# Comptes Rendus
Mathématique

Snehajit Misra and Nabanita Ray

## On Ampleness of vector bundles

Volume 359, issue 6 (2021), p. 763-772

Published online: 2 September 2021

https://doi.org/10.5802/crmath.222

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Algebraic geometry / Géométrie algébrique

# On Ampleness of vector bundles

Snehajit Misra*, a and Nabanita Raya

a Tata Institute of Fundamental Research (TIFR), Homi Bhabha Road, Colaba, Mumbai 400005, India.

E-mails: smisra@math.tifr.res.in (Snehajit Misra), nray@math.tifr.res.in (Nabanita Ray)

**Abstract.** In this article, we give a necessary and sufficient condition for ampleness of semistable vector bundles with vanishing discriminant on a smooth projective variety X. As an application, we show ampleness of some special vector bundles on certain ruled surfaces. We prove similar results for parabolic ampleness.

**Mathematical subject classification** (2010). 14J60, 14N05, 14J40, 14C17.

Manuscript received 9th February 2021, revised 16th February 2021 and 2nd May 2021, accepted 2nd May 2021.

## 1. Introduction

Let X be a complex manifold of dimension n, and E be a holomorphic vector bundle of rank r on X endowed with a hermitian metric h. The hermitian bundle (E, h) determines a unique hermitian connection compatible with the complex structure on X and E, called as Chern connection, and it is denoted by $D_E$. This connection $D_E$ in turn gives rise to a curvature tensor, called as Chern curvature tensor and denoted by $\Theta(E, h) \in C^\infty(X, \wedge^{1,1} T_X^* \otimes \text{End}(E))$ a $\text{End}(E)$-valued (1, 1) form on X. If $z_1, z_2, \dots, z_n$ are local coordinates on X, and if $(e_\lambda)_{1\le\lambda\le r}$ is a local orthonormal frame on E, then one can write

$$i\Theta(E, h) = \sum_{1\le j,k\le n, 1\le\lambda,\mu\le r} c_{jk\lambda\mu} dz_j \wedge d\bar{z}_k \otimes e_\lambda^* \otimes e_\mu,$$

where $c_{jk\lambda\mu} = c_k j\mu\lambda$. One looks at the associated quadratic form on $S = T_X \otimes E$ as follows:

$$\tilde{\Theta}_{E,h}(\xi \otimes v) = \langle \Theta_{E,h}(\xi, \bar{\xi}) \cdot v, v \rangle_h = \sum_{1\le j,k\le n, 1\le\lambda,\mu\le r} c_j k\lambda\mu\xi_j \bar{\xi}_k v_\lambda \bar{v}_\mu.$$

The hermitian bundle $(E, h)$ is said to be Griffiths positive if at any point $z \in X$, we have $\tilde{\Theta}_{E,h}(\xi \otimes v) > 0$ for all $0 \neq \xi \in T_{X,z}$ and for all $0 \neq v \in E_z$.

A holomorphic vector bundle E on a complex projective manifold is called ample in the sense of Hartshorne if the tautological line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is ample. i.e. there exists a smooth hermitian metric on $\mathcal{O}_{\mathbb{P}(E)}(1)$ with positive curvature.

<footnote>* Corresponding author.</footnote>
It is always true that if a hermitian holomorphic vector bundle $(E, h)$ on a complex projective manifold $X$ is Griffiths positive, then $E$ is ample in the sense of Hartshorne. A famous conjecture of Griffiths asks whether ample bundles in the sense of Hartshorne admit Griffiths positively curved metrics. Also it is well known that if $E$ is ample, then $\det(E)$ is ample. However, ampleness of $\det(E)$ does not ensure ampleness of $E$ in general.

For a vector bundle of rank $r$ on a complex manifold $X$, the characteristic class

$$c_2(\mathrm{End}(E)) = 2rc_2(E) - (r-1)c_1^2(E) \in H^4(X, \mathbb{Q})$$

is called the discriminant of $E$, denoted by $\Delta(E)$.

In Section 3, we prove the following.

**Theorem 1.** Let $X$ be a projective variety of dimension $n$ and $(E, h)$ be a hermitian holomorphic bundle of rank $r$ on $X$. Further assume that $E$ is a semistable vector bundle with $\Delta(E) = 0$. Then the following are equivalent:

(i) $(E, h)$ is Griffiths positive.

(ii) $E$ is ample in the sense of Hartshorne.

(iii) $\det(E)$ is ample.

The Nakai-Moishezon criterion for ampleness says that a line bundle $L$ on a projective variety $X$ is ample if and only if $L^{\dim Y} \cdot Y > 0$ for every positive dimensional subvarieties $Y$ of $X$. Mumford gave an example of a non-ample line bundle on a ruled surface whose intersection with every curve is positive (see [14, Chapter 1]). Therefore, in general, it is not sufficient to check the condition only for curves in Nakai-Moishezon criterion. However, in some special cases, to check ampleness of a line bundle $L$ on $X$, it is enough to check that $L \cdot C > 0$ for every irreducible curve $C \subset X$ (e.g., on abelian varieties [21], on flag bundles [7]). One must also note that for a globally generated vector bundle $E$ on $X$, $E$ is ample if and only if its restriction to every curve $C \subset X$ is ample. This follows easily from Gieseker's Lemma (see [15, Proposition 6.1.7]). In general, there is no straight forward way to check ampleness of a given vector bundle on a projective variety $X$. In [12], it is proved that an equivariant vector bundle on a toric variety $X$ is ample if and only if its restriction to finitely many invariant curves in $X$ are ample. Similar result holds for torus equivariant vector bundles on certain homogenous variety (see [6]). In [1], a sufficient condition is given to check ampleness of a vector bundle of rank 2 on some specific smooth surfaces with Picard rank 1.

We recall from [11, Chapter 5] that a vector bundle $W$ of rank 2 on a smooth projective curve $C$ is said to be normalized if $H^0(W) \neq 0$, but $H^0(W \otimes L) = 0$ for all line bundle $L$ on $C$ with $\deg(L) < 0$. We notice that a normalized bundle $W$ is semistable if and only if $\deg(W) \ge 0$. An important consequence of Theorem 1 is the following.

**Corollary 2.** Let $\rho: X = \mathbb{P}(W) \to C$ be a ruled surface defined by a normalized rank 2 bundle on a smooth curve $C$ such that $\mu_{\min}(W) = \deg(W)$. Let $E$ be a semistable vector bundle of rank $r$ on $X$ with discriminant $\Delta(E) = 0$. Then, $E$ is ample if and only if $E|_{\sigma}$ and $E|_{f}$ are ample, where $\sigma$ is the smooth section of $\rho$ such that $\mathcal{O}_X(\sigma) \cong \mathcal{O}_{\mathbb{P}(W)}(1)$ and $f$ is a fibre of $\rho$.

The above Corollary 2 implies the following:

**Corollary 3.** Let $\rho: X = \mathbb{P}(W) \to C$ be a ruled surface on a smooth curve $C$ defined by a normalized rank 2 bundle $W$ on $C$ with $\mu_{\min}(W) = \deg(W)$, and $E$ be a vector bundle on $C$. Then the vector bundle $E = \rho^*(V) \otimes \mathcal{O}_{\mathbb{P}(W)}(m)$ is ample on $X$ if and only if $m > 0$ and $\mu_{\min}(E) > -m \deg(W)$.

We also prove similar result for parabolic ampleness in Section 4.
## 2. Preliminaries

### 2.1. Harder-Narasimhan Filtration

A non-zero torsion-free coherent sheaf $\mathcal{G}$ on $X$ is said to be $H$-semistable if

$$ \mu_H(\mathcal{F}) = \frac{c_1(\mathcal{F}) \cdot H^{n-1}}{\mathrm{rank}(\mathcal{F})} \le \mu_H(\mathcal{G}) = \frac{c_1(\mathcal{G}) \cdot H^{n-1}}{\mathrm{rank}(\mathcal{G})} $$

for all subsheaves $\mathcal{F}$ of $\mathcal{G}$. For every vector bundle $E$ on $X$, there is a unique filtration

$$ 0 = E_0 \subsetneq E_1 \subsetneq E_2 \subsetneq \cdots \subsetneq E_{k-1} \subsetneq E_k = E $$

of subbundles of $E$, called the Harder-Narasimhan filtration of $E$, such that $E_i/E_{i-1}$ is $H$-semistable torsion free sheaf for each $i \in \{1, 2, \dots, k\}$ and $\mu_H(E_i/E_{i-1}) > \mu_H(E_{i+1}/E_i)$ for each $i \in \{1, 2, \dots, k-1\}$. We define $Q_k := E_k/E_{k-1}$ and $\mu_{\min}(E) := \mu_H(Q_k) = \mu_H(E_k/E_{k-1})$.

Let $N_1(X)_{\mathbb{R}}$ be the set of all numerical equivalence classes of real one cycles on $X$. Inside $N_1(X)_{\mathbb{R}}$, the closure of the convex cone generated by effective one cycles is called the closed cone of curves and it is denoted by $\overline{\mathrm{NE}}(X)$. By Theorem 1.4.29 of [14], a divisor $D$ is ample if and only if $D \cdot \gamma > 0$ for all $\gamma \in \overline{\mathrm{NE}}(X) - \{0\}$.

## 3. Main result and applications

We begin this section by proving our main result.

**Proof of Theorem 1.** (i) $\implies$ (ii). See Theorem 6.1.25 in [15] for a proof.

(ii) $\implies$ (iii). See Corollary 5.3 in [10] for a proof.

(iii) $\implies$ (i). There exists a filtration

$$ 0 = E_0 \subsetneq E_1 \subsetneq E_2 \subsetneq \cdots \subsetneq E_{l-1} \subsetneq E_l = E $$

such that on each $G_j = E_j/E_{j-1}$, there exists a hermitian metric $h_j$ on $G_j$ for which the curvature tensor is equal to $\frac{1}{r}\gamma \otimes \mathrm{Id}_{G_j}$ where $\gamma$ is (1,1)-form representing the first Chern class $c_1(E)$ (see [19]). Since $\det(E)$ is ample, each $(G_j, h_j)$ is Griffiths positive. As extension of two Griffiths positive bundles is again Griffiths positive, we have inductively each $E_i$ is Griffiths positive and thus $E$ is also Griffiths positive. $\square$

**Remark 4.** Theorem 1 can be thought of as a generalization of Gieseker's ampleness criterion for semistable vector bundles on smooth curves (see [13, Theorem 3.2.7]). However, the condition about vanishing discriminant is not essential for both $V$ and $\det(V)$ to be ample. For example, consider the tangent bundle $T_{\mathbb{P}^2}$. Then $T_{\mathbb{P}^2}$ sits in the following exact sequence:

$$ 0 \to \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_{\mathbb{P}^2}(1)^{\oplus 3} \to T_{\mathbb{P}^2} \to 0. $$

Hence, $T_{\mathbb{P}^2}$ being quotient of an ample bundle is ample and $\det(T_{\mathbb{P}^2}) \cong \mathcal{O}_{\mathbb{P}^2}(3)$ is also ample. But $T_{\mathbb{P}^2}$ is semistable with $\Delta(T_{\mathbb{P}^2}) \neq 0$.

**Remark 5.** Note that for a vector bundle $E$ on a smooth projective curve $C$, we have $\Delta(E) = 0$. Hence our result Theorem 1 is analogous to the result in [22]. Also one can compare our result with the results in [16] and [20].

A vector bundle $V$ on an abelian variety $X$ is called weakly-translation invariant (semi-homogeneous in the sense of Mukai) if for every closed point $x \in X$, there is a line bundle $L_x$ on $X$ depending on $x$ such that $T_x^*(V) \cong V \otimes L_x$ for all $x \in X$, where $T_x$ is the translation morphism given by $x \in X$.
**Corollary 6.** A semi-homogeneous vector bundle $E$ of rank $r$ on an abelian variety $X$ is ample if and only if $\det(E)$ is ample if and only if $\det(E) \cdot C > 0$ for all irreducible curve $C$ in $X$.

**Proof.** Mukai proved that $E$ is Gieseker semistable (see [13, Chapter 1] for definition) with respect to some polarization and it has projective Chern classes zero, i.e., if $c(E)$ is the total Chern class, then $c(E) = \{1 + c_1(E)/r\}^r$ (see [18, Theorem 5.8, p. 260], [18, Proposition 6.13, p. 266]; also see [17, p. 2]). Gieseker semistability implies slope semistability (see [13]). So, in particular, we have $E$ is slope semistable with $\Delta(E) = 2rc_2(E) - (r-1)c_1^2(E) = 0$. Hence, the result follows from Theorem 19 and Proposition 1.4 in [21]. □

**Corollary 7.** Let $W$ be a vector bundle of rank $m$ over a smooth complex projective curve $C$ and $\rho: \mathbb{P}(W) \to C$ be the projectivisation map. Let $E$ be a semistable vector bundle on $\mathbb{P}(W)$ of rank $r$ with discriminant $\Delta(E) = 0$, and $c_1(E) \equiv x\xi + yf$, where $\xi$ and $f$ are the numerical classes of $\mathcal{O}_{\mathbb{P}(W)}(1)$ and a fibre of $\rho$ respectively. Then, $E$ is ample if and only if $x > 0$ and $(x\mu_{\min}(W) + y) > 0$.

**Proof.** We note that by Lemma 2.1 of [9], the nef cone of divisors in $\mathbb{P}(W)$ is given by

$$ \mathrm{Nef}(\mathbb{P}(W)) = \{a(\xi - \mu_{\min}(W)f) + bf \mid a, b \in \mathbb{R}_{\ge 0}\}. $$

Applying duality (see [14, Proposition 1.4.28]), we get

$$ \overline{\mathrm{NE}}(\mathbb{P}(W)) = \{a(\xi^{m-1} - (\mathrm{deg}(W) - \mu_{\min}(W))\xi^{m-2}f) + b\xi^{m-2}f \mid a, b \in \mathbb{R}_{\ge 0}\}. $$

Hence, $\det(E)$ is ample if and only if

$$ \bullet \ c_1(E) \cdot \{\xi^{m-1} - (\mathrm{deg}(W) - \mu_{\min}(W))\xi^{m-2}f\} = (x\mu_{\min}(W) + y) > 0 \text{ and} $$

$$ \bullet \ c_1(E) \cdot \xi^{m-2}f = x > 0. $$

Therefore, the result follows from the previous Theorem. □

**Corollary 8.** Let $\rho: X = \mathbb{P}(W) \to C$ be a ruled surface defined by a normalized rank 2 bundle on a smooth curve $C$ such that $\mu_{\min}(W) = \deg(W)$. Let $E$ be a semistable vector bundle of rank $r$ on $X$ with discriminant $\Delta(E) = 0$. Then, $E$ is ample if and only if $E|_{\sigma}$ and $E|_f$ are ample, where $\sigma$ is the smooth section of $\rho$ such that $\mathcal{O}_X(\sigma) \simeq \mathcal{O}_{\mathbb{P}(W)}(1)$ and $f$ is a fibre of $\rho$.

**Proof.** Let $c_1(E) \equiv x\zeta + yf$, where $\zeta = [\sigma] \in N^1(X)$. Note that, by the given hypothesis, both $E|_{\sigma}$ and $E|_f$ are semistable, and hence both are ample if and only if

$$ \bullet \ \deg(E|_{\sigma}) = (x\zeta + yf) \cdot \zeta = (x\deg(W) + y) > 0, \text{ and} $$

$$ \bullet \ \deg(E|_f) = (x\zeta + yf) \cdot f = x > 0. $$

But, in that case, $(x\mu_{\min}(W) + y) = (x\deg(W) + y) > 0$. Therefore, the result follows from the previous corollary. □

**Remark 9.** Let $\rho: X = \mathbb{P}(W) \to C$ be a ruled surface on a smooth curve $C$ as in Corollary 8. Then, for any semistable vector bundle $R$ on $C$ and any integer $m$, $E := \rho^*(R) \otimes \mathcal{O}_{\mathbb{P}(W)}(m)$ is a semistable vector bundle with vanishing discriminant. Hence by Corollary 8, any semistable vector bundle $V$ on $X$ of this form $\rho^*(R) \otimes \mathcal{O}_{\mathbb{P}(W)}(m)$ is ample if and only if $E|_{\sigma}$ and $E|_f$ are ample if and only if $m > 0$ and $\deg(R) > -m \deg(W)$.

For example, we consider the ruled surface $\rho: X = \mathbb{P}(W) \to C$ over the elliptic curve $C$ defined by the nonsplit extension $0 \to \mathcal{O}_C \to W \to \mathcal{O}_C \to 0$. Then for any semistable bundle $R$ on $C$ of positive degree, $E := \rho^*(R) \otimes \mathcal{O}_X(m)$ is ample for every positive integer $m$.

**Corollary 10.** Let $\rho: X = \mathbb{P}(W) \to C$ be a ruled surface on a smooth curve $C$ defined by a normalized rank 2 bundle $W$ on $C$ with $\mu_{\min}(W) = \deg(W)$, and $V$ be a vector bundle on $C$. Then the vector bundle $E = \rho^*(V) \otimes \mathcal{O}_{\mathbb{P}(W)}(m)$ is ample on $X$ if and only if $m > 0$ and $\mu_{\min}(V) > -m \deg(W)$.
**Proof.** Let

$$0 = V_0 \subsetneq V_1 \subsetneq V_2 \subsetneq \dots \subsetneq V_{k-1} \subsetneq V_k = V$$

be the Harder-Narasimhan filtration of $V$, and $Q_i = V_i / V_{i-1}$ for each $i$. Since $\rho$ is a smooth map, in particular it is flat and hence $\rho^*$ is an exact functor. We also observe that for any ample line bundle $H$ on $X$, we have $\mu_H(\rho^*Q_i) = \mu(Q_i)(f \cdot H)$ and $f \cdot H > 0$, where $f$ denotes a fiber of $\rho$. Fix $E_i := \rho^*(V_i) \otimes \mathcal{O}_{\mathbb{P}(E)}(m)$. Then above observation and the uniqueness of Harder Narasimhan filtration imply that

$$0 = E_0 \subsetneq E_1 \subsetneq E_2 \subsetneq \dots \subsetneq E_{k-1} \subsetneq E_k = E$$

is the Harder Narasimhan filtration of $E$ with respect to any polarization $H$.

Now, suppose $m > 0$ and $\mu_{\min}(V) > -m \deg(W)$. Then by the previous remark, we conclude that each $R_i := \rho^*(Q_i) \otimes \mathcal{O}_{\mathbb{P}(W)}(m)$ is ample. Inductively, each $E_i$ is ample. In particular $E$ is also ample.

Conversely, if $V$ is ample for some $m$, then $R_k = \rho^*(Q_k) \otimes \mathcal{O}_{\mathbb{P}(W)}(m)$ is ample for each $k$. Thus $m$ must satisfy $m > 0$ and $\mu_{\min}(E) > -m \deg(W)$. $\square$

**Example 11.** Let us consider the ruled surface $\rho : X = \mathbb{P}(W) \to C$ over a curve $C$ where $W = \mathcal{O}_C \oplus \mathcal{L}$ for some line bundle $\mathcal{L}$ on $C$ with $\deg(\mathcal{L}) < 0$. Then for any vector bundle $E$ on $C$ with $\mu_{\min}(E) > -m \deg(\mathcal{L})$ for some positive integer $m$, the bundle $V = \rho^*(E) \otimes \mathcal{O}_X(m)$ is ample.

Let $\rho : \mathbb{F}_e = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-e)) \to \mathbb{P}^1$ be a Hirzebruch surface for some $e \ge 0$. Let $C_0$ be its normalized section such that $\mathcal{O}_{\mathbb{F}_e}(C_0)$ is the tautological bundle on $\mathbb{F}_e$, and $f$ be a fibre of $\rho$. We recall the following results from [11, Chapter 5].

**Theorem 12.** Let $D \sim aC_0 + bF$ be a divisor on $\mathbb{F}_e$. Then

(a) If $D$ is an irreducible curve $\neq C_0$, $f$, then $a > 0$ and $b \ge ae$.

(b) The linear system $|D|$ contains a section of $\rho$ if and only if $a = 1$ and either $b = 0$ or $b \ge e$.

(c) The linear system $|D|$ contains an irreducible non-singular curve if and only if it contains an irreducible curve if and only if $a = 0$, $b = 1$ (namely $f$); or $a = 1$, $b = 0$ (namely $C_0$); or $a > 0$, $b > ae$; or $e > 0$, $a > 0$, $b = ae$.

(d) $D$ is very ample if and only if $D$ is ample if and only if $a > 0$ and $b > ae$.

**Lemma 13.** Any irreducible curve of $\mathbb{F}_e$ other than the fibers of $\rho$ is linearly equivalent to an effective curve which is a union of sections of the map $\rho$.

**Proof.** Let $C$ be an irreducible curve in $\mathbb{F}_e$ other than a fibre and the section $C_0$. Then $C \sim xC_0 + yf$ for some $x > 0$ and $y \ge xe$. Let $y = mxe + r$ for some $m > 0$ and $0 \le r < xe$. Now, $C \sim xC_0 + yf \sim (x-1)(C_0 + ef) + (C_0 + (e(xm - x + 1) + r)f)$. This proves the result. $\square$

**Proposition 14.** Let $C$ be an irreducible curve in $\mathbb{F}_e$ and $C \sim C_1 + \dots + C_r$ where $C_i$'s are sections of the map $\rho$. Let $E$ be a vector bundle on $\mathbb{F}_e$ such that for any two curves $B$ and $B'$ in $\mathbb{F}_e$ with $B \sim B'$, we have $E|_B \cong E|_{B'}$. Then $\rho_*(E|_C) \cong \bigoplus_i \rho_*(E|_{C_i})$ as vector bundles on $\mathbb{P}^1$.

**Proof.** We first observe that for any two curves $B$ and $B'$ in $\mathbb{F}_e$ which are linearly equivalent to each other,

$$\rho_*(E \otimes \mathcal{O}_B) \cong \rho_*(E \otimes \mathcal{O}_{B'}) \quad \text{on } \mathbb{P}^1.$$

In other words, $\rho_*(E|_B) \cong \rho_*(E|_{B'})$ on $\mathbb{P}^1$. So, without loss of generality we assume that $C = C_1 + \dots + C_r$ and $C_i \to C$ be an irreducible component of it. Then, $\mathcal{O}_C \to \mathcal{O}_{C_i}$, which induces a sheaf map $\rho_*(E \otimes \mathcal{O}_C) \to \rho_*(E \otimes \mathcal{O}_{C_i})$ on $\mathbb{P}^1$ for all $i$, and hence induces a map $\rho_*(E \otimes \mathcal{O}_C) \to \bigoplus_i \rho_*(E \otimes \mathcal{O}_{C_i})$ as well.

We claim that

$$\rho_*(E \otimes \mathcal{O}_C) \to \bigoplus_i \rho_*(E \otimes \mathcal{O}_{C_i})$$
is an isomorphism on $\mathbb{P}^1$. Indeed, for any $y \in \mathbb{P}^1$,

$$ (\rho_*(E \otimes \mathcal{O}_C))_y \cong \bigoplus_{x \in \{C \cap \rho^{-1}(y)\}} E_x. $$

On the other hand,

$$ \left( \bigoplus_i \rho_*(E \otimes \mathcal{O}_{C_i}) \right)_y \cong \bigoplus_i (\rho_*(E \otimes \mathcal{O}_{C_i}))_y \cong \bigoplus_{x \in C_i, \rho(x)=y} E_x. $$

Hence, the map is isomorphic at the stalk level. This proves our claim and the result. $\square$

Any rank two vector bundle $E$ on $\mathbb{F}_e$ has two numerical invariants describing it as an extension in a canonical manner. The first invariant $d_E$ is defined by the splitting type of $E$ on a general fiber $f$, i.e., if $E|_f = \mathcal{O}_{\mathbb{P}^1}(d) \oplus \mathcal{O}_{\mathbb{P}^1}(d')$ and $d \ge d'$, then $d_E = d$. The second invariant $r_E = r = \deg(\rho_*(E(-dC_0)))$. See [8] for more information about these numerical invariants $d$ and $r$. Note that, if $E$ is globally generated, then for a generic fibre $f$, $E|_f = \mathcal{O}_{\mathbb{P}^1}(d) \oplus \mathcal{O}_{\mathbb{P}^1}(d')$, where $d \ge d' \ge 0$.

**Theorem 15.** Let $E$ be a globally generated rank two bundle on $\mathbb{F}_e$ with numerical invariants $d$ and $r$, and $E$ sits in the exact sequence

$$ 0 \to \mathcal{O}(dC_0 + rf) \to E \to \mathcal{O}(d'C_0 + r'f) \to 0. \quad (1) $$

Further assume that for any two curves $B$ and $B'$ in $\mathbb{F}_e$ with $B \sim B'$, we have $E|_B \cong E|_{B'}$. Then, $E$ is ample if and only if $E|_f$, $E|_{C_0}$ and $E|_{C_0+nf}$ are ample on a generic fibre $f$, on $C_0$ and sections of $\rho$ of the forms $C_0 + nf$ with $d(n-e) + r \le 0$ respectively.

**Proof.** Restriction of ample bundle being ample, $E|_C$ is ample for any curve $C$ in $\mathbb{F}_e$ whenever $E$ is ample.

Conversely, let $E|_f$, $E|_{C_0}$ and $E|_{C_0+nf}$ are ample on a generic fibre $f$, on $C_0$ and sections of $\rho$ of the forms $C_0 + nf$ with $d(n-e) + r \le 0$ respectively. Now, if

$$ E|_f = \mathcal{O}_{\mathbb{P}^1}(d) \oplus \mathcal{O}_{\mathbb{P}^1}(d'), $$

for a generic fibre $f$ of $\rho$ with $d \ge d' \ge 0$, then the ampleness of $E|_f$ implies that $d, d' > 0$.

Let $f'$ be a fibre among those finitely many fibre which has different splitting type of $E$ than that of a generic fibre. Restricting the exact sequence (1) to $f'$, we get

$$ 0 \to \mathcal{O}_{\mathbb{P}^1}(d) \to E|_{f'} \to \mathcal{O}_{\mathbb{P}^1}(d') \to 0 $$

Hence, $E|_{f'}$ being an extension of two ample line bundle, is also ample.

Let $C \sim C_0 + nf$ be any section of $\rho$, where either $n=0$ or $n \ge e$. Now, restricting the exact sequence (1) to $C$, we get

$$ 0 \to \mathcal{O}_{\mathbb{P}^1}(d(n-e) + r) \to E|_C \to \mathcal{O}_{\mathbb{P}^1}(d'(n-e) + r') \to 0. $$

As $E|_{C_0}$ is ample on $C_0$, and $\mathcal{O}_{\mathbb{P}^1}(-d'e + r')$ being the quotient is also ample. Hence, $(-d'e + r') > 0$, which implies $d'(n-e) + r' > 0$ for any $n \ge 1$. Note that, if $d(n-e) + r > 0$ then $E|_{C_0+nf}$ is ample, as it is then an extension of two ample bundles. If $d(n-e) + r \le 0$ then $E|_{C_0+nf}$ is also ample by the given hypothesis. Therefore, we conclude that restriction of $E$ onto each fibre and each section is ample.

Let $C$ be any curve of $\mathbb{F}_e$ other than a fibre of $\rho$, and $C \sim C_1 + \cdots + C_r$ where $C_i$'s are sections. Now, using Proposition 14, we get that $\rho_*(E|_C)$ is an ample vector bundle on $\mathbb{P}^1$.

If $E$ is not ample, then by Gieseker's lemma [15, Proposition 6.1.7], there exists an irreducible curve $C$ in $\mathbb{F}_e$ other than the fibres and a surjective homomorphism $u: E|_C \to \mathcal{O}_C$. This induces the sujection $\rho_*(E|_C) \to \rho_*(\mathcal{O}_C) \cong \mathcal{O}_{\mathbb{P}^1}$, as well as the injection $\mathcal{O}_{\mathbb{P}^1} \to (\rho_*(E|_C))^*$ which contradicts the fact that $\rho_*(E|_C)$ is an ample bundle on $\mathbb{P}^1$. Therefore, $E$ is ample. This completes the proof. $\square$
### 4. Remark about Parabolic Ampleness

Let $X$ be a connected smooth complex projective variety of dimension $d$ and $D \subset X$ be an effective divisor on $X$.

**Definition 16.** A quasi parabolic structure on a coherent sheaf $E$ with respect to $D$ is a filtration by $\mathcal{O}_X$-coherent subsheaves

$$E = \mathcal{F}_1(E) \supset \mathcal{F}_2(E) \supset \cdots \supset \mathcal{F}_l(E) \supset \mathcal{F}_{l+1}(E) = E(-D)$$

where $E(-D) = E \otimes_{\mathcal{O}_X} \mathcal{O}_X(-D)$. The integer $l$ is called the length of the filtration.

A parabolic structure is a quasi-parabolic structure, as above, together with a system of parabolic weights $\{\alpha_1, \alpha_2, \dots, \alpha_l\}$ such that $0 \le \alpha_1 < \alpha_2 < \cdots < \alpha_{l-1} < \alpha_l < 1$, where each $\alpha_i$ is attached to $\mathcal{F}_i(E)$.

We shall denote the parabolic sheaf by $(E, \mathcal{F}_*, \alpha_*)$ or simply by $E_*$ when there is no confusion. For any parabolic sheaf $E_*$ defined as above, for any $t \in \mathbb{R}$, we define the following filtration $\{E_t\}_{t \in \mathbb{R}}$ of coherent sheaves parametrized by $\mathbb{R}$:

$$E_t = \mathcal{F}_t(E)(-[t]D)$$

where $[t]$ is the integral part of $t$ and $\alpha_{i-1} < t - [t] \le \alpha_i$ with $\alpha_0 = \alpha_l - 1$ and $\alpha_{l+1} = 1$. Note that, any coherent subsheaf $M$ of $E$ has an induced parabolic structure such that if $\{M_t\}_{t \in \mathbb{R}}$ is the corresponding filtration then $M_t = E_t \cap M$ for any $t \ge 0$.

The parabolic degree of $E_*$ with respect to a fixed ample bundle $L$ on $X$, denoted by $\mathrm{par\_deg}(E_*)$ is defined as follows:

$$\mathrm{par\_deg}(E_*) := \int_{-1}^{0} \mathrm{deg}(E_t) \mathrm{d}t$$

The parabolic slope of $E_*$, denoted by $\mathrm{par\_\mu}(E_*)$ is the quotient $\mathrm{par\_deg}(E_*) / \mathrm{rank}(E)$.

**Definition 17.** The parabolic sheaf $E_*$ is called parabolic semistable (resp. parabolic stable) if for every subsheaf $M$ of $E$ with $0 < \mathrm{rank}(M) < \mathrm{rank}(E)$, and $E/M$ being torsion-free sheaf, the inequality $\mathrm{par\_\mu}(M_*) \le \mathrm{par\_\mu}(E_*)$ (resp. $\mathrm{par\_\mu}(M_*) < \mathrm{\mu}(E_*)$) is satisfied.

Consider the decomposition

$$D = \sum_{i=1}^{n} n_i D_i \qquad (2)$$

where any $D_i$ is a reduced irreducible divisor and $n_i \ge 1$. Let

$$f_i : n_i D_i \to X$$

denote the inclusion of the subscheme $n_i D_i$. For $1 \le i \le n$, let

$$0 = F_{l_i+1}^i \subset F_{l_i}^i \subset F_{l_i-1}^i \subset \cdots \subset F_1^i = f_i^* E \qquad (3)$$

Let $\alpha_j^i$, $1 \le j \le l_i + 1$ be real numbers satisfying

$$1 = \alpha_{l_i+1}^i > \alpha_{l_i}^i > \alpha_{l_i-1}^i > \cdots > \alpha_2^i > \alpha_1^i \ge 0.$$

From now on we will always impose the following three conditions on the parabolic bundles $E_*$ that we will consider:

(a) the parabolic divisor $D = \sum_{i=1}^n n_i D_i$ is a normal crossing divisor, i.e., all $n_i = 1$ and $D_i$ are smooth divisors and they intersect transversally.

(b) all $F_j^i$ on $D_i$ in sequence (3) are subbundles of $f_i^* E$ for every $i$.

(c) all the weights $\alpha_j^i$ are rational numbers; so $\alpha_j^i = m_j^i/N$, where $N$ is a fixed integer and $m_j^i \in \{0, 1, \dots, N-1\}$.
In [2], parabolic tensor product has been defined. The parabolic *m*-fold symmetric product $S^m(E_*)$, is the invariant subsheaf of the *m*-fold parabolic tensor product of $E_*$ for the natural action of the permutation group for the factors of the tensor product. The underlying sheaf of the parabolic sheaf $S^m(E_*)$ will be denoted by $S^m(E_*)_0$. We recall the definition of parabolic ampleness from [3].

**Definition 18.** The parabolic sheaf $E_*$ is called parabolic ample if for any coherent sheaf $F$ on $X$ there is an integer $m_0$ such that for any $m \ge m_0$, the tensor product $F \otimes S^m(E_*)_0$ is globally generated.

Parabolic Chern classes $c_i(E_*) \in H^{2i}(X, \mathbb{Q})$ has been introduced in [3]. For a parabolic vector bundle $E_*$ of rank $r$ we define the parabolic discriminant, denoted by $\Delta_{\text{par}}(E_*)$ as follows:

$$ \Delta_{\text{par}}(E_*) := 2rc_2(E_*) - (r-1)c_1^2(E_*) $$

**Theorem 19.** Let $E_*$ be a semistable parabolic vector bundle of rank $r$ on a smooth complex projective variety $X$ such that $\Delta_{\text{par}}(E_*) = 0$. Then, $E_*$ is parabolic ample if and only if its parabolic first Chern class $c_1(E_*)$ is in the ample cone of $X$.

**Proof.** Let $p: Y \to X$ be the Kawamata cover, $V$ be the corresponding orbifold bundle on $Y$ with $c_1(V) = p^*c_1(E_*)$ (see [2] and [4]). So if $E_*$ is ample, then $V$ is also ample (see [3]) and thus $c_1(V)$ is also ample. Using the finiteness of the surjective map $p$, we conclude that $c_1(E_*)$ is in the ample cone of $X$.

Conversely, if $c_1(E_*)$ is in the ample cone of $X$, then $\det(V)$ is also ample. Also, by the given hypothesis, $V$ is orbifold semistable and hence semistable (in the usual sense) with $\Delta(V) = p^*\Delta_{\text{par}}(E_*) = 0$. Hence $V$ is ample and thus $E_*$ is parabolic ample. $\square$

**Proposition 20.** Let $\pi: X \to Y$ be a smooth surjective morphism between two smooth connected complex projective varieties $X$ and $Y$. Let $E_*$ be a parabolic semistable bundle on $Y$ with parabolic divisor $D \subset Y$ and $\Delta_{\text{par}}(E_*) = 0$. Then, the pullback bundle $\pi^*(E_*)$ under the map $\pi$ is also parabolic semistable on $X$ with parabolic divisor $\pi^*(D) \subset X$ and $\Delta_{\text{par}}(\pi^*(E_*)) = 0$.

Conversely, if $E_*$ be a parabolic semistable bundle on projective bundle $\pi: X = \mathbb{P}(\mathscr{E}) \to Y$ with parabolic divisor $D' = \pi^{-1}(D)$, with $\Delta_{\text{par}}(E_*) = 0$ and the parabolic first Chern class $c_1(E_*) = \pi^*(\mathscr{L})$ for some line bundle $\mathscr{L}$ on $Y$, then there exists a semistable parabolic bundle $E'_*$ on $Y$ with parabolic divisor $D$ and $\Delta_{\text{par}}(E'_*) = 0$ such that $E_* = \pi^*(E'_*)$.

**Proof.** Let $D = \sum_{i=1}^n D_i$ be the normal crossing divisor on $Y$ and $D' = \pi^*(D)$. Since $\pi$ is smooth, the pullback divisor $D'$ on $X$ is also a normal crossing divisor satisfying condition (a).

Let $p: Y' \to Y$ be a Kawamata cover with Galois group $G$ such that

$$ p^*D_i = k_i N(p^*D_i)_{\text{red}} $$

for some positive integers $k_i$ and $N$. Consider the following fibre product diagram

<figure><img src="image_3.png" /></figure>

Then $\tilde{p}: X' \to X$ is a Galois cover with the same Galois group $G$. Let $V$ be the orbifold bundle on $Y'$ associated to the parabolic bundle $E_*$ on $Y$ (see [4]). The pullback orbifold bundle $V' := \pi^*(V)$ then corresponds to the parabolic pullback bundle $\pi^*(E_*)$ on $X$ with parabolic divisor $D'$.

We note that $\Delta(V) = \Delta_{\text{par}}(E_*) = 0$. Since $E_*$ is parabolic semistable, by using the correspondence in [4], $V$ is also orbifold semistable, and hence semistable (in the usual sense). Therefore
the pullback bundle $V'$ is also orbifold semistable with $\Delta(V') = 0$, proving that $\pi^*(E_*)$ is parabolic semistable with $\Delta_{\text{par}}(\pi^*(E_*)) = 0$.

Conversely, let $V$ be the orbifold bundle on $X'$ associated to the parabolic bundle $E_*$ on $X$. Then,

$$c_1(V) = \tilde{p}^* c_1(E_*) = \tilde{p}^* \pi^* (\mathcal{L}) = \tilde{\pi}^* p^* (\mathcal{L})$$

Now by the given hypothesis, $V$ is orbifold semistable and hence semistable (in the usual sense). Since $\Delta(V) = 0$, by Theorem 1.2 in [5] $V|_f$ is semistable on $f \simeq \mathbb{P}^m$ for every fibre $f$ of the map $\tilde{\pi}$ (Here rank($\mathcal{E}$) = $m+1$) and $\deg(V|_f) = 0$. This implies $V \simeq \tilde{\pi}^*(W)$ for some orbifold bundle $W$ on $Y'$ which must be semistable. Let $E'_*$ be the associated semistable parabolic bundle on $Y$. Note that $\Delta(V) = \tilde{\pi}^*(\Delta(W)) = 0$ and $\tilde{\pi}^*$ is injective. Hence $\Delta(W) = 0$. By a similar argument we have $\Delta_{\text{par}}(E'_*) = 0$. Then by the construction of $E'_*$, the result follows. $\square$

**Corollary 21.** Let $W$ be a vector bundle of rank $m$ over a smooth complex projective curve $C$ and $\rho: X = \mathbb{P}(W) \to C$ be the projectivisation map. Let $E_*$ be a semistable vector bundle on $X$ of rank $r$ with parabolic discriminant $\Delta_{\text{par}}(E_*) = 0$, and parabolic 1st Chern class $c_1(E_*) = x\xi + yf$, where $\xi$ and $f$ are the numerical classes of $\mathcal{O}_{\mathbb{P}(W)}(1)$ and a fibre of $\rho$ respectively. Then, $E_*$ is ample if and only if $x > 0$ and $(x\mu_{\min}(W) + y) > 0$.

**Proof.** We note that

$$\overline{\text{NE}}(\mathbb{P}(W)) = \{a(\xi^{m-1} - (\deg(W) - \mu_{\min}(W))\xi^{m-2}f) + b\xi^{m-2}f \mid a, b \in \mathbb{R}_{\ge 0}\}$$

Hence, $c_1(E_*)$ is in the ample cone if and only if

* $c_1(E_*) \cdot \{\xi^{m-1} - (\deg(W) - \mu_{\min}(W))\xi^{m-2}f\} = (x\mu_{\min}(W) + y) > 0$ and

* $c_1(E_*) \cdot \xi^{m-2}f = x > 0$.

Therefore, the result follows from the previous theorem. $\square$

**Example 22.** Let $\rho: X = \mathbb{P}(W) \to Y$ be a projective bundle on a smooth projective variety $Y$. Let $D \subset Y$ be a normal crossing divisor in $Y$ and $F_*$ be a semistable parabolic bundle of rank $r$ on $Y$ with parabolic divisor $D$. Then $\rho^*(F_*)$ is a parabolic semistable bundle on $X$ with parabolic divisor $D' = \rho^*(D)$. Let $D' = \sum_{i=1}^n D'_i$ be the decomposition into irreducible components of $D'$. A parabolic line bundle with parabolic divisor $D'$ is a data of the form $L_* = (L, \{\alpha_1, \dots, \alpha_n\})$, where $L$ is a line bundle on $X$ and each $0 \le \alpha_i < 1$ corresponds to the divisor $D'_i$. Assume $\alpha_i \in \mathbb{Q}$ for all $i$. Then $E_* = \rho^*(F_*) \otimes L_*$ is parabolic semistable with $\Delta_{\text{par}}(E_*) = 0$. Note that $c_1(L_*) := c_1(L) + \sum_{i=1}^n \alpha_i|_{D_i}$. One can choose $L_*$ in such a way that $c_1(E_*) = c_1(\rho^*F_*) + rc_1(L_*)$ is in the ample cone of $X$. This way one can produce parabolic ample bundles on $X$.

## Acknowledgement

The authors would like to thank Prof. Indranil Biswas for many helpful discussions.

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