dominance of $\pi_{2}$ is the content of Lemma 3.6. The dominance of $\pi_{1}$ can be proved by applying the dual map, or with the same proof as in the proof of Lemma 3.6, concluding part (i).

Now assume $r_{s}<r_{1}$ and $d_{1}/r_{1}+\gamma>d_{s}/r_{s}$. We take as $\cal{A}_{s}$ a general element of $\mathbb{M}_{X}(r_{s},d_{s})$. The existence of a stable $\cal{A}_{1}\in\mathbb{M}_{X}(r_{1},d_{1})$ with an embedding $\psi:\cal{A}_{s}\hookrightarrow\cal{A}_{1}\otimes\cal{T}_{\cal{D}}$ with $\cal{A}_{1}/\psi(\cal{A}_{s})$ stable and general in $\mathbb{M}_{X}(r_{s}-r_{1},d_{1}-\gamma r_{1})$ is proved in part (i) of the proof of Theorem 3.7.

By using part (ii) of the proof of Theorem 3.7 instead of part (i), we get the case $r_{s}>r_{1}$ and $d_{1}/r_{1}+\gamma>d_{s}/r_{s}$.

Now consider part (iii) and assume $d_{1}/r_{1}+\gamma>g-1+d_{s}/r_{s}$. Take a general $([\cal{A}_{1}],[\cal{A}_{s}])\in\mathbb{M}_{X}(r_{1},d_{1})\times\mathbb{M}_{X}(r_{s},d_{s})$ and set $\cal{B}_{1}:=\cal{A}_{1}\otimes\cal{T}_{\cal{D}}$. Then it is sufficient to find $\psi:\cal{A}_{s}\rightarrow\cal{B}_{1}$ with $\mathrm{Im}(\psi)=\min\{r_{1},r_{s}\}$. By the assumptions, we have $\mu(\cal{A}_{s}^{\vee}\otimes\cal{B}_{1})>g-1$ and so Riemann-Roch gives $\operatorname{Hom}(\cal{A}_{s},\cal{B}_{1})\neq 0$. Take a general element $\psi\in\operatorname{Hom}(\cal{A}_{s},\cal{B}_{1})$ and then it is sufficient to prove that $\psi$ has rank $\min\{r_{1},r_{s}\}$. Note that we have $h^{1}(\cal{A}_{s}^{\vee}\otimes\cal{B}_{1})=0$ and so $\cal{B}_{1}$ is an element of the following set

[TABLE]

By Riemann-Roch, we have the following, for each $[\cal{F}]\in\mathbb{W}$,

[TABLE]

Now take the relative $\operatorname{Hom}$ with $\mathbb{W}$ as its parameter space, i.e. for each $[\cal{F}]\in\mathbb{W}$, the fibre is $\operatorname{Hom}(\cal{A}_{s},\cal{F})$. The total space $\Lambda$ of this relative $\operatorname{Hom}$ is irreducible, because $h^{0}(\cal{A}_{s}^{\vee}\otimes\cal{F})$ is constant for all $[\cal{F}]\in\mathbb{W}$ by [6] and [20]. By [5, part (d) of Theorem 1.2], a general element $(\varphi:\cal{A}_{s}\rightarrow\cal{F})$ of $\Lambda$ has $\varphi$ with rank $\min\{r_{1},r_{s}\}$. When $r_{1}\geq r_{s}$, this implies that $\mathrm{ker}(\Phi)=\cal{F}_{s-1}$, because the map $\psi:\cal{F}_{s}/\cal{F}_{s-1}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ is injective if and only if it has rank $r_{s}$ . ∎

Remark 3.14**.**

As in the end of proof of Proposition 3.13, to show that the set of the co-Higgs bundles $(\cal{E},\Phi)$ with certain properties is parametrized by an irreducible variety, it sometimes works to prove that (a) the set of all bundles $\cal{E}$ is parametrized by an irreducible variety $Y$, and (b) the integer $k:=\dim\operatorname{Hom}(\cal{E}_{y},\cal{E}_{y}\otimes\cal{T}_{\cal{D}})$ is constant for all $y\in Y$. In this case, the set of all $(\cal{E},\Phi)$ with no restriction on $\Phi$ is parametrized by a vector bundle of rank $k$ on $Y$.

Example 3.15**.**

Assume $r_{1}=r_{s}$ and $d_{1}/r_{1}+\gamma=d_{s}/r_{s}$. Consider a bundle $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with an arbitrary map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$. Since $d_{i}/r_{i}>d_{1}/r_{1}+\gamma$ for all $i<s$, there is no nonzero map $\cal{F}_{i}/\cal{F}_{i-1}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ and so we have $\cal{F}_{s-1}\subseteq\mathrm{ker}(\Phi)$. On the other hand, since $d_{s}/r_{s}>d_{j}/r_{j}+\gamma$ for all $j>1$, we have $\Phi(\cal{E})\subseteq\cal{F}_{1}$. We have $\mathrm{rk}(\Phi)=r_{1}$ if and only if $\cal{F}_{s}/\cal{F}_{s-1}\cong\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ and $\Phi$ is induced by an isomorphism $\cal{F}_{s}/\cal{F}_{s-1}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$.

Example 3.16**.**

Assume $r_{1}=r_{s}$ and that

[TABLE]

If we choose $(\cal{E},\Phi)$ with $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$, then as in Example 3.15 we see that $\Phi(\cal{E})\subseteq\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ and $\cal{F}_{s-1}\subseteq\mathrm{ker}(\Phi)$, because $d_{i}/r_{i}+\gamma<d_{s}/r_{s}$ for all $i>1$ and $d_{j}/r_{j}>d_{1}/r_{1}+\gamma$ for all $j<s$. Set $k:=d_{1}-d_{2}+\gamma r_{1}$. Then we have $\mathrm{rk}(\Phi)=r_{1}$ if and only if $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ is obtained from $\cal{F}_{s}/\cal{F}_{s-1}$ by applying $k$ positive elementary transformations and $\Phi$ is induced by the associated inclusion $\cal{F}_{s}/\cal{F}_{s-1}\hookrightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$.

4. Segre invariant

In this section, we do not assume that $\cal{T}_{\cal{D}}$ has some kind of negativity, so that we may have stable $(\cal{E},\Phi)$ with nonzero $\Phi$. Let $\cal{E}$ be a torsion-free sheaf of rank $r\geq 2$ and $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ a co-Higgs field. For a fixed integer $k\in\{1,\dots,r-1\}$, let us denote by $\cal{S}(k,\cal{E},\Phi)$ the set of all subsheaves $\cal{A}\subset\cal{E}$ of rank $k$ such that $\Phi(\cal{A})\subseteq\cal{A}\otimes\cal{T}_{\cal{D}}$. Define the $k^{th}$-Segre invariant to be

[TABLE]

In case $\Phi=0$, we simply denote it by $s_{k}(\cal{E})$. This is an extension of the Segre invariant, introduced in [19] with the notation $s_{k}(\cal{E})$, to the case $n\geq 2$. Over curves this notion was used in several literatures, including [5, 7, 8, 17, 21, 25, 28, 30]. If we take $\cal{T}_{\cal{D}}^{\vee}$ instead of $\cal{T}_{\cal{D}}$, we get a definition for logarithmic Higgs fields. Note that we always have $\cal{S}(k,\cal{E},0)\neq\emptyset$ and $s_{k}(\cal{E})\leq s_{k}(\cal{E},\Phi)$.

Lemma 4.1**.**

Let $(\cal{E},\Phi)$ be a $2$-nilpotent co-Higgs bundle of rank $r$, and set $\cal{A}:=\mathrm{ker}(\Phi)$ and $\cal{B}:=\mathrm{Im}(\Phi)$ with $r^{\prime}:=\mathrm{rk}(\cal{A})$. Then we have the following:

• (i)

$\cal{A}\in\cal{S}(r^{\prime},\cal{E},\Phi)$;

• (ii)

$\cal{S}(k,\cal{A},0)\subseteq\cal{S}(k,\cal{E},\Phi)$* for $1\leq k<r^{\prime}$;*

• (iii)

$\cal{B}$* is torsion-free and $\Phi^{-1}(\cal{G})\in\cal{S}(k,\cal{E},\Phi)$ for all $\cal{G}\in\cal{S}(k-r^{\prime},\cal{B},0)$ and $r^{\prime}<k<r$;*

• (iv)

$\cal{S}(k,\cal{E},\Phi)\neq\emptyset$* for all $k$.*

Proof.

Parts (i) and (ii) are obvious. Part (iii) is true, because $\cal{B}\subseteq\cal{A}\otimes\cal{T}_{\cal{D}}$ by the definition of $2$-nilpotent. Part (iv) follows from the other ones. ∎

Example 4.2**.**

From [3, Theorem 1.1] we get a description of the set of nilpotent co-Higgs structures on a fixed stable bundle of rank two on $\mathbb{P}^{n}$. Indeed it is either trivial or an $(n+1)$-dimensional vector space, depending on the parity of the first Chern class and an invariant $x_{\cal{E}}$. We get a non-trivial set of nilpotent co-Higgs structures on $\cal{E}$ if and only if $c_{1}(\cal{E})+2x_{\cal{E}}=-3$, and in this case we get $s_{1}(\cal{E},\Phi)=1$.

4.1. Curve case

From now on we assume $n=1$ with $g=g(X)$ and $\gamma<0$. Take $(\cal{E},\Phi)$ with $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ and let (2) be the Harder-Narasimhan filtration of $\cal{E}$.

Remark 4.3**.**

From the assumption $\gamma<0$, we have $\Phi(\cal{F}_{i})\subseteq\cal{F}_{i-1}\otimes\cal{T}_{\cal{D}}$.

Remark 4.3 immediately proves the following two lemmas.

Lemma 4.4**.**

For an integer $j\in\{1,\dots,s-1\}$, set $k(j)=\sum_{i=1}^{j}r_{i}$. Then

[TABLE]

Remark 4.5**.**

We expect that the inequality in Lemma 4.4 is in fact equality, although we give the positive answers only to some special cases; see Lemma 4.6 and Proposition 4.10.

Lemma 4.6**.**

Assume $g=0$ and take $\cal{E}\cong\oplus_{i=1}^{r}\cal{O}_{\mathbb{P}^{1}}(a_{i})$ with $a_{i}\geq a_{j}$ for all $i\leq j$. Then we have

[TABLE]

Proposition 4.7**.**

Fix $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with $s\geq 2$ and $\Phi\in\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})$. Choose any $k\in\{r_{1}+1,\dots,r-r_{s}+1\}$ such that there is $h\in\{1,\dots,s-1\}$ with $r_{1}+\cdots+r_{h}<k<r_{1}+\cdots+r_{h+1}$, and set

[TABLE]

Let $\cal{B}\subset\cal{F}_{h+1}/\cal{F}_{h}$ be any subsheaf of rank $r^{\prime}$ and degree $d^{\prime}$, and set $\cal{A}:=u^{-1}(\cal{B})$, where $u$ is the surjection in the exact sequence

[TABLE]

Then $\cal{B}\in\cal{S}(k,\cal{E},\Phi)$ and $s_{k}(\cal{E},\Phi)\leq k\deg\cal{E}-k(\deg\cal{F}_{h}+e)$.

Proof.

Note that $d^{\prime}$ is the degree of all rank $r^{\prime}$ maximal degree subsheaves of $\cal{F}_{h+1}/\cal{F}_{h}$. Since $\deg\cal{A}=d^{\prime}$, we have $\deg\cal{B}=\deg\cal{F}_{h}+d^{\prime}$. Since $\Phi(\cal{F}_{h+1})\subset\cal{F}_{h}\subset\cal{B}$, we have $\cal{B}\in\cal{S}(k,\cal{E},\Phi)$. Since $\deg\cal{B}=\deg\cal{F}_{h}+\deg\cal{A}$, we get the assertion. ∎

Now Lemma 4.6 and Proposition 4.7 prove the following result.

Corollary 4.8**.**

The Segre invariant $s_{k}(\cal{E},\Phi)$ is defined for all $(\cal{E},\Phi)$, if $\gamma<0$.

Example 4.9 shows that in Proposition 4.7 we may have strict inequality; of course, to be in the set-up of Proposition 4.7 we need to have $r_{h+1}\geq 2$.

Example 4.9**.**

Assume $g\geq 5$ and fix $h\in\{1,\dots,s-2\}$ with $s\geq 3$. Set $r_{h+1}=r_{h+2}=2$ and fix $r_{i}>0$ for $i\notin\{h+1,h+2\}$ and $d_{i}\in\mathbb{Z}$, $i=1,\dots,s$ such that

• •

$d_{i}/r_{i}>d_{i+1}/r_{i+1}$ for all $i=1,\dots,s-1$ and

• •

$d_{h+1}=2d_{h+2}+1$.

By a theorem of Nagata there is a stable bundle $\cal{E}_{h+1}$ of rank $2$ with degree $d_{h+1}$ and $g-1\leq s_{1}(\cal{E}_{h+1})\leq g$. Here, $s_{1}(\cal{E}_{h+1})$ is the only integer $t$ with $g-1\leq t\leq g$ and $d_{h+1}-t$ even. For $i\neq h+1$ we choose $\cal{E}_{i}$ to be any semistable bundle of degree $d_{i}$ and rank $r_{i}$. Set $\cal{E}:=\oplus_{i=1}^{s}\cal{E}_{i}$ and then we have $\cal{F}_{i}=\oplus_{j=1}^{i}\cal{E}_{j}$ in the Harder-Narasimhan filtration (2) of $\cal{E}$.

Take any $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ with $\mathrm{ker}(\Phi)\supseteq\oplus_{i=0}^{h+1}\cal{E}_{i}$, e.g. take $\Phi=0$ or, for certain $\cal{E}_{1}$ and $\cal{E}_{s}$ so that there is a nonzero map $\cal{E}_{s}\rightarrow\cal{E}_{1}\otimes\cal{T}_{\cal{D}}$, take a $2$-nilpotent map $\Phi$ with $\mathrm{ker}(\Phi)\supseteq\cal{F}_{s-1}$. Let $\cal{A}\subset\cal{F}_{h+1}/\cal{F}_{h}$ be a line subbundle of maximal degree and then we have $\cal{B}:=u^{-1}(\cal{A})=(\sum_{i=1}^{h}\cal{E}_{i})\oplus\cal{A}$. If we set $\cal{B}_{1}:=(\sum_{i=1}^{h}\cal{E}_{i})\oplus\cal{E}_{h+2}$, then we have $\deg\cal{B}_{1}>\deg\cal{B}$.

Proposition 4.10**.**

Assume $r_{i}=1$ for all $i$. For an integer $k\in\{1,\dots,r-1\}$ and any co-Higgs bundle $(\cal{E},\Phi)$ with $[\cal{E}]\in\mathbb{U}_{X}(r;1,d_{1};\dots;1,d_{r})$, we have

[TABLE]

and $\cal{F}_{k}$ is the only bundle achieving the minimum degree in $\cal{S}(k,\cal{E},\Phi)$.

Proof.

By Remark 4.3 and Lemma 4.4, we have $[\cal{F}_{k}]\in\cal{S}(k,\cal{E},\Phi)$. Thus it is sufficient to prove that $\cal{F}_{k}$ is the only one achieving the minimum degree in $\cal{S}(k,\cal{E},0)$. Take any $[\cal{G}]\in\cal{S}(k,\cal{E},0)$ with maximal degree. The maximality condition on $\deg\cal{G}$ implies that $\cal{E}/\cal{G}$ has no torsion and so it is a vector bundle of rank $r-k$ on $X$. We use double induction on $k$ and $r$. The case $k=1$ is obvious, because $\cal{F}_{1}$ is the first step of the Harder-Narasimhan filtration of $\cal{E}$.

Assume that $k$ is at least two and the proposition holds for trivial co-Higgs fields with any $k^{\prime}\in\{1,\dots,k-1\}$ and any bundles $\cal{E}^{\prime}$ whose Harder-Narasimhan filtration has rank one bundles as subquotients.

Assume for the moment $\cal{F}_{1}\subset\cal{G}$. Since $\cal{F}_{1}$ is saturated in $\cal{E}$, i.e. $\cal{E}/\cal{F}_{1}$ has no torsion, $\cal{F}_{1}$ is saturated in $\cal{G}$ and $\cal{G}/\cal{F}_{1}$ is a rank $k-1$ subsheaf of the vector bundle $[\cal{E}/\cal{F}_{1}]\in\mathbb{U}_{X}(r-1;1,d_{2};\dots;1,d_{r})$. The inductive assumption gives $\deg\cal{G}/\cal{F}_{1}\leq\deg\cal{F}_{k}/\cal{F}_{1}$, with equality if and only if $\cal{G}/\cal{F}_{1}\cong\cal{F}_{k}/\cal{F}_{1}$, i.e. $\deg\cal{G}\leq\deg\cal{F}_{k}$ with equality if and only if $\cal{G}\cong\cal{F}_{k}$.

Now assume $\cal{F}_{1}\nsubseteq\cal{G}$. Since $\cal{G}$ is saturated in $\cal{E}$, this means that $\cal{F}_{1}+\cal{G}$ has rank $k+1$. Let $\cal{N}$ be the saturation of $\cal{F}_{1}+\cal{G}$ in $\cal{E}$, and then we have $\deg\cal{N}\geq\deg\cal{F}_{1}+\deg\cal{G}$ and $\cal{N}/\cal{F}_{1}$ is a rank $k$ subsheaf of $\cal{E}/\cal{F}_{1}$. If $r\geq k-2$, then by the inductive assumption on $r$ we have $\deg\cal{N}/\cal{F}_{1}\leq\deg\cal{F}_{k+1}/\cal{F}_{1}<\deg\cal{F}_{k}-\deg\cal{F}_{1}$ and so $\deg\cal{G}<\deg\cal{F}_{k}$, a contradiction. Thus we may assume $k=r-1$ and so $\cal{N}\cong\cal{E}$. Since $\cal{F}_{1}+\cal{G}$ has rank $k+1$, the natural map $\cal{G}\rightarrow\cal{E}/\cal{F}_{1}$ is injective. Thus we have $\deg\cal{G}\leq\deg\cal{E}-\deg\cal{F}_{1}<\deg\cal{F}_{r-1}$, a contradiction. ∎

Now for $k\in\{1,\dots,r-1\}$ set

[TABLE]

$\delta_{0}(\cal{E},\Phi):=0$ and $\delta_{r}(\cal{E},\Phi):=\deg(\cal{E})$. In case $\Phi=0$, we simply denote it by $\delta_{k}(\cal{E})$.

Proposition 4.11**.**

Fix $h\in\{1,\dots,s\}$ with $s\geq 2$ and set $\rho:=r_{1}+\cdots+r_{h}$. For $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$, we have

• (i)

$s_{\rho}(\cal{E})=\rho\deg\cal{E}-k\deg\cal{F}_{h}$* and $\cal{F}_{h}\subset\cal{E}$ is the only subsheaf of rank $\rho$ with degree $\deg\cal{F}_{h}$;*

• (ii)

$\deg\cal{E}\leq\cal{F}_{h-1}+(k-\rho)d_{h}/r_{h}$* for $k$ with $\rho-r_{h}<k<\rho$ and $[\cal{G}]\in\cal{S}(k,\cal{E},0)$;*

• (iii)

$\delta_{k}(\cal{E})-\deg(\cal{F}_{h-1})$* for $k$ with $\rho-r_{h}<k<\rho$, equals*

[TABLE]

Proof.

Set $\mu_{i}:=\mu(\cal{F}_{i}/\cal{F}_{i-1})=d_{i}/r_{i}$ for $i=1,\dots,s$ and let $\cal{G}\subseteq\cal{E}$ be a rank $\rho$ subsheaf of maximal degree. Then part (i) is trivial if $s=h$, because $\cal{G}\cong\cal{E}$ in this case. Thus we may assume that $h<s$. Set $a_{0}=0$ and

[TABLE]

for $i=1,\dots,s$. If we denote by $\cal{R}_{i}\subseteq\cal{F}_{i}/\cal{F}_{i-1}$ the image of $\cal{F}_{i}\cap\cal{G}$ by the quotient map $\pi_{i}:\cal{F}_{i}\rightarrow\cal{F}_{i}/\cal{F}_{i-1}$, then $\cal{R}_{i}$ is trivial, i.e. $\cal{F}_{i}\cap\cal{G}\subseteq\cal{F}_{i-1}$, if and only if $k_{i}=0$. Setting $S:=\{i\in\{1,\dots,s\}\mid k_{i}>0\}$, we have $\sum_{i=1}^{s}k_{i}=\sum_{i\in S}k_{i}=\rho$ and that $\cal{G}\cong\cal{F}_{h}$ if and only if $k_{i}=r_{i}$ for all $i\leq h$, or equivalently $k_{i}=0$ for all $i>h$. Since $\cal{F}_{0}$ is trivial, we have $\cal{R}_{1}\cong\cal{F}_{1}\cap\cal{G}$. Thus we have $\deg\cal{G}=\sum_{i\in S}\deg\cal{R}_{i}$. Since each $\cal{F}_{i}/\cal{F}_{i-1}$ is semistable, we have $\deg\cal{R}_{i}\leq k_{i}\mu_{i}$ for all $i\in S$ and so we may use that $\mu_{i}>\mu_{j}$ for all $i<j$ to get part (i).

For part (ii) let $\cal{G}\subset\cal{E}$ be a rank $k$ subsheaf of maximal degree and define $k_{i}$, $S\subseteq\{1,\dots,s\}$ and the sheaves $\cal{R}_{i}\subset\cal{F}_{i}/\cal{F}_{i-1}$ as above. Then we have $\sum_{i\in S}k_{i}=k$ and $\deg\cal{G}\leq\sum_{i\in S}k_{i}\mu_{i}$ and again we may use that $\mu_{i}>\mu_{j}$ for all $i<j$, to get the assertion. Part (iii) comes directly from the definition of $\delta_{k}(\cal{E})$. ∎

As immediate corollaries of Theorem 4.11 we get the following.

Corollary 4.12**.**

We have $s_{k}(\cal{E},\Phi)=s_{k}(\cal{E})=s_{k}(gr(\cal{E}))$ for all $k$.

Corollary 4.13**.**

For $k$ with $r-r_{s}<k<r$, we have

[TABLE]

4.2. Simplicity

Again let $X$ be a smooth curve of genus $g$. Fix $\cal{R}\in\mathrm{Pic}(X)$ and set $\gamma:=\deg\cal{R}$. For a map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{R}$, set

[TABLE]

where $\hat{f}$ is the map $f\otimes\mathrm{id}_{\cal{R}}:\cal{E}\otimes\cal{R}\rightarrow\cal{E}\otimes\cal{R}$.

In case $\gamma>0$, it often happens that $\operatorname{End}(\cal{E},\Phi)$ is properly contained in $\operatorname{End}(\cal{E})$ and $(\cal{E},\Phi)$ is simple with $\cal{E}$ not simple, e.g. stable Higgs fields when $g\geq 2$ or stable co-Higgs fields when $g=0$. In this short section, we consider the case $\gamma<0$ and show why this is seldom the case for $\gamma<0$.

We assume that $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\cdots;r_{s},d_{s})$ with the Harder-Narasimhan filtration (2) and that $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{R}$ is nonzero and so $s\geq 2$. Note that every endomorphism of $\cal{E}$ preserves the Harder-Narasimhan filtration of $\cal{E}$. By Remark 4.3, every endomorphism of $(\cal{E},\Phi)$ also preserves the Harder-Narasimhan filtration of $\cal{E}$. Now set $\cal{K}:=\mathrm{ker}(\Phi)$ and then we have $\cal{K}\supseteq\cal{F}_{1}$ by the case $i=1$ of Remark 4.3 or Lemma 5.3 below. For two maps $\varphi\in\operatorname{End}(\cal{E}/\cal{F}_{r-1})$ and $\psi\in\operatorname{Hom}(\cal{E}/\cal{F}_{r-1},\cal{K})$, define a map $f:\cal{E}\rightarrow\cal{E}$ to be the following composition:

[TABLE]

where the first map is the natural quotient and the second map is given by $\psi\circ\varphi$. By the definition of $\cal{K}$, we have $\Phi\circ f=0$. If $\Phi$ is $2$-nilpotent, i.e. $\mathrm{Im}(\Phi)\subseteq\cal{K}\otimes\cal{R}$, e.g. if $s=2$ by Lemma 4.1, we have $\hat{f}\circ\Phi=0$. So, if $\Phi$ is $2$-nilpotent and $\operatorname{Hom}(\cal{E}/\cal{F}_{r-1},\cal{K})\neq 0$, then we have $\operatorname{End}(\cal{E},\Phi)\not\cong\mathbb{C}$. We also see from $\cal{F}_{1}\subseteq\cal{K}$ that if $\operatorname{Hom}(\cal{E}/\cal{F}_{r-1},\cal{F}_{1})\neq 0$, then we have $\operatorname{End}(\cal{E},\Phi)\not\cong\mathbb{C}$. By Riemann-Roch, we get $\operatorname{Hom}(\cal{E}/\cal{F}_{r-1},\cal{F}_{1})\neq 0$, if $d_{s}/r_{s}<d_{1}/r_{1}+g-1$. Since $\Phi\neq 0$ and each $\cal{F}_{i}/\cal{F}_{i-1}$ is semistable, we have $d_{s}/r_{s}\leq d_{1}/r_{1}+\gamma$. Now if $\cal{R}\cong\cal{T}_{\cal{D}}$, then we have $\gamma\leq 2-2g$ and so $d_{s}/r_{s}<g-1+d_{1}/r_{1}$ for all $g\geq 2$. Thus we get the following.

Proposition 4.14**.**

For $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ with a nonzero co-Higgs field $\Phi$ on a smooth curve $X$ of genus $g\geq 2$, the pair $(\cal{E},\Phi)$ is not simple.

Remark 4.15**.**

In our set-up, adding a nonzero map $\Phi$ to an unstable bundle $\cal{E}$ does not help enough to get a semistable pair $(\cal{E},\Phi)$; usually it is not simple, e.g. any endomorphism inducing $\cal{F}_{s}\rightarrow\cal{F}_{1}$ commutes with $\Phi$.

5. Higher dimensional case

In this section we consider the case when the dimension of $X$ is at least two. Note that a coherent sheaf $\cal{E}$ on $X$ is semistable if and only if $\mu_{+}(\cal{E})=\mu_{-}(\cal{E})$.

5.1. Case of $\cal{T}_{\cal{D}}$ semistable

We fix a polarization $\cal{O}_{X}(1)$ with respect to which we consider slope, stability and semistability. We assume that $\cal{T}_{\cal{D}}$ is semistable with $\mu(\cal{T}_{\cal{D}})<0$; in case $\mu(\cal{T}_{\cal{D}})\geq 0$, we would get that the framework would be the construction of stable or semistable co-Higgs or logarithmic co-Higgs bundles as in [4]. There are several manifolds $X$ with $T_{X}$ semistable, or equivalently with the semistable cotangent bundle; see [26].

Choose a pair $(\cal{E},\Phi)$ with $\cal{E}$ a torsion-free sheaf of rank $r$ and $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ with the Harder-Narasimhan filtration (2) of $\cal{E}$. Then the sheaf $\cal{F}_{i}/\cal{F}_{i-1}$ is a torsion-free semistable sheaf for all $i$ and $\mu(\cal{F}_{i}/\cal{F}_{i-1})>\mu(\cal{F}_{i-1}/\cal{F}_{i-2})$ for every $i>1$. As in §3 on curve case, for fixed integers $r_{i}$ and $d_{i}$, we consider the set $\mathbb{U}_{X}(s;r_{1},d_{1},\dots,r_{s},d_{s})$ of torsion-free sheaves of rank $r$ on $X$ with the Harder-Narasimhan filtration (2) with subquotient $\cal{F}_{i}/\cal{F}_{i-1}$ of ranks $r_{i}$ and degrees $d_{i}$ for $i=1,\dots,s$.

Recall that in characteristic zero the tensor product of two semistable sheaves is still semistable by [22, Theorem 2.5], So if $\Phi$ is not trivial, then we get $s\geq 2$ and so $\cal{E}$ is not semistable with the Harder-Narasimhan filtration (4) for $\cal{E}\otimes\cal{T}_{\cal{D}}$. If $\cal{A}$ is a semistable torsion-free sheaf, then we have

[TABLE]

Thus if $[\cal{E}]\in\mathbb{U}_{X}(s;r_{1},d_{1};\dots;r_{s},d_{s})$ and there is a nonzero map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$, then we get $d_{1}/r_{1}+\gamma\geq d_{s}/r_{s}$; see Corollary 3.9. Now let us use the same idea in Lemma 2.3. Define

[TABLE]

and then we have $\Phi(\cal{E})\subset\cal{F}_{\ell_{2}}\otimes\cal{T}_{\cal{D}}$. From $\gamma<0$, we get $\ell_{2}\leq s-1$. On the other hand, letting

[TABLE]

the map $\Phi$ induces a nonzero map $\overline{\Phi}:\cal{E}/\cal{F}_{\ell_{1}}\xrightarrow{\ \ }\cal{F}_{\ell_{2}}\otimes\cal{T}_{\cal{D}}$. In particular, if $\ell_{1}\geq\ell_{2}$, e.g. $s=2$ or $d_{2}/r_{2}+\gamma<d_{s}/r_{s}$, which imply $\ell_{2}=1$, then any such map $\Phi$ is $2$-nilpotent.

In [4, Section 2] we consider the following exact sequence for $r\geq 2$

[TABLE]

where $\cal{A}$ is a line bundle of $\deg\cal{A}<0$ with $h^{0}(\cal{T}_{\cal{D}}\otimes\cal{A}^{\vee})\geq r-1$ and $Z\subset X$ is a locally complete intersection of codimension two. Under certain assumptions on $Z$, we may choose $\cal{E}$ to be reflexive or locally free. Then any $(r-1)$-dimensional linear subspace of $H^{0}(\cal{T}_{\cal{D}}\otimes\cal{A}^{\vee})$ produces a nonzero $2$-nilpotent co-Higgs field defined by the following composition:

[TABLE]

Assume now the existence of an endomorphism $v:\cal{E}\rightarrow\cal{E}$ such that $v^{\prime}\circ\Phi=\Phi\circ v$, where $v^{\prime}:\cal{E}\otimes\cal{T}_{\cal{D}}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ is the induces by $v$ and the identity map on $\cal{T}_{\cal{D}}$. Since we assume that $\deg\cal{A}<0$, (13) is the Harder-Narasimhan filtration of $\cal{E}$. We also assume that (13) does not split and so every automorphism of $\cal{E}$ is induced by an element of $H^{0}(\cal{A}^{\vee})^{\oplus(r-1)}$ and an $(r-1){\times}(r-1)$-matrix of constants acting on $\cal{O}_{X}^{\oplus(r-1)}$. Note that, if $r=2$, these assumptions imply $h^{0}(\mathcal{E}nd(\cal{E}))=1+h^{0}(\cal{A}^{\vee})$. In this case, the co-Higgs field $\Phi$ is obtained by composing a map $\Phi_{1}:\cal{I}_{Z}\otimes\cal{A}\rightarrow\cal{T}_{\cal{D}}$ with a map $\Phi_{2}:\cal{T}_{\cal{D}}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ induced by the inclusion in (13).

5.2. Case of $\cal{T}_{\cal{D}}$ not semistable

In this subsection we assume that $\cal{T}_{\cal{D}}$ is not semistable so that it admits the Harder-Narasimhan filtration

[TABLE]

with $h\geq 2$. Assume further that $\mu_{+}(\cal{T}_{\cal{D}})=\mu(\cal{H}_{1})<0$. Since $h\geq 2$, we have $\dim X\geq h\geq 2$.

Fix a torsion-free sheaf $\cal{E}$ of rank $r$ and degree $d$ with Harder-Narasimhan filtration (2). We assume the existence of a nonzero logarithmic co-Higgs field $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$.

Lemma 5.1**.**

If $\cal{E}$ is reflexive, then $\cal{F}_{i}$ is also reflexive for each $i$.

Proof.

In case $n=1$, the sheaf $\cal{F}_{i}$ in (2) is locally free and in particular reflexive. Now assume $n\geq 2$ and then we need to prove that $\cal{F}_{i}$ has depth at least two. This is true, because $\cal{E}$ has depth at least two and $\cal{E}/\cal{F}_{i}$ has no torsion and so it has positive depth. ∎

Remark 5.2**.**

Lemma 5.1 works for arbitrary $\cal{T}_{\cal{D}}$, even in the case $n=1$.

Lemma 5.3**.**

We have $\cal{F}_{1}\subseteq\mathrm{ker}(\Phi)$ and $s\geq 2$.

Proof.

Assume $\Phi(\cal{F}_{1})\neq 0$ and let $i_{0}$ be the minimal integer $i\in\{1,\dots,s\}$ such that $\Phi(\cal{F}_{1})\subseteq\cal{F}_{i}\otimes\cal{T}_{\cal{D}}$. By the definition of $i_{0}$, the map $\Phi$ induces a nonzero map $\varphi:\cal{F}_{1}\rightarrow(\cal{F}_{i_{0}}/\cal{F}_{i_{0}-1})\otimes\cal{T}_{\cal{D}}$. Since the tensor product of two semistable sheaves, modulo its torsion, is again semistable by [22, Theorem 2.5] and $\mu(\cal{H}_{1})<0$, the sheaf $gr((\cal{F}_{i_{0}}/\cal{F}_{i_{0}-1})\otimes\cal{T}_{\cal{D}})$ given by the Harder-Narasimhan filtration of $\cal{T}_{\cal{D}}$ has all its factors with slope less than $\mu(\cal{F}_{1})$. Thus we get $\Phi=0$, a contradiction.

Now $\Phi$ is a nonzero map with $\mathrm{ker}(\Phi)\supseteq\cal{F}_{1}$ and so we have $s\geq 2$. ∎

Remark 5.4**.**

By Lemma 5.1, the pair $(\cal{F}_{1},0)$ is a logarithmic co-Higgs subsheaf of $(\cal{E},\Phi)$ and so $(\cal{E},\Phi)$ is not semistable. In particular, $\cal{E}$ is also not semistable.

5.2.1. Rank $2$ case

In this subsection we consider the co-Higgs sheaves $(\cal{E},\Phi)$ with $\cal{E}$ reflexive of rank two and $\Phi$ nonzero.

Lemma 5.5**.**

If $\cal{E}$ is reflexive of rank two, then $\Phi$ is $2$-nilpotent.

Proof.

Since $\Phi$ is nonzero, the sheaf $\cal{F}_{1}$ has rank one by Lemma 5.3. Since $\cal{F}_{1}$ is reflexive on a smooth variety $X$ by Lemma 5.1, it is a line bundle by [16, Proposition 1.9]. Now we get that $\cal{E}/\cal{F}_{1}\cong\cal{I}_{Z}\otimes\cal{A}$ for some line bundle $\cal{A}$ and some closed subscheme $Z\subset X$ with $\dim Z\leq n-2$. By definition of Harder-Narasimhan filtration, we have $\deg\cal{A}<\deg\cal{F}_{1}$. Let $\psi:\cal{E}\rightarrow(\cal{E}/\cal{F}_{1})\otimes\cal{T}_{\cal{D}}$ be the map induced by $\Phi$. Since $\mathrm{ker}(\Phi)\supseteq\cal{F}_{1}$ by Lemma 5.3, it is sufficient to prove that $\Phi(\cal{E})\subseteq\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$, i.e. $\psi=0$. Note that $\psi$ induces a map $\widetilde{\psi}:(\cal{E}/\cal{F}_{1})\rightarrow(\cal{E}/\cal{F}_{1})\otimes\cal{T}_{\cal{D}}$ with $\mathrm{Im}(\psi)=\mathrm{Im}(\widetilde{\psi})$, due to $\cal{F}_{1}\subseteq\mathrm{ker}(\Phi)$. Since $(\cal{E}/\cal{F}_{1})$ has rank one and it is torsion-free, it is semistable. Again as in the proof of Lemma 5.3, since $\mu(\cal{H}_{1})<0$ and the tensor product of two semistable sheaves, modulo its torsion, is semistable by [22, Theorem 2.5], we have $\mu((\cal{E}/\cal{F}_{1})\otimes\cal{H}_{1})<\mu(\cal{E}/\cal{F}_{1})$ and so $\widetilde{\psi}=0$. Thus we have $\psi=0$. ∎

Now we describe all pairs $(\cal{E},\Phi)$ with $\cal{E}$ reflexive of rank two and $\Phi$ nonzero. By Lemma 5.3 and assumption that $\Phi$ is nonzero, the sheaf $\cal{E}$ is not semistable and $s=2$. By Lemmas 5.1, 5.3, 5.5 and [16, Proposition 1.9], the map $\Phi$ is $2$-nilpotent and it fits into an exact sequence

[TABLE]

with $Z$ a closed subscheme of $X$ with either $Z=\emptyset$ or $\dim Z=n-2$. Moreover, $\Phi$ is uniquely determined by a map $u:\det(\cal{E})\otimes\cal{F}_{1}^{\vee}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$. Thus the set of all logarithmic co-Higgs structures on $\cal{E}$ is parametrized by

[TABLE]

The trivial element $0\in V(\cal{E})$ corresponds to the trivial co-Higgs field $\Phi=0$. Note that $\Phi=0$ also exists for stable sheaves.

Now we reverse the construction. Fix two line bundles $\cal{L}_{1}$ and $\cal{L}_{2}$ on $X$ with $\deg\cal{L}_{1}>\deg\cal{L}_{2}$ and a closed subscheme $Z\subset X$ such that a general extension

[TABLE]

is reflexive. We just observed that any co-Higgs field $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ is $2$-nilpotent and that $\operatorname{Hom}(\cal{E},\cal{E}\otimes\cal{T}_{\cal{D}})\cong H^{0}(\cal{L}_{1}\otimes\cal{L}_{2}^{\vee}\otimes\cal{T}_{\cal{D}})$. We may see [16, Theorem 4.1] for a description about the conditions on $\cal{L}_{1}$, $\cal{L}_{2}$, $\omega_{X}$ and $Z$ assuring the existence of a reflexive sheaf fitting in (15) when $n=3$. Since (15) is the Harder-Narasimhan filtration of any $\cal{E}$ fitting into (15), so the family of the co-Higgs sheaves $(\cal{E},\Phi)$ with $gr(\cal{E})=\cal{L}_{1}\oplus(\cal{I}_{Z}\otimes\cal{L}_{2})$ is parametrized by a fibration over $\mathbb{P}\operatorname{Ext}_{X}^{1}(\cal{I}_{Z}\otimes\cal{L}_{2},\cal{L}_{1})$ whose fibre over $[\cal{E}]$ is $H^{0}(\cal{L}_{1}\otimes\cal{L}_{2}^{\vee}\otimes\cal{T}_{\cal{D}})$.

Remark 5.6**.**

Assume $s=2$ and $\mu_{+}(\cal{T}_{\cal{D}})<0$. For a torsion-free coherent sheaf $\cal{E}$ of rank at least $2$, as in the proof of Lemma 5.5 we see that every logarithmic co-Higgs field $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$ is integrable and $2$-nilpotent with

[TABLE]

where $\cal{F}_{1}$ is semistable, and $\cal{E}/\cal{F}_{1}$ is torsion-free and semistable. Recall that if $\cal{E}$ is reflexive, then so is $\cal{F}_{1}$ by Lemma 5.1. Take any exact sequence

[TABLE]

Any such extension in (16) is torsion-free. For any fixed $\cal{G}$ fitting into (16), not necessarily reflexive, the proof of Lemma 5.5 shows that every logarithmic co-Higgs field $\Phi:\cal{G}\rightarrow\cal{G}\otimes\cal{T}_{\cal{D}}$ is integrable and $2$-nilpotent with $\operatorname{Hom}(\cal{G},\cal{G}\otimes\cal{T}_{\cal{D}})\cong\operatorname{Hom}(\cal{E}/\cal{F}_{1},\cal{F}_{1}\otimes\cal{T}_{\cal{D}})$.

5.2.2. Rank $3$ case

We assume $r=3$ and that $\cal{E}$ is reflexive. Since we assume $\mu_{+}(\cal{T}_{\cal{D}})<0$, we get $s\geq 2$ by Lemma 5.3 and so $s\in\{2,3\}$.

Remark 5.7**.**

The case $s=2$ is dealt in Remark 5.6. In this case, the sheaf $\cal{F}_{1}$ is either a line bundle or a semistable reflexive sheaf of rank two with $\cal{E}/\cal{F}_{1}\cong\cal{I}_{Z}\otimes\cal{A}$ for some line bundle $\cal{A}$ and a closed subscheme $Z\subset X$ with $\dim Z\leq n-2$. In both cases, we may apply Remark 2.8.

From now on we assume $s=3$ and so the sheaf $\cal{F}_{i}$ in (2) has rank $i$ for each $i$. By Lemma 5.1, the sheaf $\cal{F}_{1}$ is a line bundle and $\cal{F}_{2}$ is reflexive so that $\cal{F}_{2}/\cal{F}_{1}\cong\cal{I}_{Z_{1}}\otimes\cal{A}_{1}$ and $\cal{E}/\cal{F}_{2}\cong\cal{I}_{Z_{2}}\otimes\cal{A}_{2}$ with $\cal{A}_{1},\cal{A}_{2}$ line bundles and $Z_{1},Z_{2}$ closed subschemes of $X$ with dimension at most $n-2$. Here we have $\deg\cal{F}_{1}>\deg\cal{A}_{1}>\deg\cal{A}_{2}$. Set

[TABLE]

where $\mu_{-}(\cal{F}_{2})=\deg\cal{A}_{1}$ and $\mu_{+}(\cal{E}\otimes\cal{T}_{\cal{D}})=\deg\cal{F}_{1}+\mu_{+}(\cal{T}_{\cal{D}})$

(a) Assume $\delta(\cal{E})>0$ and then we have $\Phi_{|\cal{F}_{2}}=0$, i.e. $\Phi$ is uniquely induced by a map $u_{1}:\cal{I}_{Z_{2}}\otimes\cal{A}_{2}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$. Since $\cal{E}/\cal{F}_{2}\cong\cal{I}_{Z_{2}}\otimes\cal{A}_{2}$ is of rank one and $\mu_{+}(\cal{T}_{\cal{D}})<0$, the composition of $u_{1}$ with the quotient map $\cal{E}\otimes\cal{T}_{\cal{D}}\rightarrow(\cal{E}/\cal{F}_{2})\otimes\cal{T}_{\cal{D}}$ is trivial, implying $\mathrm{Im}(u_{1})\subseteq\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$. Thus $\Phi$ is uniquely determined by a map $u:\cal{I}_{Z_{2}}\otimes\cal{A}_{2}\rightarrow\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$. Conversely, any map $u:\cal{I}_{Z_{2}}\otimes\cal{A}_{2}\rightarrow\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$ induces a $2$-nilpotent logarithmic co-Higgs field on $\cal{E}$ by taking the composition $u\circ\pi$, where $\pi:\cal{E}\rightarrow\cal{E}/\cal{F}_{2}$ is the quotient map.

(b) Assume now $\delta(\cal{E})\leq 0$. Set $\cal{B}:=\mathrm{Im}(\Phi_{|\cal{F}_{2}})$ and $\cal{G}:=\mathrm{Im}(\Phi)$. Since we have

[TABLE]

the composition of $\Phi$ with the quotient map $\cal{E}\otimes\cal{T}_{\cal{D}}\rightarrow(\cal{E}/\cal{F}_{2})\otimes\cal{T}_{\cal{D}}$ is trivial and so we have $\cal{G}\subseteq\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$. If $\cal{B}$ is trivial, then we may apply part (a), i.e. $\Phi$ is $2$-nilpotent and it is uniquely induced by $u:\cal{I}_{Z_{2}}\otimes\cal{A}_{2}\rightarrow\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$. Now we assume that $\cal{B}$ is not trivial. Since $\Phi(\cal{F}_{1})=0$ and $\cal{F}_{2}/\cal{F}_{1}$ is a torsion-free sheaf of rank one, we have $\cal{B}\cong\cal{F}_{2}/\cal{F}_{1}$ and so $\mathrm{rk}(\cal{G})\in\{1,2\}$. Note that we have $\cal{B}\subseteq\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ from $\mu_{+}(\cal{E}/\cal{F}_{2})+\mu_{+}(\cal{T}_{\cal{D}})<\mu_{+}(\cal{E}/\cal{F}_{2})$.

(b-i) First assume $\mathrm{rk}(\cal{G})=1$ and then $\cal{B}$ is a subsheaf of $\cal{G}$ with the same rank. Since $\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$ is a saturated subsheaf of $\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$, we have $\cal{G}\subseteq\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$. Thus $\Phi$ is uniquely determined by a map $\cal{E}/\cal{F}_{1}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$, i.e. by an element of $H^{0}(\cal{T}_{\cal{D}}\otimes\cal{F}_{1}\otimes\cal{A}^{\vee})$; the converse also holds, but we cannot guarantee the integrability of the associated logarithmic co-Higgs field.

(b-ii) Now assume $\mathrm{rk}(\cal{G})=2$. Since we have $\cal{G}=\psi(\cal{E}/\cal{F}_{1})$ for the map $\psi:\cal{E}/\cal{F}_{1}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$, the map $\psi$ is injective as a map of sheaves and $\cal{G}\cong\cal{E}/\cal{F}_{1}$. In this case we also have $\cal{F}_{1}=\mathrm{ker}(\Phi)$. We get that $\cal{E}$ is a reflexive sheaf fitting into an exact sequence

[TABLE]

with $\cal{F}_{1}$ a line bundle and $\cal{G}$ a torsion-free unstable sheaf of rank two with $\deg\cal{F}_{1}>\mu_{+}(\cal{G})$. The map $\Phi$ is determined by a unique injective map $v:\cal{G}\rightarrow\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$. Conversely, set $\cal{G}_{1}\subset\cal{G}$ to be the Harder-Narasimhan filtration of $\cal{G}$ and $\cal{F}_{2}=f^{-1}(\cal{G}_{1})$, where $f$ is the surjection in (17). Then the composition of the quotient map $\cal{E}\rightarrow\cal{E}/\cal{F}_{1}$ with an injective map $\cal{G}\rightarrow\cal{F}_{2}\otimes\cal{T}_{\cal{D}}$ induces a logarithmic co-Higgs field $\Phi$ with the given data $(\cal{F}_{1},\cal{F}_{2},\cal{G})$, which does not necessarily satisfy the integrability condition. Note that if $\cal{G}\subset\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$, i.e. $\Phi$ comes from an injective map $\cal{G}\rightarrow\cal{F}_{1}\otimes\cal{T}_{\cal{D}}$, then $\Phi$ is $2$-nilpotent and so it is integrable.

Example 5.8**.**

Assume that $\cal{T}_{\cal{D}}$ is not semistable with Harder-Narasimhan filtration (14) and set $\mu_{2}(\cal{T}_{\cal{D}}):=\mu(\cal{H}_{2}/\cal{H}_{1})$. Let $\cal{E}$ be a torsion-free sheaf of rank $r$ with (2) as its Harder-Narasimhan filtration and assume $\mu_{+}(\cal{E})-\mu_{-}(\cal{E})<\mu_{2}(\cal{T}_{\cal{D}})$. In this case, for any map $\Phi:\cal{E}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$, the sheaf $\mathrm{Im}(\Phi)$ is contained in the subsheaf $\overline{\cal{E}\otimes\cal{H}_{1}}$ of $\cal{E}\otimes\cal{T}_{\cal{D}}$, which is the image of the natural map $\cal{E}\otimes\cal{H}_{1}\rightarrow\cal{E}\otimes\cal{T}_{\cal{D}}$. We have $\overline{\cal{E}\otimes\cal{H}_{1}}\cong\cal{E}\otimes\cal{H}_{1}$ if either $\cal{E}$ or $\cal{H}_{1}$ is locally free. Note that $\cal{H}_{1}$ is locally free, if it has rank one, because $\cal{H}_{1}$ is reflexive and $X$ is smooth; see [16, Proposition 1.9]. In particular, if $n=2$, then $\cal{H}_{1}$ is a line bundle and $\mu_{2}(\cal{T}_{\cal{D}})=\mu_{-}(\cal{T}_{\cal{D}})$. Thus under these assumptions we may repeat the observations given in the case $\cal{T}_{\cal{D}}$ semistable using $\cal{H}_{1}$ instead of $\cal{T}_{\cal{D}}$. Without any assumption on $\mu_{2}(\cal{T}_{\cal{D}})$ we may see at least a part of the logarithmic co-Higgs fields of $\cal{E}$ in this way.